Wikiversity talk:Main Page

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

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== The Wikiversity:Main page learning project ==

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

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

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

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

:{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:29, 12 August 2018 (UTC)
::What about zulu language [[User:Lucky Shabalala|Lucky Shabalala]] ([[User talk:Lucky Shabalala|discuss]] • [[Special:Contributions/Lucky Shabalala|contribs]]) 05:57, 30 April 2025 (UTC)

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

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

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

== Georgian (ka) wikiversity ==
PLEASE 

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

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

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

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

== new langueages ==

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

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

== How to change an username? ==

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

==Religious user names allowed in Wikiversity?==

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

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

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

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

== LinkedIn ==

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

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

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

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

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

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

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

== Add New Language ==

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

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

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

== I'm learning Turkish🤩 ==

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

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

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

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

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

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

== Action in the earliest? ==

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

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

== Biology ==

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

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

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

== Physics ==

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

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

== Popularize ==

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

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

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

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

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

== Can you please add isiZulu plz ==

Because all othere languages her so i can umderstand batter  [[User:Lucky Shabalala|Lucky Shabalala]] ([[User talk:Lucky Shabalala|discuss]] • [[Special:Contributions/Lucky Shabalala|contribs]]) 06:06, 30 April 2025 (UTC)

:Add it how? Add more resources to learn the language? I think that would be fantastic, but it's very labor-intensive and I doubt anyone here has the competence to add that kind of material. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:40, 30 April 2025 (UTC)

== banner ==

says set learning free, propare grammer would be Start learning for free [[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 20:21, 3 February 2026 (UTC)

:I'm a native American English speaker and this banner is grammatical. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:52, 4 February 2026 (UTC)
::That’s not the problem. I’m wondering if we should more clearly emphasize what Wikiversity is on this banner. Idk maybe it’s fine as it is I would just like it to be clearer[[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 16:15, 4 February 2026 (UTC)
:::nevermind i just got the banner thought it was supposed to say start learning free, but its actually set learning free(like release learning) [[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 16:12, 6 February 2026 (UTC)

== Set Learning Free with Wikiversity ==

Wikiversity is a global learning community dedicated to free education, collaborative research, and open knowledge sharing. From academic courses and research projects to professional development and lifelong learning, Wikiversity empowers educators, students, and curious minds to learn, teach, and grow together without barriers. With thousands of learning resources available across multiple disciplines, it continues to make quality education accessible to everyone, everywhere.

In the era of digital education and AI-driven learning, visibility and innovation are essential for educational platforms and growing organizations. '''BizisTech''' helps businesses and digital projects scale through SEO, AI-powered marketing, web development, and growth strategies built for long-term success.

'''Wikiversity — Setting Learning Free for the World.'''

'''BizisTech — Empowering the Future of Digital Growth.''' [[User:Hamadbizistech01|Hamadbizistech01]] ([[User talk:Hamadbizistech01|discuss]] • [[Special:Contributions/Hamadbizistech01|contribs]]) 04:19, 16 July 2026 (UTC)</text>
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{{attention}} To request an edit to the [[Wikiversity:Page protection|protected]] Main Page, add {{tl|editprotected}} to your request. Such requests should either be obvious or uncontroversial, or be discussed to show consensus, so please do not make vague requests here. If possible, describe exactly what changes should be made so that any custodian can quickly satisfy the request.&lt;br&gt;
{{attention}} To raise general topics about [[Wikiversity]], make general suggestions about Wikiversity, to ask questions, or to talk about anything else of a general nature, use the [[Wikiversity:Colloquium|Colloquium]].&lt;br&gt;
{{attention}} To discuss the structure, appearance, etc. of the [[Wikiversity:Main Page|Main Page]], go to the [[Wikiversity:Main page learning project]] and the [[Wikiversity talk:Main page learning project|talk page for the main page learning project]].
&lt;/div&gt;

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

{{archive box|
{{center top}}'''List of talk archives'''{{center bottom}}
{{Col list|3|
{{Special:Prefixindex/Wikiversity talk:Main Page/Archive |hideredirects=1|stripprefix=1}}
}}
{{SearchWithPrefix|prefix=Wikiversity talk:Main Page/|resourceName=talk archive}}
}}

== The Wikiversity:Main page learning project ==

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

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

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

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

:{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:29, 12 August 2018 (UTC)
::What about zulu language [[User:Lucky Shabalala|Lucky Shabalala]] ([[User talk:Lucky Shabalala|discuss]] • [[Special:Contributions/Lucky Shabalala|contribs]]) 05:57, 30 April 2025 (UTC)

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

&lt;nowiki&gt;{{editprotected}}&lt;/nowiki&gt;
&lt;!-- Begin request --&gt;


&lt;!-- End request --&gt;
[[Special:Contributions/204.234.101.112|204.234.101.112]] ([[User talk:204.234.101.112|discuss]]) 21:17, 14 February 2019 (UTC)

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

== Georgian (ka) wikiversity ==
PLEASE 

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

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

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

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

== new langueages ==

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

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

== How to change an username? ==

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

==Religious user names allowed in Wikiversity?==

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

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

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

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

== LinkedIn ==

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

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

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

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

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

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

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

== Add New Language ==

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

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

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

== I'm learning Turkish🤩 ==

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

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

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

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

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

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

== Action in the earliest? ==

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

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

== Biology ==

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

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

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

== Physics ==

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

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

== Popularize ==

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

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

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

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

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

== Can you please add isiZulu plz ==

Because all othere languages her so i can umderstand batter  [[User:Lucky Shabalala|Lucky Shabalala]] ([[User talk:Lucky Shabalala|discuss]] • [[Special:Contributions/Lucky Shabalala|contribs]]) 06:06, 30 April 2025 (UTC)

:Add it how? Add more resources to learn the language? I think that would be fantastic, but it's very labor-intensive and I doubt anyone here has the competence to add that kind of material. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:40, 30 April 2025 (UTC)

== banner ==

says set learning free, propare grammer would be Start learning for free [[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 20:21, 3 February 2026 (UTC)

:I'm a native American English speaker and this banner is grammatical. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:52, 4 February 2026 (UTC)
::That’s not the problem. I’m wondering if we should more clearly emphasize what Wikiversity is on this banner. Idk maybe it’s fine as it is I would just like it to be clearer[[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 16:15, 4 February 2026 (UTC)
:::nevermind i just got the banner thought it was supposed to say start learning free, but its actually set learning free(like release learning) [[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 16:12, 6 February 2026 (UTC)</text>
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    <title>Advanced Placement and Pre-AP Chemistry projects</title>
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Welcome to this learning project about '''{{PAGENAME}}'''!

This is a suggested list of headings to organise a learning project - please see [[Template_talk:Learning_project_boilerplate#Directions for use|Directions for use]] for more information. Please delete any headings (etc.) if you feel they are not relevant, and rearrange as you see fit.

==Learning Project Summary==
* '''Suggested Prerequisites:[http://apcentral.collegeboard.com/apc/members/program/initiatives/22794.html AP Equity Statement]'''
** ...
* '''Assessment suggestions:Keep an inquisitive and open mind'''
* '''[[Wikiversity:Major portals|Portal]]:(List main portal/s here)'''
* '''[[Wikiversity:Schools|School]]:(List main school/s here)'''
* '''Department: Education and Literature'''
* '''Stream'''
* '''Level:'''

==Content summary==
Advanced Placement (AP) Chemistry is a college-level course offered to students in higher secondary school. Pre-AP Chemistry is not a class but a set of strategies (covering grades K-12) to help students prepare to take AP Chemistry or a college general chemistry course.

This category has been created to allow teachers of AP and Pre-AP Chemistry to exchange ideas in an open environment. You may not post copyrighted materials unless you hold the copyright and grant others free and fair use of your material. Other participants are free to change and edit these materials. It is this &quot;evolution&quot; of materials, as demonstrated by Wikipedia, that leads to the best and most accurate information.

This page is under development. Any and all are free to help.

==Goals==
This learning project aims to create a moral ground on which to teach. To apply morals in the classroom is the ultimate goal.

==Contents==

===Learning materials===
Add learning materials here - see also the box below

===Readings and other resources===
{{Sisterprojectsearch}}
Each project/lesson/activity may have a suggested reading selection, eg:

* '''Wikipedia article:'''
* '''Wikibooks textbook:'''
* etc.

===Labs===
* ...
Chemistry Labs 
Advanced Projects in Chemistry- Analysis of Alloys 
Author -Anjali Gharpure (anjali@podar.net)
Chemistry Projects at the K-11 and K-12 grades require application of the theoretical concepts studied in class.  Transition metals are studied under the d-block. One of the peculiar characteristics of transition metals is formation of alloys.  Alloys are solid homogeneous solutions of two or more solids.  In an alloy, two or more elements combine such that at least one is a metal, and the resultant material has metallic properties. Alloys are designed to have properties that are more desirable than those of pure metals. Steel is stronger than iron, while brass, an alloy of copper and zinc is more durable than copper and more attractive than zinc in appearance.
Unlike pure metals, alloys do not have a single melting point. They possess a melting range. Alloys improve properties of metals, by increasing hardness, tensile strength, chemical resistance and attractive appeal by modifying color.
Alloys of Zn and Copper are widely used in everyday life and known to all of us.  Detection of the presence of the metals used to prepare these alloys can be achieved through semi-micro qualitative analysis. These tests are useful both for the student of chemistry as well as for a forensic laboratory chemist. Laboratory work in alloy analysis  involves a three step protocol
1. Grinding the alloy sample to a fine particulate powder 
2. Digesting the sample in an appropriate acid 
3. Detection of the metal ions in the alloy.

Brass	Zn-20-40% &amp; Cu-60-80%
Zn was detected with potassium ferrocyanide test and the Sodium hydroxide test
Cu-60-80% Potassium ferrocyanide, potassium iodide and liquid ammonia tests  
Bronze 	Zn-10%	Zn was detected with potassium ferrocyanide test and the Sodium hydroxide test
	Cu -90%	Potassium ferrocyanide, potassium iodide and liquid ammonia tests  

Chemical Reactions –
Acknowledgements- The experiments for  the analysis of  the metal ions as well digesting alloys of bronze and brass were carried out by Anchit R.Giri  and Shashwat Kishore of Class XII, R.N Podar CBSE Senior Secondary High School , Santacruz(W).   
For Copper Analysis :
Liquid ammonia test -
Cu(NO3)2+4NH4OH--&gt; [Cu(NH3)4][NO3]2+4H2O-Deep blue solution

Potassium ferrocyanide test-
[Cu(NH3)4]SO4+4CH3COOH--&gt; CuSO4+ 4CH3COONH4
2CuSO4+ K4 [Fe(CN)6]--&gt;Cu2[Fe(CN)6]+2K2SO4 - Chocolate brown precipitate

Potassium iodide test
2CuSO4+ 4KI--&gt;Cu2I2 (white precipitate) +I2 (brown coloration)+2K2SO4

For Zn Analysis 
1.	Sodium Hydroxide test 
ZnCl2+2NaOH--&gt; Zn(OH)2 white precipitate + 2NaCl
Zn(OH)2+2NaOH--&gt; Na2ZnO2 soluble in excess NaOH+2H2O

2. Potassium ferrocyanide test
2ZnCl2+K4 [Fe(CN)6]--&gt;Zn2[Fe(CN)6]  bluish white precipitate +4KCl

A transition metal forms an alloy with another transition metal ion with ease because of the similarity in atomic size.  The lattice site in the crystal structure of one transition metal can be occupied by other transition metals giving a homogeneous solid solution termed as an alloy.  Mutual substitution of one or more metals leads to the formation binary, ternary or tertiary alloys.  Hardness, resistance to corrosion, tensile strength, load bearing capacity are some of the properties which can be bettered by alloy formation.  Actual lab analysis impresses the composition of the alloy for the student.

Advanced Chemistry Projects – Acids and Bases
Contributing Author: Anjali Gharpure 
To make your Science Project significant choose topics from your curriculum that you have studied in theory.  A science project at the K-11 or K-12 grade or for your A levels, requires demonstration of your understanding the concept taught in class supported with meaningful experimental work.   It is important that your experiments are simple and non-hazardous.  Irrespective of the boards for which you appear, you all learn about acids and bases.
Theoretically, acids and bases are learnt according to different theories put forth by Arrhenius, Bronsted and Lowry as well as G.N.Lewis.  In contrast, we also study the classical concept of acids and bases based on qualitative tests.  If it tastes sour –it is an acid, but if it tastes slippery and bitter on the tongue, then it is surely a base.  An acid reacts with a base to give salt and water and vice versa a base reacts with an acid also to give salt and water.    According to the Arrhenius concept, an acid releases a proton in solution and a base releases hydroxyl ions in solution.  
The Bronsted and Lowry concept defines an acid as a proton donor but a base as a proton acceptor.  The Lewis concept of an acid and base is that of an electron pair acceptor and donor with the formation of a coordinate covalent bond.
The project at hand is to explain these concepts through simple experimentation.  
Strength versus Concentration
The Lewis concept of acids and bases is incapable of explaining strengths of acids and bases.  The strength of an acid or a base is directly proportional to the extent to which it produces hydronium or hydroxide ions in solution. Strong species produce stoichiometric equivalents of  hydroxide ions while weak species produce less than a stoichiometric equivalent of  hydroxide ions. Concentration on the other hand refers to the molarity or molar concentration in moles per litre of the solute in solution.  The terms concentration and strength are not interchangeable.
A 0.001molar solution of HCl is 100 % dissociated in solution and so it is a strong acid as it furnishes almost all its protons in solution.  However a 2 M solution of acetic acid is a more concentrated solution than 0.001M HCl but is not stronger than HCl as the number  of protons dissociated are very few.
Here a meaningful experiment would be to demonstrate the difference between a weak strength acid and a strong strength acid.
The simplest determination is carrying out a pH measurement.  From the pH measurement the H3O+ ion concentration can be determined.  
pH= - log10 H3O+  from this relationship the H3O+ ion concentration for a weak as well as a strong acid can be determined.
The stronger acid will have a lower pH value and thus a higher hydronium ion concentration.
Ka , Kb and Acid/Base Strength
Most acids, like acetic acid, are weak. They do not completely ionize in water to produce a stoichiometric equivalent of hydronium ions. This means aqueous acetic acid molecules are in equilibrium with hydronium and acetate ions.
   
For any system at equilibrium, the law of chemical equilibrium results in an expression with the equilibrium constant K. 
Including the equilibrium concentrations of aqueous species in equilibrium constant expressions, the K expression for acetic acid in water is: 
 
where the subscript &quot;a&quot; indicates that an acid ionizes in water to form hydronium. Ka is the acid ionization constant or the equilibrium constant for the ionization of a weak acid in water to produce hydronium ions and the conjugate base of the acid.

An equilibrium constant expression may also be written for a weak base like ammonia in water. 
 
 
The subscript &quot;b&quot; indicates that a base hydrolyses water to form hydroxide. Kb is the base ionization constant or the equilibrium constant for the ionization of a weak base in water to produce hydroxide and the conjugate acid of the base. 
Ka and Kb values indicate acid strength and base strength respectively. For example, a higher Ka value indicates higher hydronium ion concentrations or greater ionization. The equilibrium position lies further to the right when Ka is higher. A higher Kb value means a higher hydroxide ion concentration.  

The strength of acetic acid is known to you.  From the pH measurement the H3O+ ion concentration is known.  The number of H3O+ ions in the solution will equal the number of acetate ions and so Ka = [H3O+]2/[CH3COOH]
Thus an experimental determination of the Ka values for different acids and Kb values for different bases can be undertaken
This project conveys through an experimental determination 
•	the understanding of the concepts of acids and bases
•	 the difference between strengths of acids and bases
•	pH+ pOH=14
•	 the calculation of Ka or Kb and its significance, as well as,
•	 pH and its use to determine the hydronium ion concentration.  
Contact: anjali@podar.net


Advanced Chemistry Project -
Study of Bromination Reactions in Phenol, Aniline and cinnamic Acid 
Contributing Author : Anjali Gharpure (anjali@podar.net)
The theoretical study of electrophilic substitution reactions is best studied by actually carrying out laboratory reactions and making a comparative study of the yields of the products under the available conditions.  At the K-11-12 level, the instruction of electrophilic substitution reactions is superficial, until it is impressed upon by lab work.   
An electrophile is an electron loving substituent. An electrophile is attracted to electrons and participates in a chemical reaction by accepting an electron pair.  The benzene ring acts as a source of electrons.  In an electrophilic substitution reaction the hydrogen atoms on the benzene ring are replaced by the attacking reagent which is deficient in electrons.  Bromination is an electrophilic substitution reaction.   Benzene, C6H6, is a planar molecule containing a ring of six carbon atoms each with a hydrogen atom attached. There are delocalised electrons above and below the plane of the ring. The presence of the delocalised electrons makes benzene particularly stable. Benzene resists addition reactions because that would involve breaking the delocalisation and losing that stability.
Two effects work in electrophilic substitution reactions. One is induction, which is an effect that occurs through the sigma- bond system. Electron withdrawing groups will &quot;pull&quot; electron density away from the ring making the electrons of the ring less available for attack by an electrophile. Electron donating groups do the opposite. Induction also plays a role in some of the directional effects.  The other effect is resonance, which occurs through the pi system of bonds  in the molecule. Resonance plays an important role in the stability of intermediates of the reactions.
Sometimes these two effects are opposite to one another. One effect may be stronger and exert greater influence. The ease with which a compound suddenly becomes receptive to bromination due to presence of electron donating groups like –OH in phenol and –NH2 in aniline is studied.  Phenols are potentially very reactive towards electrophilic aromatic substitution.  In aqueous solutions, phenol undergoes ionization to give the phenoxide ion.   The negative charge on the oxygen of the phenoxide ion donates electrons into the benzene ring to a large extent.  The strong activation often means that milder reaction conditions than those used for benzene and also a trisubstituted reaction product.   The reaction is carried out in the laboratory without a strong Lewis acid like Friedel Crafts catalyst.  If bromine water is added to a solution of phenol in water, the bromine water is decolorized and a white precipitate is formed which smells of antiseptic.  The precipitate is 2,4,6-tribromophenol. 
Ar-OH+3Br2 Ar-OH Br3 +3HBr 
The NH2 group in aniline strongly activates the aromatic ring through the delocalization of the lone pair of electrons of the N-atom over the entire aromatic ring.  Aromatic amines undergo the reaction readily and it is impossible to stop the reaction at the monosubstitution stage.  Aniline too on bromination gives a tribromosubstituted derivative.
Ar –NH2+ 3Br2--&gt; 3HBr+ Ar-NH2-Br3
Cinnamic acid is an alkenoic acid.  It undergoes an electrophilic addition reaction.  The yield of the product is fairly high as the reaction mechanism does not involve large change in bond energies.  The double bond present in cinnamic acid consists of one sigma and one pi-bond.  Pi- electrons form an electron cloud which lies above and below the plane of the sigma bonded carbon atoms.  The pi electron cloud is thus more exposed and less tightly held by the two carbon atoms.  The pi electrons attract electrophiles and undergoes electrophilic addition reactions.  Bromine molecule is non-polar but when it comes in the vicinity of a double bond, the pi electrons of the double bond begin to repel the bromine molecule.  The bromine molecule gets polarized and the positive end of the bromine molecule is attracted towards the pi-electrons forming a carbocation.  This is the slow and rate determining step of the reaction.  The carbocation is higly reactive and undergoes a nucleophilic attack by the bromide ion giving a dibromo addition product across the double bond.     One gm of cinnamic acid  was placed in a conical flask and dissolved in10mL of hot water. The hot solution was added to 10 mL of a strong solution of 20% bromine.  The derivative was filtered and purified in hot alcohol.  The melting point of the di- bromo derivative of cinnamic acid was determined to be 195oC
Ar –CH=CHCOOH+3Br2--&gt; Ar-CHBr-CHBr-COOH
Acknowledgements-The bromo derivatives were patiently prepared by Manasvi Lalvani, Shreya Krishnan and Sanjana Siddhra, Grade 12,  of R.N. Podar Sr. Secondary CBSE High School, Santacruz (W).

===Activities and POGIL's===
*Activity 1. 
*etc.


===Assignments===
*AP: Book Reading Movie Assignment
*Pre-AP: Make your own mole.

===Tests and Quizzes===
*Quiz 1
*Quiz 2
*Test 1
*etc.

==Subpages==
Subpages can be created - like [[/concepts]] (for concepts involved in this learning project)


==Active participants==
Active participants in this [[Portal:Learning Projects#Learning Groups|Learning Group]]
* ...


{| width=&quot;100%&quot; style=&quot;background:#FFFCCF;{{text color default}}; border:1px solid #DAA520&quot;
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==[[Portal:Learning Materials|Learning materials]]==
Learning materials and [[Portal:Learning Projects|learning projects]] are located in the main Wikiversity namespace. Simply make a [[link]] to the name of the lesson (lessons are independent pages in the [[Wikiversity:Namespaces|main namespace]]) and start writing! 

You should also read about the [[Portal:Education/Wikiversity model|Wikiversity:Learning model]]. Lessons should center on learning activities for Wikiversity participants. Learning materials and learning projects can be used by multiple projects - and you are encouraged to cooperate with other departments that use the same learning resource.

|}

[[Category:Chemistry]]
[[Category:Projects]]
[[Category:High School]]


----


Advanced Placement, AP and Pre-AP are registered trademarks of the [http://www.collegeboard.com College Board]

All AP teachers and administrators are encouraged to read the College Board's [http://apcentral.collegeboard.com/apc/members/program/initiatives/22794.html Equity Statement]</text>
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{{TOCright}}

Welcome to this learning project about '''{{PAGENAME}}'''!

This is a suggested list of headings to organise a learning project - please see [[Template_talk:Learning_project_boilerplate#Directions for use|Directions for use]] for more information. Please delete any headings (etc.) if you feel they are not relevant, and rearrange as you see fit.

==Learning Project Summary==
* '''Suggested Prerequisites:[http://apcentral.collegeboard.com/apc/members/program/initiatives/22794.html AP Equity Statement]'''
** ...
* '''Assessment suggestions:Keep an inquisitive and open mind'''
* '''[[Wikiversity:Major portals|Portal]]:(List main portal/s here)'''
* '''[[Wikiversity:Schools|School]]:(List main school/s here)'''
* '''Department: Education and Literature'''
* '''Stream'''
* '''Level:'''

==Content summary==
Advanced Placement (AP) Chemistry is a college-level course offered to students in higher secondary school. Pre-AP Chemistry is not a class but a set of strategies (covering grades K-12) to help students prepare to take AP Chemistry or a college general chemistry course.

This category has been created to allow teachers of AP and Pre-AP Chemistry to exchange ideas in an open environment. You may not post copyrighted materials unless you hold the copyright and grant others free and fair use of your material. Other participants are free to change and edit these materials. It is this &quot;evolution&quot; of materials, as demonstrated by Wikipedia, that leads to the best and most accurate information.

This page is under development. Any and all are free to help.

==Goals==
This learning project aims to create a moral ground on which to teach. To apply morals in the classroom is the ultimate goal.

==Contents==

===Learning materials===
Add learning materials here - see also the box below

===Readings and other resources===
{{Sisterprojectsearch}}
Each project/lesson/activity may have a suggested reading selection, eg:

* '''Wikipedia article:'''
* '''Wikibooks textbook:'''
* etc.

===Labs===
* ...
Chemistry Labs 
Advanced Projects in Chemistry- Analysis of Alloys 
Author -Anjali Gharpure (anjali@podar.net)
Chemistry Projects at the K-11 and K-12 grades require application of the theoretical concepts studied in class.  Transition metals are studied under the d-block. One of the peculiar characteristics of transition metals is formation of alloys.  Alloys are solid homogeneous solutions of two or more solids.  In an alloy, two or more elements combine such that at least one is a metal, and the resultant material has metallic properties. Alloys are designed to have properties that are more desirable than those of pure metals. Steel is stronger than iron, while brass, an alloy of copper and zinc is more durable than copper and more attractive than zinc in appearance.
Unlike pure metals, alloys do not have a single melting point. They possess a melting range. Alloys improve properties of metals, by increasing hardness, tensile strength, chemical resistance and attractive appeal by modifying color.
Alloys of Zn and Copper are widely used in everyday life and known to all of us.  Detection of the presence of the metals used to prepare these alloys can be achieved through semi-micro qualitative analysis. These tests are useful both for the student of chemistry as well as for a forensic laboratory chemist. Laboratory work in alloy analysis  involves a three step protocol
1. Grinding the alloy sample to a fine particulate powder 
2. Digesting the sample in an appropriate acid 
3. Detection of the metal ions in the alloy.

Brass	Zn-20-40% &amp; Cu-60-80%
Zn was detected with potassium ferrocyanide test and the Sodium hydroxide test
Cu-60-80% Potassium ferrocyanide, potassium iodide and liquid ammonia tests  
Bronze 	Zn-10%	Zn was detected with potassium ferrocyanide test and the Sodium hydroxide test
	Cu -90%	Potassium ferrocyanide, potassium iodide and liquid ammonia tests  

Chemical Reactions –
Acknowledgements- The experiments for  the analysis of  the metal ions as well digesting alloys of bronze and brass were carried out by Anchit R.Giri  and Shashwat Kishore of Class XII, R.N Podar CBSE Senior Secondary High School , Santacruz(W).   
For Copper Analysis :
Liquid ammonia test -
Cu(NO3)2+4NH4OH--&gt; [Cu(NH3)4][NO3]2+4H2O-Deep blue solution

Potassium ferrocyanide test-
[Cu(NH3)4]SO4+4CH3COOH--&gt; CuSO4+ 4CH3COONH4
2CuSO4+ K4 [Fe(CN)6]--&gt;Cu2[Fe(CN)6]+2K2SO4 - Chocolate brown precipitate

Potassium iodide test
2CuSO4+ 4KI--&gt;Cu2I2 (white precipitate) +I2 (brown coloration)+2K2SO4

For Zn Analysis 
1.	Sodium Hydroxide test 
ZnCl2+2NaOH--&gt; Zn(OH)2 white precipitate + 2NaCl
Zn(OH)2+2NaOH--&gt; Na2ZnO2 soluble in excess NaOH+2H2O

2. Potassium ferrocyanide test
2ZnCl2+K4 [Fe(CN)6]--&gt;Zn2[Fe(CN)6]  bluish white precipitate +4KCl

A transition metal forms an alloy with another transition metal ion with ease because of the similarity in atomic size.  The lattice site in the crystal structure of one transition metal can be occupied by other transition metals giving a homogeneous solid solution termed as an alloy.  Mutual substitution of one or more metals leads to the formation binary, ternary or tertiary alloys.  Hardness, resistance to corrosion, tensile strength, load bearing capacity are some of the properties which can be bettered by alloy formation.  Actual lab analysis impresses the composition of the alloy for the student.

Advanced Chemistry Projects – Acids and Bases
Contributing Author: Anjali Gharpure 
To make your Science Project significant choose topics from your curriculum that you have studied in theory.  A science project at the K-11 or K-12 grade or for your A levels, requires demonstration of your understanding the concept taught in class supported with meaningful experimental work.   It is important that your experiments are simple and non-hazardous.  Irrespective of the boards for which you appear, you all learn about acids and bases.
Theoretically, acids and bases are learnt according to different theories put forth by Arrhenius, Bronsted and Lowry as well as G.N.Lewis.  In contrast, we also study the classical concept of acids and bases based on qualitative tests.  If it tastes sour –it is an acid, but if it tastes slippery and bitter on the tongue, then it is surely a base.  An acid reacts with a base to give salt and water and vice versa a base reacts with an acid also to give salt and water.    According to the Arrhenius concept, an acid releases a proton in solution and a base releases hydroxyl ions in solution.  
The Bronsted and Lowry concept defines an acid as a proton donor but a base as a proton acceptor.  The Lewis concept of an acid and base is that of an electron pair acceptor and donor with the formation of a coordinate covalent bond.
The project at hand is to explain these concepts through simple experimentation.  
Strength versus Concentration
The Lewis concept of acids and bases is incapable of explaining strengths of acids and bases.  The strength of an acid or a base is directly proportional to the extent to which it produces hydronium or hydroxide ions in solution. Strong species produce stoichiometric equivalents of  hydroxide ions while weak species produce less than a stoichiometric equivalent of  hydroxide ions. Concentration on the other hand refers to the molarity or molar concentration in moles per litre of the solute in solution.  The terms concentration and strength are not interchangeable.
A 0.001molar solution of HCl is 100 % dissociated in solution and so it is a strong acid as it furnishes almost all its protons in solution.  However a 2 M solution of acetic acid is a more concentrated solution than 0.001M HCl but is not stronger than HCl as the number  of protons dissociated are very few.
Here a meaningful experiment would be to demonstrate the difference between a weak strength acid and a strong strength acid.
The simplest determination is carrying out a pH measurement.  From the pH measurement the H3O+ ion concentration can be determined.  
pH= - log10 H3O+  from this relationship the H3O+ ion concentration for a weak as well as a strong acid can be determined.
The stronger acid will have a lower pH value and thus a higher hydronium ion concentration.
Ka , Kb and Acid/Base Strength
Most acids, like acetic acid, are weak. They do not completely ionize in water to produce a stoichiometric equivalent of hydronium ions. This means aqueous acetic acid molecules are in equilibrium with hydronium and acetate ions.
   
For any system at equilibrium, the law of chemical equilibrium results in an expression with the equilibrium constant K. 
Including the equilibrium concentrations of aqueous species in equilibrium constant expressions, the K expression for acetic acid in water is: 
 
where the subscript &quot;a&quot; indicates that an acid ionizes in water to form hydronium. Ka is the acid ionization constant or the equilibrium constant for the ionization of a weak acid in water to produce hydronium ions and the conjugate base of the acid.

An equilibrium constant expression may also be written for a weak base like ammonia in water. 
 
 
The subscript &quot;b&quot; indicates that a base hydrolyses water to form hydroxide. Kb is the base ionization constant or the equilibrium constant for the ionization of a weak base in water to produce hydroxide and the conjugate acid of the base. 
Ka and Kb values indicate acid strength and base strength respectively. For example, a higher Ka value indicates higher hydronium ion concentrations or greater ionization. The equilibrium position lies further to the right when Ka is higher. A higher Kb value means a higher hydroxide ion concentration.  

The strength of acetic acid is known to you.  From the pH measurement the H3O+ ion concentration is known.  The number of H3O+ ions in the solution will equal the number of acetate ions and so Ka = [H3O+]2/[CH3COOH]
Thus an experimental determination of the Ka values for different acids and Kb values for different bases can be undertaken
This project conveys through an experimental determination 
•	the understanding of the concepts of acids and bases
•	 the difference between strengths of acids and bases
•	pH+ pOH=14
•	 the calculation of Ka or Kb and its significance, as well as,
•	 pH and its use to determine the hydronium ion concentration.  
Contact: anjali@podar.net


Advanced Chemistry Project -
Study of Bromination Reactions in Phenol, Aniline and cinnamic Acid 
Contributing Author : Anjali Gharpure (anjali@podar.net)
The theoretical study of electrophilic substitution reactions is best studied by actually carrying out laboratory reactions and making a comparative study of the yields of the products under the available conditions.  At the K-11-12 level, the instruction of electrophilic substitution reactions is superficial, until it is impressed upon by lab work.   
An electrophile is an electron loving substituent. An electrophile is attracted to electrons and participates in a chemical reaction by accepting an electron pair.  The benzene ring acts as a source of electrons.  In an electrophilic substitution reaction the hydrogen atoms on the benzene ring are replaced by the attacking reagent which is deficient in electrons.  Bromination is an electrophilic substitution reaction.   Benzene, C6H6, is a planar molecule containing a ring of six carbon atoms each with a hydrogen atom attached. There are delocalised electrons above and below the plane of the ring. The presence of the delocalised electrons makes benzene particularly stable. Benzene resists addition reactions because that would involve breaking the delocalisation and losing that stability.
Two effects work in electrophilic substitution reactions. One is induction, which is an effect that occurs through the sigma- bond system. Electron withdrawing groups will &quot;pull&quot; electron density away from the ring making the electrons of the ring less available for attack by an electrophile. Electron donating groups do the opposite. Induction also plays a role in some of the directional effects.  The other effect is resonance, which occurs through the pi system of bonds  in the molecule. Resonance plays an important role in the stability of intermediates of the reactions.
Sometimes these two effects are opposite to one another. One effect may be stronger and exert greater influence. The ease with which a compound suddenly becomes receptive to bromination due to presence of electron donating groups like –OH in phenol and –NH2 in aniline is studied.  Phenols are potentially very reactive towards electrophilic aromatic substitution.  In aqueous solutions, phenol undergoes ionization to give the phenoxide ion.   The negative charge on the oxygen of the phenoxide ion donates electrons into the benzene ring to a large extent.  The strong activation often means that milder reaction conditions than those used for benzene and also a trisubstituted reaction product.   The reaction is carried out in the laboratory without a strong Lewis acid like Friedel Crafts catalyst.  If bromine water is added to a solution of phenol in water, the bromine water is decolorized and a white precipitate is formed which smells of antiseptic.  The precipitate is 2,4,6-tribromophenol. 
Ar-OH+3Br2 Ar-OH Br3 +3HBr 
The NH2 group in aniline strongly activates the aromatic ring through the delocalization of the lone pair of electrons of the N-atom over the entire aromatic ring.  Aromatic amines undergo the reaction readily and it is impossible to stop the reaction at the monosubstitution stage.  Aniline too on bromination gives a tribromosubstituted derivative.
Ar –NH2+ 3Br2--&gt; 3HBr+ Ar-NH2-Br3
Cinnamic acid is an alkenoic acid.  It undergoes an electrophilic addition reaction.  The yield of the product is fairly high as the reaction mechanism does not involve large change in bond energies.  The double bond present in cinnamic acid consists of one sigma and one pi-bond.  Pi- electrons form an electron cloud which lies above and below the plane of the sigma bonded carbon atoms.  The pi electron cloud is thus more exposed and less tightly held by the two carbon atoms.  The pi electrons attract electrophiles and undergoes electrophilic addition reactions.  Bromine molecule is non-polar but when it comes in the vicinity of a double bond, the pi electrons of the double bond begin to repel the bromine molecule.  The bromine molecule gets polarized and the positive end of the bromine molecule is attracted towards the pi-electrons forming a carbocation.  This is the slow and rate determining step of the reaction.  The carbocation is higly reactive and undergoes a nucleophilic attack by the bromide ion giving a dibromo addition product across the double bond.     One gm of cinnamic acid  was placed in a conical flask and dissolved in10mL of hot water. The hot solution was added to 10 mL of a strong solution of 20% bromine.  The derivative was filtered and purified in hot alcohol.  The melting point of the di- bromo derivative of cinnamic acid was determined to be 195oC
Ar –CH=CHCOOH+3Br2--&gt; Ar-CHBr-CHBr-COOH
Acknowledgements-The bromo derivatives were patiently prepared by Manasvi Lalvani, Shreya Krishnan and Sanjana Siddhra, Grade 12,  of R.N. Podar Sr. Secondary CBSE High School, Santacruz (W).

===Activities and POGIL's===
*Activity 1. 
*etc.


===Assignments===
*AP: Book Reading Movie Assignment
*Pre-AP: Make your own mole.

===Tests and Quizzes===
*Quiz 1
*Quiz 2
*Test 1
*etc.

==Subpages==
Subpages can be created - like [[/concepts]] (for concepts involved in this learning project)


==Active participants==
Active participants in this [[Portal:Learning Projects#Learning Groups|Learning Group]]
* ...


{| width=&quot;100%&quot; style=&quot;background:#FFFCCF;{{text color default}}; border:1px solid #DAA520&quot;
|
==[[Portal:Learning Materials|Learning materials]]==
Learning materials and [[Portal:Learning Projects|learning projects]] are located in the main Wikiversity namespace. Simply make a [[link]] to the name of the lesson (lessons are independent pages in the [[Wikiversity:Namespaces|main namespace]]) and start writing! 

You should also read about the [[Portal:Education/Wikiversity model|Wikiversity:Learning model]]. Lessons should center on learning activities for Wikiversity participants. Learning materials and learning projects can be used by multiple projects - and you are encouraged to cooperate with other departments that use the same learning resource.

|}

[[Category:Chemistry]]
[[Category:Projects]]
[[Category:High School]]


----


Advanced Placement, AP and Pre-AP are registered trademarks of the [http://www.collegeboard.com College Board]

All AP teachers and administrators are encouraged to read the College Board's [http://apcentral.collegeboard.com/apc/members/program/initiatives/22794.html Equity Statement]</text>
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  </page>
  <page>
    <title>Dangme</title>
    <ns>0</ns>
    <id>86044</id>
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      <text bytes="4294" sha1="k6xyi3wyt4vyrgec48uj5a8454d4is6" xml:space="preserve">Dangme (Danbe, Ga-adangme, Ga-adangbe, Adangme, Adangbe)  is one of the Ga-Dangme languages within the Kwa branch of the Niger-Congo language family. It is spoken in Greater Accra, in south-east Ghana.


== Alphabets ==
ABCDEɛFGHIJKLMNOɔPQRSTUVWXYZ

== Expressions ==
'''I suɔ mo'''.  - I love you.

'''O pee nɔ'''. -  Thank you.

'''Mo tsumi.''' - Thank you.

'''I kpa mo pɛɛ.'''  - Please.

'''Mo-hee-e!''' -  Welcome.

'''Kusiɛ.''' -  I'm sorry.


'''si lafa'''		-		hundred times

'''sidi lafa'''		-	hundred cedis.


'''ligbi'''	-			day

'''otsi'''		-		week

'''lingmi'''	-			day before yesterday

'''hlami''' 	-			month

'''lingmi hlami'''		-	last month

'''jeha'''		-		year

'''momo'''	-			already

'''lolo'''		-		not yet

'''gble'''	-			never

'''bloonya se'''	-		after Christmas

'''gbidii'''		-		forever

'''mla'''			-	quickly

'''Osabu sisije'''	-		beginning of June

'''Osabu nyagbe''' -			end of June

'''Koo ya lolo.'''		-	Don't go yet.


J'''e mi pee kusuu'''.	-	It's a cloudy day.

'''Diblii wo kpii.'''	-		It's very dark.

'''Afua wo.	'''	-	It's misty.

'''Ahlabata be su ta.'''	-	The harmattan is near.

'''Pu da geii.'''		-	The sun is overhead.


'''I nga mo.'''		-	Good morning.

'''Mo he manye.'''	-		Good morning.

'''I haa mo.'''		-	Good afternoon.

'''I taa mo ama.'''	-		Good evening.

'''Mo hee-e!'''		-	Welcome!

'''Yaa ba!'''		-	Good bye. (to guest)

'''mo tsumi'''	-		thank you

'''Saminya, mo tsumi.''' 	-	 Very well, thank you.

'''I be he wami.'''		-	I'm not feeling well.

'''Ma na mo ekohu.'''	-	I shall see you again.

'''E hi kaa i na mo.'''	-	It's good to see you.
 
'''kɛ a tsɛɛ mo kɛ?'''       -what is your name(literally what do they call you)?


'''yumu'''			-	black

'''bluu'''			-	blue

'''asla su'''			-	brown

'''ba mumu'''		-	green

'''lazu'''			-	grey

'''tsutsu'''		-		red

'''huatahuata'''	-		spotted

'''futa	'''		-	white


'''na ku'''		-		bull

'''adadee'''		-		cat

'''na yo'''		-		cow

'''gbe/ ala'''			-	dog

'''teji'''			-	donkey

'''apletsi'''		-		goat

'''kpotoo'''		-		pig

'''nyaka'''		-		crocodile

'''kpi'''			-	grasscutter

'''jata'''			-	lion

'''kua'''			-	monkey

'''aya	'''		-	python

'''okpoe'''		-		rat

'''baasaki'''		-	scorpion

'''solue'''		-		squirrel

'''kolue'''		-		tortoise

'''hlami'''		-		turtle

'''ogbetee'''		-	wolf


'''titla'''			-	canary

'''agbaa'''		-		dove

'''kpakpahe'''	-		duck

'''klakumi'''		-	turkey

ako			-	parrot

kuakualabite	-		crow

'''oka'''			-	hawk

'''papalotsui'''	-		kite

'''patu'''		-		owl

'''otoklii'''		-		sparrow

'''okpoku'''		-	vulture	

'''nga'''			-	weaver bird


'''sangmlekiti'''	-		millipede

'''ahua'''		-		snail

'''tatu'''			-	ant

'''sa ngmo'''		-	bed bug

'''hwo'''			-	bee

'''aga'''			-	grasshopper

'''anunu'''		-		house fly

'''gbesle'''		-		red ant

'''baba'''		-		termite

'''adondongmatsi'''	-	wasp


'''kaawi'''		-		crab

'''ma'''			-	herring

'''osa'''			-	shrimp

'''gbolue'''		-		shark


'''I nui sisi.'''	-		I don't understand it.

'''Mo de ekohu.''' -			Say it again.

'''I li.'''			-	I don't know.


'''E susu ja.'''	-		She thinks so.

'''I susuu we ja.'''	-		I do not think so.

'''E ji ga womi kpakpa.'''	-	It's a good idea.


'''Jije o je?	'''	-	Where do you come from?

'''I je Osudoku'''.	-		I come from Osudoku.

'''Jije o yaa?'''		-	Where are you going to?

'''I yaa we mi.'''		-	I am going home.


'''O muklii mi de fu?'''	-	Are you angry?

'''Ye mi mi fu.'''		-	I am angry.

'''I yi gbeye.'''		-	I am not afraid.

'''I hao.'''			-	I am worried.


'''Mo na tsui.'''		-	Be patient.

'''Koo pee basabasa.''' -		Don't misbehave yourself.

'''Mo ko pia mi'''.	-		Don't blame me.


'''Awusabi ji mi.'''	-		I am an orphan.

'''E ji laami sane'''.	-	This is a secret.

'''E muklii sane.'''	-		She has had a miscarriage.

'''Waa je.'''		-	Let's go.

'''De o le mi?'''	-		Do you know me?

'''aloo	'''		-	or

'''Ma he amadaa sau kake.''' -	I want to buy a bunch of plantain.

'''I be Ga sue mla.'''	-	I shall not reach Accra early.

'''O be tikiti lo?'''	-		Don't you have a ticket?

'''Ni tsumi saisaa.'''	-	Any kind of work.

'''Wo klala duku futa a.'''	-	Take the white handkerchief. 

'''Mo ko je ye he.'''	-	Don't leave me.

== See Also ==
* [[/Common Phrases/]]

== Links ==
* [[Wikipedia: Dangme language]]

[[Category:Ghana]]</text>
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  <page>
    <title>Understanding Arithmetic Circuits</title>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260715.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260715.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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  <page>
    <title>Wikiversity:Wikipedia</title>
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      <contributor>
        <username>Atcovi</username>
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      <comment>/* See also */ +[[Wikiversity:Differences between Wikiversity and Wikipedia]]</comment>
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      <text bytes="636" sha1="qtsrh6969y2uusr00ppmcc2aamm91x1" xml:space="preserve">This page describes interwiki-relations between Wikiversity and [[Wikipedia]].

==Using a Wikiversity page as reference in Wikipedia==
As content of a wiki that anyone can edit, a Wikiversity page is generally not [[Wikipedia:Wikipedia:Identifying reliable sources|identified as a reliable source in Wikipedia]]. Therefore, usage of Wikiversity as a reference in Wikipedia is generally not advisable.

==See also==
*[[Wikiversity:Differences between Wikiversity and Wikipedia]], the differences in scopes between Wikiversity and Wikipedia. 
*[[Wikipedia:Wikipedia:Wikiversity|Wikipedia:Wikiversity]], the corresponding page in Wikipedia</text>
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  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
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      <contributor>
        <username>Young1lim</username>
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      <comment>/* Geometric Series Examples */</comment>
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      <text bytes="5296" sha1="8l14jxp7qifoyfus2esi2w9oe65y38p" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260715.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
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      <comment>This user has been globally banned</comment>
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      <text bytes="784" sha1="td0h2lt34ya6f86fqjpu6f2rec128f5" xml:space="preserve">__NOINDEX__ &lt;table style=&quot;border: 1px solid #aaa; margin: 4px 10%; border-collapse: collapse; background: #f9f9f9;&quot; class=&quot;plainlinks&quot; role=&quot;presentation&quot;&gt;&lt;tr&gt;&lt;td style=&quot;border:none; padding:2px 0 2px 0.9em;&quot;&gt;[[File:Wikimedia Foundation logo - vertical.svg|45px|alt=Wikimedia Foundation Logo]]&lt;/td&gt;&lt;td style=&quot;border:none; padding: 0.25em 0.9em; text-align:center;&quot;&gt;'''Consistent with the Terms of Use, {{#ifexpr:floor({{NAMESPACENUMBER}}/2)=1|{{BASEPAGENAME}}|this user}} has been banned by the Wikimedia Foundation from editing Wikimedia sites.''' &lt;br /&gt; Please address any questions to ca[[File:At sign.svg|x15px|middle|link=|alt=@]]wikimedia.org.&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt; {{#ifeq:{{NAMESPACENUMBER}}|3|[[Category:Opted-out of message delivery]]}}[[Category:Wikimedians banned by the WMF]]</text>
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  <page>
    <title>History of Topics in Special Relativity/relsource</title>
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      <contributor>
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      <comment>/* Historical relativity sources */ url</comment>
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      <text bytes="53943" sha1="lq7bkn7z1m8fomggpbq73svmq1ste4v" xml:space="preserve">===Historical relativity sources===
&lt;onlyinclude&gt;
&lt;section begin=abra1905 /&gt;
*{{Citation | last=Abraham, M.| year=1905 | chapter=§ 42. Die Lichtzeit in einem gleichförmig bewegten System|title= Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung | publisher=Teubner | location=Leipzig}}
::{{icon|wikisource}} See also the transcription [[s:de:Elektromagnetische Theorie der Strahlung (1905)|§ 42. Die Lichtzeit in einem gleichförmig bewegten System]] on German Wikisource&lt;section end=abra1905 /&gt;
&lt;section begin=abra08elek /&gt;
*{{Citation | last=Abraham, M.| year=1908 |title= Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung; 2. Auflage| publisher=Teubner | location=Leipzig|url=https://www.archive.org/details/theoriederelekt00fpgoog}}&lt;section end=abra08elek /&gt;
&lt;section begin=abra09elek /&gt;
*{{Citation |author=Abraham, M.|year=1909a |title=Zur Elektrodynamik bewegter Körper |journal=Rendiconti del Circolo Matematico di Palermo|volume=28|pages=1-28|url=http://hdl.handle.net/2027/mdp.39015040422019}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Electrodynamics of Moving Bodies (Abraham)|On the Electrodynamics of Moving Bodies]] on English Wikisource&lt;section end=abra09elek /&gt;
&lt;section begin=abra09mech /&gt;
*{{Citation |author=Abraham, M.|year=1909b |title=Zur elektromagnetischen Mechanik |journal=Physikalische Zeitschrift |volume=10|issue=21|pages=737-741|url=http://jvr.freewebpage.org/TableOfContents/Volume5/Issue1/Abraham1909.pdf}}&lt;section end=abra09mech /&gt;
&lt;section begin=abra10elek /&gt;
*{{Citation |author=Abraham, M. |year=1910 |title=Sull'Elletrodinamica di Minkowski |journal=Rendiconti del Circolo Matematico di Palermo |volume=30|pages=33-46|url=http://hdl.handle.net/2027/mdp.39015040422035}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Electrodynamics of Minkowski|On the Electrodynamics of Minkowski]] on English Wikisource&lt;section end=abra10elek /&gt;
&lt;section begin=abra12ener /&gt;
*{{Citation |author=Abraham, M. |year=1912 |title=Die Erhaltung der Energie und der Materie im Schwerkraftfelde |journal=Physikalische Zeitschrift |volume=13|pages=311-314|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/341}}&lt;section end=abra12ener /&gt;
&lt;section begin=bate10elec /&gt;
*{{Citation|author=Bateman, H.|year=1910|origyear=1909|title=The Transformation of the Electrodynamical Equations|journal=Proceedings of the London Mathematical Society|volume=8|pages=223-264|url=http://hdl.handle.net/2027/inu.30000021006535}}
::{{icon|wikisource}} See also the transcription [[s:The Transformation of the Electrodynamical Equations|The Transformation of the Electrodynamical Equations]] on English Wikisource&lt;section end=bate10elec /&gt;
&lt;section begin=bate12 /&gt;
*{{Citation|author=Bateman, H.|year=1912|origyear=1910|title=Some geometrical theorems connected with Laplace's equation and the equation of wave motion |journal=American Journal of Mathematics|volume=34|issue=3|doi=10.2307/2370223|pages=325–360|url=https://archive.org/details/jstor-2370223|jstor=2370223}}&lt;section end=bate12 /&gt;
&lt;section begin=bor14 /&gt;
*{{Citation|author=Borel, É.|year=1914|title=Introduction Geometrique à quelques Théories Physiques|location=Paris|url=http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=04710001}}&lt;section end=bor14 /&gt;
&lt;section begin=born09elek /&gt;
*{{Citation |author=Born, M. |year=1909 |title=Die träge Masse und das Relativitätsprinzip |journal=Annalen der Physik|volume=333|issue=3|pages=571-584|url=http://hdl.handle.net/2027/uc1.b5622478}}&lt;section end=born09elek /&gt;
&lt;section begin=brill09 /&gt;
*{{Citation|author=Brill, A.|title=Vorlesungen zur Einführung in die Mechanik raumerfüllender Massen|publisher=B.G. Teubner|location=Leipzig|year=1909|url=https://rcin.org.pl/dlibra/publication/13858}}&lt;section end=brill09 /&gt;
&lt;section begin=brill25 /&gt;
*{{Citation |author=Brill, J.|year=1925 |journal=Proceedings of the Cambridge Philosophical Society|title= Note on the Lorentz group|pages=630|volume=22|issue=5 |doi=10.1017/S030500410000949X|bibcode=1925PCPS...22..630B}}&lt;section end=brill25 /&gt;
&lt;section begin=buch04 /&gt;
*{{Citation |author=Bucherer, A. H. |year=1904 |title=Mathematische Einführung in die Elektronentheorie |publisher=Teubner |location=Leipzig |url=https://archive.org/details/mathematischeei02buchgoog}}&lt;section end=buch04 /&gt;
&lt;section begin=buch08 /&gt;
*{{Citation |author=Bucherer, A. H. |year=1908 |title=Messungen an Becquerelstrahlen. Die experimentelle Bestätigung der Lorentz-Einsteinschen Theorie. |journal=Physikalische Zeitschrift |volume=9 |issue=22 |pages=755–762|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0009/page/785}}. For Minkowski's and Voigt's statements see p.&amp;nbsp;762.
::{{icon|wikisource}} See also the transcription [[s:de:Messungen an Becquerelstrahlen|Messungen an Becquerelstrahlen]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:Measurements of Becquerel rays|Measurements of Becquerel rays]] on English Wikisource&lt;section end=buch08 /&gt;
&lt;section begin=car12 /&gt;
*{{Citation|author=Cartan, É.|year=1912|journal=Société de Mathématique the France – Comptes Rendus des Séances|title=Sur les groupes de transformation de contact et la Cinématique nouvelle|pages=23|url=https://archive.org/details/bulletinsocit40soci/page/422/mode/2up}}&lt;section end=car12 /&gt;
&lt;section begin=cohn04a /&gt;
*{{Citation |author=Cohn, E. |year=1904a |title=Zur Elektrodynamik bewegter Systeme I |journal= Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |volume=1904/2 |issue=40 |pages =1294–1303|url=https://archive.org/details/sitzungsberichte1904deut/page/1294/mode/2up}}
::{{icon|wikisource}} See also the transcription [[s:de:Zur Elektrodynamik bewegter Systeme I|Zur Elektrodynamik bewegter Systeme I]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Electrodynamics of Moving Systems I|On the Electrodynamics of Moving Systems I]] on English Wikisource&lt;section end=cohn04a /&gt;
&lt;section begin=cohn04b /&gt;
*{{Citation |author=Cohn, E. |year=1904b |title=Zur Elektrodynamik bewegter Systeme II |journal= Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften |volume=1904/2 |issue=43 |pages =1404–1416|url=https://archive.org/details/sitzungsberichte1904deut/page/1404/mode/2up}} 
::{{icon|wikisource}} See also the transcription [[s:de:Zur Elektrodynamik bewegter Systeme II|Zur Elektrodynamik bewegter Systeme II]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Electrodynamics of Moving Systems II|On the Electrodynamics of Moving Systems II]] on English Wikisource&lt;section end=cohn04b /&gt;
&lt;section begin=con11qua /&gt;
*{{Citation|author=Conway, A. W.|year=1911|title=On the application of quaternions to some recent developments of electrical theory|journal=Proceedings of the Royal Irish Academy, Section A|volume=29|pages=1–9|url= https://archive.org/download/proceedingsofro29roya}}&lt;section end=con11qua /&gt;
&lt;section begin=cunn10 /&gt;
*{{Citation|author=Cunningham, E.|year=1910|origyear=1909|title=The principle of Relativity in Electrodynamics and an Extension Thereof|journal=Proceedings of the London Mathematical Society |volume=8|pages=77–98|doi=10.1112/plms/s2-8.1.77}}
::{{icon|wikisource}} See also the transcription [[s:en:The principle of Relativity in Electrodynamics and an Extension Thereof|The principle of Relativity in Electrodynamics and an Extension Thereof]] on English Wikisource&lt;section end=cunn10 /&gt;
&lt;section begin=cunn14princ /&gt;
*{{Citation |author=Cunningham, E. |year=1914 |title=The principle of relativity|publisher=Cambridge: University Press|url=https://archive.org/details/principleofrelat00cunniala}}&lt;section end=cunn14princ /&gt;
&lt;section begin=einst05elek /&gt;
*{{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].&lt;section end=einst05elek /&gt;
&lt;section begin=einst07pri /&gt;
*{{Citation|author=Einstein, A.|date=1908|orig-date=Submitted December 1907|title=Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen|journal=Jahrbuch der Radioaktivität und Elektronik|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 2, Document 47, and in the corresponding English translation volume|volume=4|pages=411-462|url=https://archive.org/details/jahrbuch-der-radioaktivitat-und-elektronik-4.1907/page/410/mode/2up}}.&lt;section end=einst07pri /&gt;
&lt;section begin=einst11zurb /&gt;
*{{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu &quot;Die Relativitäts-Theorie&quot;|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}&lt;br /&gt; While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX].&lt;section end=einst11zurb /&gt;
&lt;section begin=einst12manu /&gt;
*{{Citation |author=Einstein, A. |year=1912-14 |chapter=Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|chapter-url=https://einsteinpapers.press.princeton.edu/vol4-doc/25}}&lt;section end=einst12manu /&gt;
&lt;section begin=einst13ent /&gt;
*{{Citation |author=Einstein, A. &amp; Grossmann, M. |year=1913 |title=Entwurf einer verallgemeinerten Relativitätstheorie und eine Theorie der Gravitation|journal=Zeitschrift für Mathematik und Physik|volume=62|pages=225-261|url=https://einsteinpapers.press.princeton.edu/vol4-doc/324}}&lt;section end=einst13ent /&gt;
&lt;section begin=einst14grund /&gt;
*{{Citation |author=Einstein, A. |year=1914 |title=Die Formale Grundlage der allgemeinen Relativitätstheorie|journal=Berliner Sitzungsberichte|volume=1914 (2)|pages=1030–1085|url=https://einsteinpapers.press.princeton.edu/vol6-doc/100}}&lt;section end=einst14grund /&gt;
&lt;section begin=einst16grund /&gt;
*{{Citation |author=Einstein, A. |year=1916 |title=Die Grundlage der allgemeinen Relativitätstheorie|journal=Annalen der Physik|volume=354 (7)|pages=769-822|url=https://einsteinpapers.press.princeton.edu/vol6-doc/311}}&lt;section end=einst16grund /&gt;
&lt;section begin=einst17pop /&gt;
*{{Cite book |last=Einstein  |first= Albert|year=1917-1954 |title=Über die spezielle und die allgemeine Relativitätstheorie (fifth edition) |publisher=Vieweg |place=Braunschweig |url=https://archive.org/details/relativitythespe00einsuoft}} – See also the English translation [[s:Relativity (1931)|Relativity: The Special and the General Theory]]; For information on different versions, see Collected papers of Albert Einstein, Vol 6, pp. 417ff.&lt;section end=einst17pop /&gt;
&lt;section begin=einst22kyoto /&gt;
*{{Citation |author=Einstein, A. |year=1922 |orig-date=2012|chapter=Einstein's lecture at the University of Kyoto – How I Created the Theory of Relativity|title=The collected papers of Albert Einstein|volume=13|pages=624ff|chapter-url=https://einsteinpapers.press.princeton.edu/vol13-doc/720}}&lt;section end=einst22kyoto /&gt;
&lt;section begin=frank09a /&gt;
*{{Citation|author=Frank, P. |year=1909 |title=Die Stellung des Relativitätsprinzips im System der Mechanik und Elektrodynamik|journal=Wiener Sitzungsberichte IIa |volume=118 |pages=373-446|url=http://hdl.handle.net/2027/mdp.39015073682224}}&lt;section end=frank09a /&gt;
&lt;section begin=fra11 /&gt;
*{{Citation|author1=Frank, P. |author2=Rothe, H.|year=1911|title=Über die Transformation der Raum-Zeitkoordinaten von ruhenden auf bewegte Systeme|journal=Annalen der Physik|volume=339|issue=5|pages=825–855|url=http://gallica.bnf.fr/ark:/12148/bpt6k15337j/f845.table|doi=10.1002/andp.19113390502|bibcode = 1911AnP...339..825F }}&lt;section end=fra11 /&gt;
&lt;section begin=fra12 /&gt;
*{{Citation|author1=Frank, P. |author2=Rothe, H.|title=Zur Herleitung der Lorentztransformation|journal=Physikalische Zeitschrift|volume=13|year=1912|pages=750–753|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/782}}&lt;section end=fra12 /&gt;
&lt;section begin=gans05 /&gt;
*{{Citation |author=Gans, R. |year=1905 |title=H. A. Lorentz. Elektromagnetische Vorgänge |journal= Beiblätter zu den Annalen der Physik |volume=29 |issue=4 |pages =168–170|url=https://archive.org/details/beibltterzudena18pockgoog/page/168/mode/2up}} 
::{{icon|wikisource}} See also the transcription [[s:Translation:H.A. Lorentz: Electromagnetic Phenomena|H.A. Lorentz: Electromagnetic Phenomena]] on English Wikisource&lt;section end=gans05 /&gt;
&lt;section begin=gram13 /&gt;
*{{Citation|author=Grammel, R.|year=1913|title=Zur relativitätstheoretischen Elektrodynamik bewegter Körper|journal=Annalen der Physik|volume=346|issue=8|pages=570-580|doi=10.1002/andp.19133460805}}&lt;section end=gram13 /&gt;
&lt;section begin=gbaum /&gt;
*{{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|year=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}&lt;section end=gbaum /&gt;
&lt;section begin=grun21a /&gt;
*{{Citation|author1=Gruner, P.; Sauter, J. |title=Représentation géométrique élémentaire des formules de la théorie de la relativité|journal=Archives des sciences physiques et naturelles|series=5|volume=3|pages=295–296|year=1921a|url=http://gallica.bnf.fr/ark:/12148/bpt6k2991536/f295.image}} 
::{{icon|wikisource}} See also the transcription [[s:Translation:Elementary geometric representation of the formulas of the special theory of relativity|Elementary geometric representation of the formulas of the special theory of relativity]] on English Wikisource&lt;section end=grun21a /&gt;
&lt;section begin=grun21b /&gt;
*{{Citation|author=Gruner, P.|title=Eine elementare geometrische Darstellung der Transformationsformeln der speziellen Relativitätstheorie|journal=Physikalische Zeitschrift|volume=22|pages=384–385|year=1921b|url=https://archive.org/details/physikalische-zeitschrift-vol-22/page/384/mode/2up}}
::{{icon|wikisource}} See also the transcription [[s:Translation:An elementary geometrical representation of the transformation formulas of the special theory of relativity|An elementary geometrical representation of the transformation formulas of the special theory of relativity]] on English Wikisource&lt;section end=grun21b /&gt;
&lt;section begin=hahn /&gt;
*{{Citation |author=Hahn, E. |year=1912 |title=Grundlagen zu einer Theorie der Lorentztransformationen (Dissertation)|publisher=B. G. Teubner|location=Leipzig|url=https://hdl.handle.net/2027/uc1.b2648072}}; Reprinted 1913 in ''Archiv der Mathematik und Physik 21: 1-42''&lt;section end=hahn /&gt;
&lt;section begin=heav89 /&gt;
*{{Citation |author=Heaviside, O. |year=1889 |title=On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric |journal=Philosophical Magazine |series=5 |volume=27 |issue=167 |pages=324–339 |doi=10.1080/14786448908628362 |url=https://zenodo.org/record/1431195/files/article.pdf }}&lt;section end=heav89 /&gt;
&lt;section begin=hen13tens /&gt;
*{{Citation|author=Henschke, E.|year=1913|origyear=1912|title=Über eine Form des Prinzips der kleinsten Wirkung in der Elektrodynamik des Relativitätsprinzips|journal=Annalen der Physik|volume=345|issue=5|pages=887-934|doi=10.1002/andp.19133450505}}&lt;section end=hen13tens /&gt;
&lt;section begin=herg04pot /&gt;
*{{Citation |author=Herglotz, G. |year=1904 |title=Über die Berechnung retardierter Potentiale|journal=Gött. Nachr. |issue=6 |pages=549–556|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1904&amp;DMDID=DMDLOG_0056}}&lt;section end=herg04pot /&gt;
&lt;section begin=herg10 /&gt;
*{{Citation|author=Herglotz, G.|year=1910|origyear=1909|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper]|journal=Annalen der Physik|volume=336|issue=2 |pages=393–415|doi=10.1002/andp.19103360208|bibcode = 1910AnP...336..393H}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On bodies that are to be designated as &quot;rigid&quot;|On bodies that are to be designated as &quot;rigid&quot; from the standpoint of the relativity principle]] on English Wikisource&lt;section end=herg10 /&gt;
&lt;section begin=herg11ela /&gt;
*{{Citation|author=Herglotz, G. |year=1911 |title=Über die Mechanik des deformierbaren Körpers vom Standpunkte der Relativitätstheorie|journal=Annalen der Physik |volume=341 (13) |pages=493-533|url=http://gallica.bnf.fr/ark:/12148/bpt6k153397.image.f509}}; English translation by David Delphenich: [http://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/herglotz_-_rel._cont._mech..pdf On the mechanics of deformable bodies from the standpoint of relativity theory].&lt;section end=herg11ela /&gt;
&lt;section begin=igna10 /&gt;
*{{Citation|author=Ignatowsky, W. v.|title=Einige allgemeine Bemerkungen über das Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=11|year=1910|pages=972–976|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0011/page/1048}}
::{{icon|wikisource}} See also the transcription [[s:de:Einige allgemeine Bemerkungen über das Relativitätsprinzip|Einige allgemeine Bemerkungen über das Relativitätsprinzip]] on German Wikisource &lt;section end=igna10 /&gt;
&lt;section begin=ignat10prin2 /&gt;
*{{Citation |author=Ignatowsky, W. v. |year=1910 |title=Das Relativitätsprinzip |journal=Archiv der Mathematik und Physik 17: 1-24, 18: 17-40|url=http://hdl.handle.net/2027/mdp.39015085215708}}
::{{icon|wikisource}} See also the transcription [[s:de:Das Relativitätsprinzip (Ignatowski)|Das Relativitätsprinzip]] on German Wikisource &lt;section end=ignat10prin2 /&gt;
&lt;section begin=ignat11hyd /&gt;
*{{Citation|author=Ignatowsky, W. v. |year=1911 |title=Zur Hydrodynamik vom Standpunkte des Relativitätsprinzips |journal=Physikalische Zeitschrift |volume=12 |pages=441-442|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/477}}&lt;section end=ignat11hyd /&gt;
&lt;section begin=igna11 /&gt;
*{{Citation|author=Ignatowski, W. v.|title=Eine Bemerkung zu meiner Arbeit: &quot;Einige allgemeine Bemerkungen zum Relativitätsprinzip&quot;|journal=Physikalische Zeitschrift|volume=12|year=1911|pages=779|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/831}}
::{{icon|wikisource}} See also the transcription [[s:de:Eine Bemerkung zu meiner Arbeit: &quot;Einige allgemeine Bemerkungen zum Relativitätsprinzip&quot;|Eine Bemerkung zu meiner Arbeit: &quot;Einige allgemeine Bemerkungen zum Relativitätsprinzip&quot;]] on German Wikisource &lt;section end=igna11 /&gt;
&lt;section begin=ishi10med1 /&gt;
*{{Citation |author=Ishiwara, J. |year=1910 |title=Zur Theorie der elektromagnetischen Vorgänge in bewegten Körpern |journal=Proceedings of the Tokyo Mathematico-Physical Society. 2nd Series|volume=5|issue=18|pages=310-327|url=https://www.jstage.jst.go.jp/article/ptmps1907/5/18/5_18_310/_article/-char/en}}&lt;section end=ishi10med1 /&gt;
&lt;section begin=ishi11imp /&gt;
*{{Citation |author=Ishiwara, J. |year=1911 |title=Ueber die elektromagnetischen Impulsgleichungen in der Relativitätstheorie |journal=Proceedings of the Tokyo Mathematico-Physical Society. 2nd Series|volume=6|issue=11|pages=164-176|url=https://www.jstage.jst.go.jp/article/ptmps1907/6/11/6_11_164/_article/-char/en}}&lt;section end=ishi11imp /&gt;
&lt;section begin=kir57elek /&gt;
*{{Citation |author=Kirchhoff, G. |year=1857 |title=Ueber die Bewegung der Elektricität in Leitern |journal=Annalen der Physik|volume=178|issue=12 |pages=529-544|url=https://books.google.com/books?id=hum2dY0EwrEC&amp;pg=PA529}}&lt;section end=kir57elek /&gt;
&lt;section begin=klein08 /&gt;
*{{Citation|author=Klein, F.|editor=Hellinger, E.|year=1908|title=Elementarmethematik vom höheren Standpunkte aus. Teil I. Vorlesung gehalten während des Wintersemesters 1907-08|publisher=Teubner|location=Leipzig|url=https://archive.org/details/elementarmathem00kleigoog}}&lt;section end=klein08 /&gt;
&lt;section begin=klein10 /&gt;
*{{Citation|author=Klein, F.|year=1921|origyear=1910|journal=Gesammelte Mathematische Abhandlungen |title=Über die geometrischen Grundlagen der Lorentzgruppe|volume=1|pages=533–552|doi=10.1007/978-3-642-51960-4_31|isbn=978-3-642-51898-0}}
::{{icon|wikisource}} See also the transcription [[s:de:Über die geometrischen Grundlagen der Lorentzgruppe|Über die geometrischen Grundlagen der Lorentzgruppe]] on German Wikisource&lt;section end=klein10 /&gt;
&lt;section begin=klein10b /&gt;
*{{Citation|author=Klein, F. |author2=Sommerfeld A.|editor=Noether, Fr.|year=1910|title=Über die Theorie des Kreisels. Heft IV|location=Leipzig|publisher=Teuber|url=https://archive.org/details/fkleinundasommer019696mbp}}&lt;section end=klein10b /&gt;
&lt;section begin=klein11 /&gt;
*{{Citation|author=Klein, F.|editor=Hellinger, E.|year=1911|title=Elementarmethematik vom höheren Standpunkte aus. Teil I (Second Edition). Vorlesung gehalten während des Wintersemesters 1907-08|publisher=Teubner|location=Leipzig|hdl=2027/mdp.39015068187817}}&lt;section end=klein11 /&gt;
&lt;section begin=kott12mink /&gt;
*{{Citation|author=Kottler, F.|year=1912|title=Über die Raumzeitlinien der Minkowski'schen Welt|journal=Wiener Sitzungsberichte 2a|volume=121|pages=1659–1759|url=https://hdl.handle.net/2027/mdp.39015051107277}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On the spacetime lines of a Minkowski world|On the spacetime lines of a Minkowski world]] on English Wikisource&lt;section end=kott12mink /&gt;
&lt;section begin=kott14bes /&gt;
*{{Citation|author=Kottler, F.|year=1914a|title=Relativitätsprinzip und beschleunigte Bewegung|journal=Annalen der Physik|volume=349|issue=13|pages=701–748|url=http://gallica.bnf.fr/ark:/12148/bpt6k15347v.image.f737|doi=10.1002/andp.19143491303|bibcode=1914AnP...349..701K}}&lt;section end=kott14bes /&gt;
&lt;section begin=lamla12hyd /&gt;
*{{Citation |author=Lamla, E. |year=1912 |title=Über die Hydrodynamik des Relativitätsprinzips (Dissertation)|publisher=Berlin: Trowitsch|url=http://resolver.sub.uni-goettingen.de/purl?PPN31487514X}}&lt;br&gt;Summary in: {{Citation|author=Lamla, E. |year=1912 |title=Über die Hydrodynamik des Relativitätsprinzips|journal=Annalen der Physik |volume=342 (4) |pages=772-796|url=http://gallica.bnf.fr/ark:/12148/bpt6k15340f.image.f790}}&lt;section end=lamla12hyd /&gt;
&lt;section begin=laemmel11 /&gt;
*{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&amp;d=NZZ19110428-01.2.4.1}}&lt;section end=laemmel11 /&gt;
&lt;section begin=laemmel21 /&gt;
*{{Citation|author=Lämmel, R.|date=1921|orig-date=December 1920|title=Die Grundlagen der Relativitätstheorie|location=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}&lt;section end=laemmel21 /&gt;
&lt;section begin=Lampa /&gt;
*{{cite journal | author = Lampa, A. | year = 1924 | title = Wie erscheint nach der Relativitätstheorie ein bewegter Stab einem ruhenden Beobachter? | language = German | journal = Zeitschrift für Physik | volume = 27 | issue = 1 | pages = 138–148 | doi = 10.1007/BF01328021 |bibcode = 1924ZPhy...27..138L | s2cid = 119547027|url=https://archive.org/details/zeitschrift-fuer-physik-a-atoms-and-nuclei_1924_27/page/138/mode/2up}}&lt;section end=Lampa /&gt;
&lt;section begin=lanc25 /&gt;
*{{Citation |author=Lanczos, C. |year=1925 |title=Review of: ''Anton Lampa. Wie erscheint nach der Relativitätstheorie ein bewegter Stab einem ruhenden Beobachter?'' |journal=Physikalische Berichte |volume=6|issue=4|pages=251|url=https://archive.org/details/physics-briefs-physikalische-berichte_1925-02-15_6_4/page/250/mode/2up}}&lt;section end=lanc25 /&gt;
&lt;section begin=laue08 /&gt;
*{{Citation |author=Laue, M. v. |year=1908 |title=Die Wellenstrahlung einer bewegten Punktladung nach dem Relativitätsprinzip |journal=Berichte der Deutschen Physikalischen Gesellschaft|pages=838-844|url=http://www.archive.org/details/verhandlungende39unkngoog}}
::{{icon|wikisource}} See also the transcription [[s:Translation:The Wave Radiation of a Moving Point Charge in Accordance with the Principle of Relativity|The Wave Radiation of a Moving Point Charge in Accordance with the Principle of Relativity]] on English Wikisource&lt;section end=laue08 /&gt;
&lt;section begin=laue11gru /&gt;
*{{Citation |author=Laue, M. v. |year=1911 |title=Review of: ''F. Grünbaum. Über einige ideelle Versuche zum Relativitätsprinzip'' |journal=Beiblätter zu den Annalen der Physik |volume=35|pages=1187|url=https://books.google.com/books?id=n5ZLAAAAMAAJ}}&lt;section end=laue11gru /&gt;
&lt;section begin=laue11prin /&gt;
*{{Citation |author=Laue, M. v. |year=1911 |title=Das Relativitätsprinzip |publisher=Vieweg |location=Braunschweig|url=https://archive.org/details/dasrelativittsp00lauegoog}}&lt;section end=laue11prin /&gt;
&lt;section begin=laue13prin /&gt;
*{{Citation |author=Laue, M. v. |date=December 1912|publication-date=1913 |title=Das Relativitätsprinzip (2. Edition) |publisher=Vieweg |location=Braunschweig|url=http://digitale.beic.it/primo_library/libweb/action/dlDisplay.do?vid=BEIC&amp;docId=39bei_digitool6467296}}&lt;section end=laue13prin /&gt;
&lt;section begin=laue21prin /&gt;
*{{Citation |author=Laue, M. v. |year=1921 |title=Das Relativitätsprinzip (4. edition) |publisher=Vieweg |location=Braunschweig|url=http://www.archive.org/details/dierelativitts01laueuoft}}&lt;section end=laue21prin /&gt;
&lt;section begin=lar97 /&gt;
*{{Citation |author=Larmor, J. |year=1897 |title=On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with material media |journal=Philosophical Transactions of the Royal Society |volume=190 |pages=205–300 |doi=10.1098/rsta.1897.0020|bibcode = 1897RSPTA.190..205L |doi-access=free }}
::{{icon|wikisource}} See also the transcription [[s:Dynamical Theory of the Electric and Luminiferous Medium III|Dynamical Theory of the Electric and Luminiferous Medium III]] on English Wikisource&lt;section end=lar97 /&gt;
&lt;section begin=lar29 /&gt;
*{{Citation |author=Larmor, J. |year=1929 |origyear=1897|title=Mathematical and Physical Papers: Volume II |chapter=On a Dynamical Theory of the Electric and Luminiferous Medium. Part 3: Relations with material media|pages=2–132|publisher=Cambridge University Press|isbn=978-1-107-53640-1|url=https://archive.org/details/mathematicalphys0002jose/page/38}} (Reprint of Larmor (1897) with new annotations by Larmor, see particularly p. 39.)&lt;section end=lar29 /&gt;
&lt;section begin=lar00 /&gt;
*{{Citation |author=Larmor, J. |year=1900 |title=Aether and Matter |publisher=Cambridge University Press|url=http://www.archive.org/details/aethermatterdeve00larmuoft}}
::{{icon|wikisource}} See also the transcription [[s:Aether and Matter/Chapter 10|Aether and Matter]] on English Wikisource&lt;section end=lar00 /&gt;
&lt;section begin=lar04a /&gt;
*{{Citation|author=Larmor, J.|title=On the intensity of the natural radiation from moving bodies and its mechanical reaction|journal=Philosophical Magazine|year=1904a|volume=7|issue=41|pages=[https://archive.org/details/londonedinburgh671904lond/page/578 578]–586|url=https://archive.org/details/londonedinburgh671904lond|doi=10.1080/14786440409463149}}&lt;section end=lar04a /&gt;
&lt;section begin=lar04b /&gt;
*{{Citation|author=Larmor, J.|title=On the ascertained Absence of Effects of Motion through the Aether, in relation to the Constitution of Matter, and on the FitzGerald-Lorentz Hypothesis|journal=Philosophical Magazine|volume=7|issue=42|year=1904b|pages=621–625|doi=10.1080/14786440409463156}}
::{{icon|wikisource}} See also the transcription [[s:en:Absence of Effects of Motion through the Aether|Absence of Effects of Motion through the Aether]] on English Wikisource&lt;section end=lar04b /&gt;
&lt;section begin=lehmann /&gt; 
*{{Citation |author=Lehmann, Otto |orig-date=September 1910 |date=1911|title=Das Relativitätsprinzip, der neue Fundamentalsatz der Physik |journal=Verhandlungen des naturwissenschaften Vereins in Karlsruhe |volume=23 |pages=49-74}} 
::{{icon|wikisource}} See also the transcription [[s:de:Das Relativitätsprinzip der neue Fundamentalsatz der Physik|Das Relativitätsprinzip der neue Fundamentalsatz der Physik]] on German Wikisource&lt;section end=lehmann /&gt;
&lt;section begin=lew09 /&gt;
*{{Citation|author=Lewis, G. N.; Tolman, R. C. | date=1909|title=The Principle of Relativity, and Non-Newtonian Mechanics |journal=Proceedings of the American Academy of Arts and Sciences|volume=44 |pages=709–726|doi=10.2307/20022495 |issue=25 |title-link=s:The Principle of Relativity, and Non-Newtonian Mechanics |jstor=20022495 }}&lt;section end=lew09 /&gt;
&lt;section begin=lew10vec /&gt;
*{{Citation |author=Lewis, G. N. |year=1910 |title=On Four-Dimensional Vector Analysis, and Its Application in Electrical Theory |journal=Proceedings of the American Academy of Arts and Sciences|volume=43|issue=7|pages=165-181|url=https://www.jstor.org/stable/20022625}}&lt;section end=lew10vec /&gt;
&lt;section begin=lewis12non /&gt;
*{{Citation |author=Lewis, G. N. &amp; Wilson, E. B. |year=1912 |title=The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics |journal=Proceedings of the American Academy of Arts and Sciences|volume=48|pages=387–507|url=https://www.jstor.org/stable/20022840}}&lt;section end=lewis12non /&gt;
&lt;section begin=lor86 /&gt;
*{{Citation |author=Lorentz, H. A. |year=1886 |title=Over den invloed, dien de beweging der aarde op de lichtverschijnselen uitoefent |journal=Versl. Kon. Ak. Wet.|volume=2|issue=3|pages=297–372 |url=https://books.google.com/books?id=fkFnAAAAcAAJ&amp;pg=PA297}}&lt;section end=lor86 /&gt;
&lt;section begin=lor92elek /&gt;
*{{Citation |author=Lorentz, H. A. |year=1892a |title=La Théorie electromagnétique de Maxwell et son application aux corps mouvants |journal=Archives Néerlandaises des Sciences Exactes et Naturelles |volume=25 |pages=363–552 |url=https://archive.org/details/lathorielectrom00loregoog}}&lt;section end=lor92elek /&gt;
&lt;section begin=lor92b /&gt;
*{{Citation |last=Lorentz |first=H. A. |year=1892b |title=De relatieve beweging van de aarde en den aether |journal=Zittingsverlag Akad. V. Wet. |pages=74–79 |volume=1}}
::{{icon|wikisource}} See also the transcription [[s:Translation:The Relative Motion of the Earth and the Aether|The Relative Motion of the Earth and the Aether]] on English Wikisource&lt;section end=lor92b /&gt;
&lt;section begin=lor95 /&gt;
*{{Citation |author=Lorentz, H. A. |year=1895 |title=Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern |location=Leiden |publisher=E.J. Brill }}
::{{icon|wikisource}} See also the transcription [[s:de:Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern|Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies|Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies]] on English Wikisource&lt;section end=lor95 /&gt;
&lt;section begin=lor99 /&gt;
*{{Citation |author=Lorentz, H. A. |year=1899 |title=Simplified Theory of Electrical and Optical Phenomena in Moving Systems |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=1 |pages=427–442 |bibcode=1898KNAB....1..427L }}
::{{icon|wikisource}} See also the transcription [[s:Simplified Theory of Electrical and Optical Phenomena in Moving Systems|Simplified Theory of Electrical and Optical Phenomena in Moving Systems]] on English Wikisource&lt;section end=lor99 /&gt;
&lt;section begin=lor04 /&gt;
*{{Citation |author=Lorentz, H. A. |year=1904 |title=Electromagnetic phenomena in a system moving with any velocity smaller than that of light |journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences |volume=6 |pages=809–831 |bibcode=1903KNAB....6..809L }}
::{{icon|wikisource}} See also the transcription [[s:Electromagnetic phenomena|Electromagnetic phenomena in a system moving with any velocity smaller than that of light]] on English Wikisource&lt;section end=lor04 /&gt;
&lt;section begin=lor16 /&gt;
*{{Citation |author=Lorentz, H. A. |year=1916|origyear=1915|title=The theory of electrons and its applications to the phenomena of light and radiant heat |url=https://archive.org/details/electronstheory00lorerich |place=Leipzig &amp; Berlin |publisher=B.G. Teubner}}&lt;section end=lor16 /&gt;
&lt;section begin=lor21 /&gt;
*{{Citation|author=Lorentz, H. A.|year=1921|origyear=1914|title=Deux Mémoires de Henri Poincaré sur la Physique Mathématique|journal=Acta Mathematica|volume=38|issue=1|pages=293–308|doi=10.1007/BF02392073}}
::{{icon|wikisource}} See also the transcription [[s:fr:Deux Mémoires de Henri Poincaré sur la Physique Mathématique|Deux Mémoires de Henri Poincaré sur la Physique Mathématique]] on French Wikisource 
::{{icon|wikisource}} See also the transcription [[s:Translation:Two Papers of Henri Poincaré on Mathematical Physics|Two Papers of Henri Poincaré on Mathematical Physics]] on English Wikisource&lt;section end=lor21 /&gt;
&lt;section begin=lorentz14 /&gt;
*See §11 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}&lt;section end=lorentz14 /&gt;
&lt;section begin=lorentz22 /&gt;
*See pp. 95ff in: {{Citation|author=Lorentz, H. A.|date=1927|orig-date=Lectures from 1922|title=Problems of modern physics; a course of lectures delivered in the California Institute of Technology|place=Boston|publisher=Ginn and Company|url=https://archive.org/details/in.ernet.dli.2015.155685}}&lt;section end=lorentz22 /&gt;
&lt;section begin=lor67pot /&gt;
*{{Citation |author=Lorenz, L. |year=1867 |title=On the identity of the vibrations of light with electrical currents |journal=Phil. Mag|series=4|volume=34 |pages=287–301|url=https://books.google.com/books?id=k5zJkz4f0MAC&amp;pg=PA287}}&lt;section end=lor67pot /&gt;
&lt;section begin=mang10 /&gt;
*{{Citation|author=Mangoldt, H. v.|title=Längen- und Zeitmessung in der Relativitätstheorie.|journal=Physikalische Zeitschrift|volume=11|pages=737–744|year=1910|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0011/page/799}}&lt;section end=mang10 /&gt;
&lt;section begin=marc06elek /&gt;
*{{Citation |author=Marcolongo, R. |year=1906 |title=Sugli integrali delle equazioni dell’elettrodinamica |journal=Atti della Reale Accademia dei Lincei Rendiconti |volume=15 |pages=344-349|url=http://villafarnesina.it/pubblicazioni/rendicontiFMN/rol/pdf/S5V15T1A1906P344_349.pdf}}&lt;section end=marc06elek /&gt;
&lt;section begin=mich81 /&gt;
*{{Citation |author=Michelson, A. |title=The Relative Motion of the Earth and the Luminiferous Ether|journal = American Journal of Science |volume = 22 |issue=128 |year = 1881 |pages = 120–129 |doi=10.2475/ajs.s3-22.128.120|title-link=s:On the Relative Motion of the Earth and the Luminiferous Ether |bibcode=1881AmJS...22..120M |s2cid=130423116 }}&lt;section end=mich81 /&gt;
&lt;section begin=mich82 /&gt;
*{{Citation |author=Michelson, A. |year=1882 |title=Sur le mouvement relatif de la Terre et de l'éther |journal=Comptes Rendus |volume=94 |pages=520-523|url=https://books.google.com/books?id=qK8Z3OosFrgC&amp;pg=PA520}}&lt;section end=mich82 /&gt;
&lt;section begin=mich87 /&gt;
*{{Citation |author=Michelson, A.; Morley, E.|title=On the Relative Motion of the Earth and the Luminiferous Ether |journal=American Journal of Science |volume=34 |issue=203 |year=1887 |pages=333–345 |doi=10.2475/ajs.s3-34.203.333|title-link=s:On the Relative Motion of the Earth and the Luminiferous Ether |bibcode=1887AmJS...34..333M |s2cid=124333204 }}&lt;section end=mich87 /&gt;
&lt;section begin=mink07a /&gt;
*{{citation |author=Minkowski, H. |origyear=1907|year=1915 |title=Das Relativitätsprinzip |journal=Annalen der Physik |volume=352 |issue=15 |pages=927–938|url=http://gallica.bnf.fr/ark:/12148/bpt6k15350r.image.f951}}
::{{icon|wikisource}} See also the transcription [[s:de:Das Relativitätsprinzip (Minkowski)|Das Relativitätsprinzip]] on German Wikisource&lt;section end=mink07a /&gt;
&lt;section begin=mink07b /&gt;
*{{Citation |author=Minkowski, H. |year=1908 |origyear=1907 |title=Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern |journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |pages=53–111|url=https://archive.org/details/nachrichten09klasgoog}}
::{{icon|wikisource}} See also the transcription [[s:de:Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern|Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern]] on German Wikisource 
::{{icon|wikisource}} See also the transcription [[s:Translation:The Fundamental Equations for Electromagnetic Processes in Moving Bodies|The Fundamental Equations for Electromagnetic Processes in Moving Bodies]] on English Wikisource&lt;section end=mink07b /&gt;
&lt;section begin=mink08 /&gt;
*{{Citation|author=Minkowski, H. |year=1909 |origyear=1908 |title=Raum und Zeit |journal=Physikalische Zeitschrift |volume=10 |pages=75–88|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0010/page/103}}
::{{icon|wikisource}} See also the transcription [[s:de:Raum und Zeit (Minkowski)|Raum und Zeit]] on German Wikisource 
::{{icon|wikisource}} See also the transcription [[s:Space and Time|Space and Time]] on English Wikisource&lt;section end=mink08 /&gt;
&lt;section begin=mueller11 /&gt;
*{{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=Das Zeitproblem|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&amp;issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&amp;issuepage=11 Part 2 published 23 October 1911]}}; An alternative, partial English translation of some sections from Müller's paper can be found on pp. 300ff. in ''Gilbert, L. (1914). [https://www.jstor.org/stable/27900487 A satire on the principle of relativity]. The Monist, 288-309.'' 
::{{icon|wikisource}} See also the transcription [[s:de:Das Zeitproblem (1911)|Das Zeitproblem]] on German Wikisource&lt;section end=mueller11 /&gt;
&lt;section begin=nord10mech /&gt;
*{{Citation|author=Nordström, G. |year=1910 |title=Zur elektromagnetischen Mechanik |journal=Physikalische Zeitschrift |volume=11 |pages=440-445|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0011/page/490}}&lt;section end=nord10mech /&gt;
&lt;section begin=nord11mech /&gt;
*{{Citation|author=Nordström, G. |year=1911 |title=Zur Relativitätsmechanik deformierbarer Körper |journal=Physikalische Zeitschrift |volume=12 |pages=854-857|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/906}}&lt;section end=nord11mech /&gt;
&lt;section begin=nord13mass /&gt;
*{{Citation|author=Nordström, G. |year=1913 |title=Träge und schwere Masse in der Relativitätsmechanik |journal=Annalen der Physik |volume=345 (5) |pages=856-878|url=http://gallica.bnf.fr/ark:/12148/bpt6k15343g.image.f863}}&lt;section end=nord13mass /&gt;
&lt;section begin=Penrose /&gt;
*{{citation |author=Penrose, R. |date=January 1959 |orig-date=July 1958|title=The Apparent Shape of a Relativistically Moving Sphere |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=55 |issue=1 |pages=137–139 |doi=10.1017/S0305004100033776|bibcode = 1959PCPS...55..137P |s2cid=123023118 }}&lt;section end=Penrose /&gt;
&lt;section begin=petz /&gt;
*{{citation |author=Petzoldt, J.|title=Die Relativitätstheorie der Physik|date=1914 |journal=Zeitschrift für positivistische Philosophie|volume=2|pages=1-56|url=https://hdl.handle.net/2027/uc1.b3111792}};
::{{icon|wikisource}} See also the transcription [[s:de:Die Relativitätstheorie der Physik|Die Relativitätstheorie der Physik]] on German Wikisource&lt;section end=petz /&gt;
&lt;section begin=planck07dyn /&gt;
*{{Citation|author=Planck, M. |year=1907 |title=Zur Dynamik bewegter Systeme|journal=Berliner Sitzungsberichte, Erster Halbband |volume=29 |pages=542-570}}&lt;br&gt;Reprinted in: {{Citation|author=Planck, M. |origyear=1907|year=1908 |title=Zur Dynamik bewegter Systeme|journal=Annalen der Physik |volume=331 (6) |pages=1-34|url=https://gallica.bnf.fr/ark:/12148/bpt6k15332t/f9.image}}
::{{icon|wikisource}} See also the transcription [[s:de:Zur Dynamik bewegter Systeme|Zur Dynamik bewegter Systeme]] on German Wikisource 
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Dynamics of Moving Systems|On the Dynamics of Moving Systems]] on English Wikisource&lt;section end=planck07dyn /&gt;
&lt;section begin=plum10 /&gt;
*{{citation|author=Plummer, H.C.K.|year=1910 |title=On the Theory of Aberration and the Principle of Relativity|journal=Monthly Notices of the Royal Astronomical Society|volume=40 |pages=252–266|bibcode=1910MNRAS..70..252P}}
::{{icon|wikisource}} See also the transcription [[s:On the Theory of Aberration and the Principle of Relativity|On the Theory of Aberration and the Principle of Relativity]] on English Wikisource&lt;section end=plum10 /&gt;
&lt;section begin=poi00 /&gt;
*{{Citation |author=Poincaré, H. |year=1900 |title=La théorie de Lorentz et le principe de réaction |journal=Archives Néerlandaises des Sciences Exactes et Naturelles |volume=5 |pages=252–278 }}. See also the [http://www.physicsinsights.org/poincare-1900.pdf English translation].
::{{icon|wikisource}} See also the transcription [[s:fr:La théorie de Lorentz et le principe de réaction|La théorie de Lorentz et le principe de réaction]] on French Wikisource&lt;section end=poi00 /&gt;
&lt;section begin=poi04 /&gt;
*{{Citation |author=Poincaré, H. |year=1906 |origyear=1904 |chapter=The Principles of Mathematical Physics |title=Congress of arts and science, universal exposition, St. Louis, 1904 |volume=1 |pages=604–622 |publisher=Houghton, Mifflin and Company |location=Boston and New York}}
::{{icon|wikisource}} See also the transcription [[s:The Principles of Mathematical Physics|The Principles of Mathematical Physics]] on English Wikisource&lt;section end=poi04 /&gt;
&lt;section begin=poinc05a /&gt;
*{{Citation |author=Poincaré, H. |year=1905 |title=Sur la dynamique de l'électron |journal=Comptes Rendus |volume=140 |pages=1504–1508}}
::{{icon|wikisource}} See also the transcription [[s:fr:Sur la dynamique de l’électron (juin)|Sur la dynamique de l’électron]] on French Wikisource 
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Dynamics of the Electron (June)|On the Dynamics of the Electron]] on English Wikisource&lt;section end=poinc05a /&gt;
&lt;section begin=poinc05b /&gt;
*{{Citation |author=Poincaré, H. |year=1906 |origyear=1905 |title=Sur la dynamique de l'électron |journal=Rendiconti del Circolo Matematico di Palermo |volume=21 |pages=129–176}}
::{{icon|wikisource}} See also the transcription [[s:fr:Sur la dynamique de l’électron|Sur la dynamique de l’électron]] on French Wikisource 
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Dynamics of the Electron (July)|On the Dynamics of the Electron]] on English Wikisource&lt;section end=poinc05b /&gt;
&lt;section begin=poi21 /&gt;
*{{Citation|author=Poincaré, Henri|year=1921|origyear=1912|title=Rapport sur les travaux de M. Cartan (fait à la Faculté des sciences de l'Université de Paris)|journal=Acta Mathematica|volume=38|issue=1|pages=137–145|doi=10.1007/bf02392064 |url=https://archive.org/stream/actamathematica38upps#page/n153/mode/2up|doi-access=free}} Written by Poincaré in 1912, printed in Acta Mathematica in 1914 though belatedly published in 1921.&lt;section end=poi21 /&gt;
&lt;section begin=riem67pot /&gt;
*{{Citation |author=Riemann, B. |year=1867 |origyear=1858|title=Ein Beitrag zur Elektrodynamik |journal=Annalen der Physik|volume=207|issue=6 |pages=237-243|url=https://books.google.com/books?id=2bxZAAAAcAAJ&amp;pg=PA237}}&lt;section end=riem67pot /&gt;
&lt;section begin=sea97 /&gt;
*{{Citation |author=Searle, G. F. C. |year=1897 |title=On the Steady Motion of an Electrified Ellipsoid |journal=Philosophical Magazine |series=5 |volume=44 |issue=269 |pages=329–341 |doi=10.1080/14786449708621072 }}
::{{icon|wikisource}} See also the transcription [[s:On the Steady Motion of an Electrified Ellipsoid|On the Steady Motion of an Electrified Ellipsoid]] on English Wikisource&lt;section end=sea97 /&gt;
&lt;section begin=silber06vec /&gt;
*{{Citation |last=Silberstein |first=L. |year=1907 |origyear=1906|title=Elektromagnetische Grundgleichungen in bivectorieller Behandlung |url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_em_equations_in_bivector_fomr.pdf |journal=[[Annalen der Physik]] |volume=327 |issue= 3|pages=579–586 |bibcode=1907AnP...327..579S |doi=10.1002/andp.19073270313 }}&lt;section end=silber06vec /&gt;
&lt;section begin=silber07vec /&gt;
*{{Citation |last=Silberstein |first=L. |year=1907 |title=Nachtrag zur Abhandlung über 'Elektromagnetische Grundgleichungen in bivectorieller Behandlung' |url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/silberstein_-_addendum.pdf |journal=[[Annalen der Physik]] |volume=329 |pages=783–784 |bibcode=1907AnP...329..783S |doi=10.1002/andp.19073291409 |issue=14 }}&lt;section end=silber07vec /&gt;
&lt;section begin=silber11quat /&gt;
*{{Citation | author=Silberstein, L. | year=1912 | origyear=1911|title=Quaternionic form of relativity| journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science | volume =23 | issue=137|pages =790–809|url=https://archive.org/details/londonedinburg6231912lond | doi=10.1080/14786440508637276}}&lt;section end=silber11quat /&gt;
&lt;section begin=silber12quat /&gt;
*{{Citation | author=Silberstein, L. | year=1913 | origyear=1912|title=Second memoir on quaternionic relativity| journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science | volume =25 | issue=CXLV|pages =135–144|url=https://archive.org/details/londonedinburg6251913lond}}&lt;section end=silber12quat /&gt;
&lt;section begin=silber14quat /&gt;
*{{Citation|author=Silberstein, L.|year=1914|title=The Theory of Relativity|publisher=Macmillan|location=London|url=https://archive.org/details/theoryofrelativi00silbrich}}&lt;section end=silber14quat /&gt;
&lt;section begin=sitter11grav /&gt;
*{{Citation |author=deSitter, W. |year=1911 |title=On the bearing of the Principle of Relativity on Gravitational Astronomy |journal=Monthly Notices of the Royal Astronomical Society|volume=71|pages=388-415|url=http://hdl.handle.net/2027/mdp.39015019246357}}&lt;section end=sitter11grav /&gt;
&lt;section begin=som09 /&gt;
*{{Citation|author=Sommerfeld, A.|year=1909|title=Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie|journal=Verh. Der DPG|volume=21|pages=577–582}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Composition of Velocities in the Theory of Relativity|On the Composition of Velocities in the Theory of Relativity]] on English Wikisource&lt;section end=som09 /&gt;
&lt;section begin=som10alg /&gt;
*{{citation |author=Sommerfeld, A. |year=1910a |title=Zur Relativitätstheorie I: Vierdimensionale Vektoralgebra |journal=Annalen der Physik |volume=337 |issue=9 |pages=749–776|url=https://zenodo.org/record/1424173}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Theory of Relativity I: Four-dimensional Vector Algebra|On the Theory of Relativity I: Four-dimensional Vector Algebra]] on English Wikisource &lt;section end=som10alg /&gt;
&lt;section begin=som10ana /&gt;
*{{citation |author=Sommerfeld, A. |year=1910b |title=Zur Relativitätstheorie II: Vierdimensionale Vector Analysis|journal=Annalen der Physik |volume=338 |issue=14 |pages=649–689 |url=https://zenodo.org/record/1424179}}
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Theory of Relativity II: Four-dimensional Vector Analysis|On the Theory of Relativity II: Four-dimensional Vector Analysis]] on English Wikisource &lt;section end=som10ana /&gt;
&lt;section begin=tamaki11b /&gt;
*{{Citation|author=Tamaki, K.|title=On a four-dimensional vector treatment of the electromagnetic field in a moving body|journal=Memoirs of the College of Science and Engineering Kyôto Imperial University|volume=3|year=1911|pages=141-153|url=http://hdl.handle.net/2027/uc1.b3091810}}&lt;section end=tamaki11b /&gt;
&lt;section begin=tamaki13b /&gt;
*{{Citation|author=Tamaki, K.|title=On the symmetrical expressions of relations between the physical quantities in a stationary and moving system|journal=Memoirs of the College of Science and Engineering Kyôto Imperial University|volume=5|year=1913|pages=235-252|url=http://hdl.handle.net/2027/uc1.b3091812}}&lt;section end=tamaki13b /&gt;
&lt;section begin=Terrell /&gt;
*{{citation |author=Terrell, J. |date=November 1959 |orig-date=June 1959|title=Invisibility of the Lorentz Contraction |journal=Physical Review |volume=116 |issue=4 |pages=1041–1045 |doi=10.1103/PhysRev.116.1041|bibcode = 1959PhRv..116.1041T }}&lt;section end=Terrell /&gt;
&lt;section begin=thom89 /&gt;
*{{Citation |author=Thomson, J. J. |year=1889 |title=On the Magnetic Effects produced by Motion in the Electric Field |journal=Philosophical Magazine |volume=28 |series=5 |issue=170 |pages=1–14 |doi=10.1080/14786448908619821}}
::{{icon|wikisource}} See also the transcription [[s:On the Magnetic Effects produced by Motion in the Electric Field|On the Magnetic Effects produced by Motion in the Electric Field]] on English Wikisource&lt;section end=thom89 /&gt;
&lt;section begin=var10 /&gt;
*{{Citation | author=Varićak, V. | year=1910 | title=Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie | journal = Physikalische Zeitschrift| volume =11| pages =93–6|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0011/page/123}}
::{{icon|wikisource}} See also the transcription [[s:de:Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie|Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:Application of Lobachevskian Geometry in the Theory of Relativity|Application of Lobachevskian Geometry in the Theory of Relativity]] on English Wikisource&lt;section end=var10 /&gt;
&lt;section begin=var12 /&gt;
*{{Citation | author=Varičak, V. | year=1912 | title=Über die nichteuklidische Interpretation der Relativtheorie | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume =21 | pages =103–127 }}
::{{icon|wikisource}} See also the transcription [[s:de:Über die nichteuklidische Interpretation der Relativtheorie|Über die nichteuklidische Interpretation der Relativtheorie]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Non-Euclidean Interpretation of the Theory of Relativity|On the Non-Euclidean Interpretation of the Theory of Relativity]] on English Wikisource&lt;section end=var12 /&gt;
&lt;section begin=voi87 /&gt;
*{{Citation |author=Voigt, W. |year=1887 |title=Ueber das Doppler'sche Princip |journal=Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen |issue=2 |pages=41–51 }}
::{{icon|wikisource}} See also the transcription [[s:de:Ueber das Doppler'sche Princip|Ueber das Doppler'sche Princip]] on German Wikisource
::{{icon|wikisource}} See also the transcription [[s:Translation:On the Principle of Doppler|On the Principle of Doppler]] on English Wikisource&lt;section end=voi87 /&gt;
&lt;section begin=weber00rie /&gt;
*{{Citation |author=Weber, H. M.|year=1900|title=Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen (4. edition, volume I)|location=Braunschweig|publisher=Vieweg|url=https://archive.org/details/diepartiellendi03webegoog}}&lt;section end=weber00rie /&gt;
&lt;section begin=weber01rie /&gt;
*{{Citation |author=Weber, H. M.|year=1901|title=Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen (4. edition, volume II)|location=Braunschweig|publisher=Vieweg|url=https://archive.org/details/diepartiellendi01riemgoog}}&lt;section end=weber01rie /&gt;
&lt;section begin=weyl18raum1 /&gt;
*{{Citation |author=Weyl, H. |date=March 1918 |title=Raum-Zeit-Relativität|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}&lt;section end=weyl18raum1 /&gt;
&lt;section begin=weyl19erw /&gt;
*{{Citation |author=Weyl, H. |year=1919 |title=Eine neue Erweiterung der Relativitätstheorie |journal=Annalen der Physik |volume=364 |issue=10 |pages=101-133|url=http://gallica.bnf.fr/ark:/12148/bpt6k15361d.image.f109}}&lt;section end=weyl19erw /&gt;
&lt;section begin=weyl19raum3 /&gt;
*{{Citation |author=Weyl, H. |year=1919 |title=Raum-Zeit-Relativität (3. Auflage)|publisher=Berlin: Springer|url=https://archive.org/details/raumzeitmateriev00weyl}}&lt;section end=weyl19raum3 /&gt;
&lt;section begin=wien04 /&gt;
*{{Citation |last1=Wien |first1=W. |year=1904 |title=Zur Elektronentheorie |journal=Physikalische Zeitschrift |volume=5 |issue=14 |pages=393–395 |bibcode= |doi= }}
::{{icon|wikisource}} See also the transcription [[s:de:Zur Elektronentheorie|Zur Elektronentheorie]] on German Wikisource&lt;section end=wien04 /&gt;
&lt;section begin=wein60 /&gt;
*{{Citation |author=Weinstein, R. |date=October 1960 |orig-date=December 1959|title=Observation of length by a single observer |journal=American Journal of Physics |volume=28 |issue=7 |pages=607-610 |doi=10.1119/1.1935916}}&lt;section end=wein60 /&gt;&lt;/onlyinclude&gt;</text>
      <sha1>lq7bkn7z1m8fomggpbq73svmq1ste4v</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2818346</id>
      <parentid>2818309</parentid>
      <timestamp>2026-07-15T13:56:27Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2818346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="rflhp43ku4o94b1wagavo3rts4be2gi" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260715.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>rflhp43ku4o94b1wagavo3rts4be2gi</sha1>
    </revision>
  </page>
  <page>
    <title>Module:Params</title>
    <ns>828</ns>
    <id>308015</id>
    <revision>
      <id>2818395</id>
      <parentid>2818013</parentid>
      <timestamp>2026-07-16T06:50:18Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>Update from [[d:Special:GoToLinkedPage/mediawikiwiki/Q122696746|master]] using [[mw:Synchronizer| #Synchronizer]]</comment>
      <origin>2818395</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="88000" sha1="jm8x5ug81bre69dpt2csm0n3owsmu2i" xml:space="preserve">require[[strict]]


	---                                        ---
	---     PRIVATE ENVIRONMENT                ---
	---    ________________________________    ---
	---                                        ---


	--[[ Abstract utilities ]]--
	----------------------------


-- Helper function for `string.gsub()` (for managing zero-padded numbers)
local function zero_padded (str)
	return ('%03d%s'):format(#str, str)
end


-- Helper function for `table.sort()` (for natural sorting)
local function natural_sort (var1, var2)
	return var1:gsub('%d+', zero_padded) &lt; var2:gsub('%d+', zero_padded)
end


-- Return a copy or a reference to a table
local function copy_or_ref_table (src, refonly)
	if refonly then return src end
	local newtab = {}
	for key, val in pairs(src) do newtab[key] = val end
	return newtab
end


-- Copy at most N items (of all kinds) from `src` to `dest` and return `dest`
local function copy_table_maxn (dest, src, len)
	local idx = 1
	for key, val in pairs(src) do
		dest[key], idx = val, idx + 1
		if idx &gt; len then break end
	end
	return dest
end


-- Remove some numeric elements from a table, shifting everything to the left
local function remove_numeric_keys (tbl, idx, len)
	local cache, tmp = {}, idx + len - 1
	for key, val in pairs(tbl) do
		if type(key) == 'number' and key &gt;= idx then
			if key &gt; tmp then cache[key - len] = val end
			tbl[key] = nil
		end
	end
	for key, val in pairs(cache) do tbl[key] = val end
end


-- Make a reduced copy of a table (shifting in both directions if necessary)
local function copy_table_reduced (tbl, idx, len)
	local ret, tmp = {}, idx + len - 1
	if idx &gt; 0 then
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &lt; idx then
				ret[key] = val
			elseif key &gt; tmp then ret[key - len] = val end
		end
	elseif tmp &gt; 0 then
		local nshift = 1 - idx
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' then ret[key] = val
			elseif key &gt; tmp then ret[key - tmp] = val
			elseif key &lt; idx then ret[key + nshift] = val end
		end
	else
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &gt; tmp then
				ret[key] = val
			elseif key &lt; idx then ret[key + len] = val end
		end
	end
	return ret
end


-- Make an expanded copy of a table (shifting in both directions if necessary)
local function copy_table_expanded (tbl, idx, len)
	local ret, tmp = {}, idx + len - 1
	if idx &gt; 0 then
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &lt; idx then
				ret[key] = val
			else ret[key + len] = val end
		end
	elseif tmp &gt; 0 then
		local nshift = idx - 1
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' then ret[key] = val
			elseif key &gt; 0 then ret[key + tmp] = val
			elseif key &lt; 1 then ret[key + nshift] = val end
		end
	else
		for key, val in pairs(tbl) do
			if type(key) ~= 'number' or key &gt; tmp then
				ret[key] = val
			else ret[key - len] = val end
		end
	end
	return ret
end


-- Given a table, create two new tables containing the sorted list of keys
local function get_key_list_sorted (tbl, sort_fn)
	local nums, words, nn, nw = {}, {}, 0, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			nn = nn + 1
			nums[nn] = key
		else
			nw = nw + 1
			words[nw] = key
		end
	end
	table.sort(nums)
	table.sort(words, sort_fn)
	return nums, words, nn, nw
end


-- Parse a parameter name string and return it as a string or a number
local function get_parameter_name (par_str)
	local ret = par_str:match'^%s*(.-)%s*$'
	if ret ~= '0' and ret:find'^%-?[1-9]%d*$' == nil then return ret end
	return tonumber(ret)
end


-- Move a key from a table to another, but only if under a different name and
-- always parsing numeric strings as numbers
local function steal_if_renamed (val, src, skey, dest, dkey)
	local realkey = get_parameter_name(dkey)
	if skey ~= realkey then dest[realkey], src[skey] = val, nil end
end


	--[[ Public strings ]]--
	------------------------


-- Special match keywords (functions and modifiers MUST avoid these names)
local mkeywords = {
	['or'] = 0,
	pattern = 1,
	plain = 2,
	strict = 3
}


-- Sort functions (functions and modifiers MUST avoid these names)
local sortfunctions = {
	alphabetically = false,
	naturally = natural_sort
}


-- Callback styles for the `mapping_*` and `renaming_*` class of modifiers
-- (functions and modifiers MUST avoid these names)
--[[

Meanings of the columns:

  col[1] = Loop type (0-3)
  col[2] = Number of module arguments that the style requires (1-3)
  col[3] = Minimum number of sequential parameters passed to the callback
  col[4] = Name of the callback parameter where to place each parameter name
  col[5] = Name of the callback parameter where to place each parameter value
  col[6] = Argument in the modifier's invocation that will override `col[4]`
  col[7] = Argument in the modifier's invocation that will override `col[5]`

A value of `-1` indicates that no meaningful value is stored (i.e. `nil`)

]]--
local mapping_styles = {
	names_and_values = { 3, 2, 2, 1, 2, -1, -1 },
	values_and_names = { 3, 2, 2, 2, 1, -1, -1 },
	values_only = { 1, 2, 1, -1, 1, -1, -1 },
	names_only = { 2, 2, 1, 1, -1, -1, -1 },
	names_and_values_as = { 3, 4, 0, -1, -1, 2, 3 },
	names_only_as = { 2, 3, 0, -1, -1, 2, -1 },
	values_only_as = { 1, 3, 0, -1, -1, -1, 2 },
	blindly = { 0, 2, 0, -1, -1, -1, -1 }
}


-- Memory slots (functions and modifiers MUST avoid these names)
local memoryslots = {
	h = 'header',
	f = 'footer',
	i = 'itersep',
	l = 'lastsep',
	n = 'ifngiven',
	p = 'pairsep',
	s = 'oxfordsep'
}


-- Possible trimming modes for the `parsing` modifier
local trim_parse_opts = {
	trim_none = { false, false },
	trim_positional = { false, true },
	trim_named = { true, false },
	trim_all = { true, true }
}


-- Possible string modes for the iteration separator in the `parsing` and
-- `reinterpreting` modifiers
local isep_parse_opts = {
	splitter_pattern = false,
	splitter_string = true
}


-- Possible string modes for the key-value separator in the `parsing` and
-- `reinterpreting` modifiers
local psep_parse_opts = {
	setter_pattern = false,
	setter_string = true
}


-- Possible position references for the `splicing` modifier
local position_references = {
	add_nothing = 0,
	add_smallest_number = 1,
	add_last_of_sequence = 2,
	add_largest_number = 3
}


-- Possible modes for the `reassigning` modifier
local a_modes = {
	duplicate = 0,
	transfer = 1,
	provide = 2,
	spare = 3,
	sacrifice = 4
}


-- Functions and modifiers MUST avoid these names too: `here`, `in_substack`,
-- `let`, `expose`, `use`, `with_flushed_glue`, `without_flushed_glue`
-- `without_sorting`


	--[[ Private constants ]]--
	---------------------------


-- Hard-coded name of the module (to avoid going through `frame:getTitle()`)
local modulename = 'Module:Params'


-- The functions listed here declare that they don't need the `frame.args`
-- metatable to be copied into a regular table; if they are modifiers they also
-- guarantee that they will make their own (modified) copy available
local refpipe = {
	call_for_each_group = true,
	--coins = true,
	count = true,
	evaluating = true,
	for_each = true,
	list = true,
	list_values = true,
	list_maybe_with_names = true,
	value_of = true
}


-- The functions listed here declare that they don't need the
-- `frame:getParent().args` metatable to be copied into a regular table; if 
-- they are modifiers they also guarantee that they will make their own
-- (modified) copy available
local refparams = {
	call_for_each_group = true,
	combining = true,
	combining_by_calling = true,
	combining_values = true,
	concat_and_call = true,
	concat_and_invoke = true,
	concat_and_magic = true,
	count = true,
	grouping_by_calling = true,
	mixing_names_and_values = true,
	keeping_at_most = true,
	renaming_by_mixing = true,
	renaming_to_sequence = true,
	renaming_to_uppercase = true,
	renaming_to_lowercase = true,
	--renaming_to_values = true,
	shifting = true,
	splicing = true,
	--swapping_names_and_values = true,
	value_of = true,
	with_name_matching = true
}


-- Maximum number of numeric parameters that can be filled, if missing (we
-- chose an arbitrary number for this constant; you can discuss about its
-- optimal value at Module talk:Params)
local maxfill = 1024


-- The private table of functions
local library = {}


-- Functions and modifiers that can only be invoked in first position
local static_iface = {}


	--[[ Private functions ]]--
	---------------------------


-- Create a new context
local function context_new (child_frame)
	local main_frame = child_frame:getParent()
	return {
		frame = main_frame,
		opipe = child_frame.args,
		oparams = main_frame.args,
		firstposonly = static_iface,
		iterfunc = pairs,
		sorttype = 0,
		n_parents = 0,
		n_children = 0,
		n_available = maxfill
	}
end


-- Move to the next action within the user-given list
local function context_iterate (ctx, n_forward)
	local nextfn
	if ctx.pipe[n_forward] ~= nil then
		nextfn = ctx.pipe[n_forward]:match'^%s*(.*%S)'
	end
	if nextfn == nil then error(modulename ..
		': You must specify a function to call', 0) end
	if library[nextfn] == nil then
		if ctx.firstposonly[nextfn] == nil then error(modulename ..
			': The function ‘' .. nextfn .. '’ does not exist', 0)
		else error(modulename .. ': The ‘' .. nextfn ..
			'’ directive can only appear in first position', 0)
		end
	end
	remove_numeric_keys(ctx.pipe, 1, n_forward)
	return library[nextfn]
end


-- Main loop
local function main_loop (ctx, start_with)
	local fn = start_with
	repeat fn = fn(ctx) until not fn
	if ctx.n_parents &gt; 0 then error(modulename ..
		': One or more ‘merging_substack’ directives are missing', 0) end
	if ctx.n_children &gt; 0 then error(modulename ..
		', For some of the snapshots either the ‘flushing’ directive is missing or a group has not been properly closed with ‘merging_substack’', 0) end
end


-- Load a `setting`-like directive string into the `dest` table
local function set_strings_from_opts (dest, opts, start_from)
	local cmd
	if opts[start_from] == nil then return start_from - 1 end
	cmd = opts[start_from]:gsub('%s+', ''):gsub('/+', '/')
		:match'^/*(.*[^/])'
	if cmd == nil then return start_from end
	local vname, chr
	local amap, sep, argc = {}, string.byte('/'), start_from + 1
	for idx = 1, #cmd do
		chr = cmd:byte(idx)
		if chr == sep then
			for key, val in ipairs(amap) do
				dest[val], amap[key] = opts[argc], nil
			end
			argc = argc + 1
		else
			vname = memoryslots[string.char(chr)]
			if vname == nil then error(modulename ..
				', ‘setting’: Unknown slot ‘' ..
				string.char(chr) .. '’', 0) end
			table.insert(amap, vname)
		end
	end
	for key, val in ipairs(amap) do dest[val] = opts[argc] end
	return argc
end


-- Add a new stack of parameters to `ctx.children`
local function new_substack (ctx)
	local currsnap, newparams = ctx.n_children + 1, {}
	if ctx.children == nil then ctx.children = { newparams }
	else ctx.children[currsnap] = newparams end
	ctx.n_children = currsnap
	return newparams
end


-- Parse a raw argument containing a `sortfunctions` directive, or
-- `'without_sorting'`, or `nil`
local function load_sort_opt (raw_arg)
	if raw_arg == nil then return nil, 1, false end
	local trarg = raw_arg:match'^%s*(.-)%s*$'
	if trarg == 'without_sorting' then return nil, 2, false, trarg end
	local tmp = sortfunctions[trarg]
	if tmp == nil then return nil, 1, false, trarg end
	return tmp or nil, 2, true, trarg
end


-- Parse optional user arguments of type `...|[let/use]|[...]|[let/use]|[...]|
-- [number of additional parameters]|[parameter 1]|[parameter 2]|[...]`
local function load_child_opts (src, start_from, append_after, params)
	local tnamed, tmp1, tmp2
	local pin, tbl, mem = start_from, {}, {}
	while src[pin] ~= nil and src[pin + 1] ~= nil do
		tmp1 = src[pin]:match'^%s*(.*%S)'
		if tmp1 == 'let' and src[pin + 2] ~= nil then
			tmp1 = get_parameter_name(src[pin + 1])
			mem[tmp1], tbl[tmp1], pin = nil, src[pin + 2], pin + 3
--[[
		elseif tmp1 == 'expose' then
			tmp1 = get_parameter_name(src[pin + 1])
			tmp2 = params[tmp1]
			mem[tmp1], tbl[tmp1], pin = tmp2, tmp2, pin + 2
]]--
		elseif tmp1 == 'use' and src[pin + 2] ~= nil then
			tmp1 = get_parameter_name(src[pin + 2])
			tmp2 = params[tmp1]
			mem[tmp1], tbl[get_parameter_name(src[pin + 1])], pin =
				tmp2, tmp2, pin + 3
		else break end
	end
	local tnew = copy_or_ref_table(params, next(mem) == nil)
	for key in pairs(mem) do tnew[key] = nil end
	if pin ~= start_from then tnamed, tbl = tbl, {} end
	tmp1 = tonumber(src[pin])
	if tmp1 ~= nil and math.floor(tmp1) == tmp1 then
		if tmp1 &lt; 0 then tmp1 = -1 end
		tmp2 = append_after - pin
		for idx = pin + 1, pin + tmp1 do tbl[idx + tmp2] = src[idx] end
		pin = pin + tmp1 + 1
	end
	if tnamed ~= nil then
		for key, val in pairs(tnamed) do tbl[key] = val end
	end
	return tbl, pin, tnew, mem
end


-- Load the optional arguments of some of the `mapping_*` and `renaming_*`
-- class of modifiers
local function load_callback_opts (src, n_skip, default_style, params)
	local style, shf
	local tmp = src[n_skip + 1]
	if tmp ~= nil then style = mapping_styles[tmp:match'^%s*(.-)%s*$'] end
	if style == nil then style, shf = default_style, n_skip - 1
	else shf = n_skip end
	local n_exist, karg, varg = style[3], style[4], style[5]
	tmp = style[6]
	if tmp &gt; -1 then
		karg = src[tmp + shf]:match'^%s*(.-)%s*$'
		if karg == '0' or karg:find'^%-?[1-9]%d*$' ~= nil then
			karg = tonumber(karg)
			n_exist = math.max(n_exist, karg)
		end
	end
	tmp = style[7]
	if tmp &gt; -1 then
		varg = src[tmp + shf]:match'^%s*(.-)%s*$'
		if varg == '0' or varg:find'^%-?[1-9]%d*$' ~= nil then
			varg = tonumber(varg)
			n_exist = math.max(n_exist, varg)
		end
	end
	local dest, argc, tnew, mem = load_child_opts(src, style[2] + shf,
		n_exist, params)
	tmp = style[1]
	if (tmp == 3 or tmp == 2) and dest[karg] ~= nil then
		tmp = tmp - 2 end
	if (tmp == 3 or tmp == 1) and dest[varg] ~= nil then
		tmp = tmp - 1 end
	return dest, argc, tmp, karg, varg, tnew, mem
end


-- Parse the arguments of some of the `mapping_*` and `renaming_*` class of
-- modifiers
local function load_replace_args (opts, whoami)
	if opts[1] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No pattern string was given', 0) end
	if opts[2] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No replacement string was given', 0) end
	local ptn, repl, nmax, argc = opts[1], opts[2], tonumber(opts[3]), 3
	if nmax ~= nil or (opts[3] or ''):match'^%s*$' ~= nil then argc = 4 end
	local flg = opts[argc]
	if flg ~= nil then flg = mkeywords[flg:match'^%s*(.-)%s*$'] end
	if flg == 0 then flg = nil elseif flg ~= nil then argc = argc + 1 end
	return ptn, repl, nmax, flg, argc, (nmax ~= nil and nmax &lt; 1) or
		(flg == 3 and ptn == repl)
end


-- Parse the arguments of the `with_*_matching` class of modifiers
local function load_pattern_args (opts, whoami)
	local keyw
	local ptns, state, nptns, cnt = {}, 0, 0, 1
	for _, val in ipairs(opts) do
		if state == 0 then
			nptns, state = nptns + 1, -1
			ptns[nptns] = { val, false, false }
		else
			keyw = val:match'^%s*(.*%S)'
			if keyw == nil or mkeywords[keyw] == nil or (
				state &gt; 0 and mkeywords[keyw] &gt; 0
			) then break
			else
				state = mkeywords[keyw]
				if state &gt; 1 then ptns[nptns][2] = true end
				if state == 3 then ptns[nptns][3] = true end
			end
		end
		cnt = cnt + 1
	end
	if state == 0 then error(modulename .. ', ‘' .. whoami ..
		'’: No pattern was given', 0) end
	return ptns, nptns, cnt
end


-- Load the optional arguments of the `parsing`, `reinterpreting` and
-- `evaluating` modifiers
local function load_parse_opts (opts, start_from, isp, psp)
	local tmp
	local optslots, noptslots, argc, trimn, trimu, iplain, pplain =
		{ true, true, true }, 3, start_from, true, false, true, true
	repeat
		noptslots, tmp = noptslots - 1, opts[argc]
		if tmp == nil then break end
		tmp = tmp:match'^%s*(.-)%s*$'
		if optslots[1] ~= nil and trim_parse_opts[tmp] ~= nil then
			tmp = trim_parse_opts[tmp]
			trimn, trimu, optslots[1] = tmp[1], tmp[2], nil
		elseif optslots[2] ~= nil and isep_parse_opts[tmp] ~= nil then
			argc = argc + 1
			iplain, isp, optslots[2] = isep_parse_opts[tmp],
				opts[argc], nil
		elseif optslots[3] ~= nil and psep_parse_opts[tmp] ~= nil then
			argc = argc + 1
			pplain, psp, optslots[3] = psep_parse_opts[tmp],
				opts[argc], nil
		else break end
		argc = argc + 1
	until noptslots &lt; 1
	return isp, iplain, psp, pplain, trimn, trimu, argc
end


-- Map parameters' values using a custom callback and a referenced table
local value_maps = {
	[0] = function (tbl, margs, karg, varg, fn)
		for key in pairs(tbl) do tbl[key] = fn() end
	end,
	[1] = function (tbl, margs, karg, varg, fn)
		for key, val in pairs(tbl) do
			margs[varg] = val
			tbl[key] = fn()
		end
	end,
	[2] = function (tbl, margs, karg, varg, fn)
		for key in pairs(tbl) do
			margs[karg] = key
			tbl[key] = fn()
		end
	end,
	[3] = function (tbl, margs, karg, varg, fn)
		for key, val in pairs(tbl) do
			margs[karg], margs[varg] = key, val
			tbl[key] = fn()
		end
	end
}


-- Private table for `map_names()`
local name_thieves = {
	[0] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[1] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[varg] = val
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[2] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[karg] = key
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end,
	[3] = function (cache, tbl, rargs, karg, varg, fn)
		for key, val in pairs(tbl) do
			rargs[karg], rargs[varg] = key, val
			steal_if_renamed(val, tbl, key, cache, fn())
		end
	end
}


-- Map parameters' names using a custom callback and a referenced table
local function map_names (tbl, rargs, karg, varg, looptype, fn)
	local cache = {}
	name_thieves[looptype](cache, tbl, rargs, karg, varg, fn)
	for key, val in pairs(cache) do tbl[key] = val end
end


-- Return a new table that contains `src` regrouped according to the numeric
-- suffixes in its keys
local function make_groups (src)
	-- NOTE: `src` might be the original metatable!
	local prefix, gid
	local groups = {}
	for key, val in pairs(src) do
		-- `key` must only be a string or a number...
		if type(key) == 'string' then
			prefix, gid = key:match'^%s*(.-)%s*(%-?%d*)%s*$'
			gid = tonumber(gid) or ''
		else
			prefix, gid = '', key
		end
		if groups[gid] == nil then groups[gid] = {} end
		if prefix == '0' or prefix:find'^%-?[1-9]%d*$' ~= nil then
			prefix = tonumber(prefix)
			if prefix &lt; 1 then prefix = prefix - 1 end
		end
		groups[gid][prefix] = val
	end
	return groups
end


-- Split into parts a string containing the `$#` and `$@` placeholders and
-- return the information as a skeleton table, a canvas table and a length
local function parse_placeholder_string (target)
	local idx, s_pos, skel, canvas = 1, 1, {}, {}
	local e_pos = string.find(target, '%$[@#]', 1, false)
	while e_pos ~= nil do
		canvas[idx] = target:sub(s_pos, e_pos - 1)
		skel[idx + 1] = target:sub(e_pos, e_pos + 1) == '$@'
		idx = idx + 2
		s_pos = e_pos + 2
		e_pos = string.find(target, '%$[@#]', s_pos, false)
	end
	if (s_pos &gt; target:len()) then idx = idx - 1
	else canvas[idx] = target:sub(s_pos) end
	return skel, canvas, idx
end


-- Populate a table by parsing a parameter string (heavy lifting for `parsing`,
-- `reinterpreting` and `evaluating`)
local function parse_parameter_string (tbl, str, isp, ipl, psp, ppl, trn, tru)
	local key, val, spos1, spos2, pos1, pos2
	local pos3, idx, lenplone = 0, 1, #str + 1
	if isp == nil or isp == '' then
		if psp == nil or psp == '' then
			if tru then tbl[idx] = str:match'^%s*(.-)%s*$'
			else tbl[idx] = str end
			return idx
		end
		spos1, spos2 = str:find(psp, 1, ppl)
		if spos1 == nil then
			key = idx
			if tru then val = str:match'^%s*(.-)%s*$'
			else val = str end
			idx = idx + 1
		else
			key = get_parameter_name(str:sub(1, spos1 - 1))
			val = str:sub(spos2 + 1)
			if trn then val = val:match'^%s*(.-)%s*$' end
		end
		tbl[key] = val
		return idx
	end
	if psp == nil or psp == '' then
		repeat
			pos1 = pos3 + 1
			pos2, pos3 = str:find(isp, pos1, ipl)
			val = str:sub(pos1, (pos2 or lenplone) - 1)
			if tru then val = val:match'^%s*(.-)%s*$' end
			tbl[idx], idx = val, idx + 1
		until pos2 == nil
		return idx
	end
	repeat
		pos1 = pos3 + 1
		pos2, pos3 = str:find(isp, pos1, ipl)
		val = str:sub(pos1, (pos2 or lenplone) - 1)
		spos1, spos2 = val:find(psp, 1, ppl)
		if spos1 == nil then
			key = idx
			if tru then val = val:match'^%s*(.-)%s*$' end
			idx = idx + 1
		else
			key = get_parameter_name(val:sub(1, spos1 - 1))
			val = val:sub(spos2 + 1)
			if trn then val = val:match'^%s*(.-)%s*$' end
		end
		tbl[key] = val
	until pos2 == nil
	return idx
end


-- Heavy lifting for `snapshotting` and `remembering`
local function make_child (ctx, src, whoami)
	local len = tonumber(ctx.pipe[1])
	if len == nil or len == 0 then
		local stack = new_substack(ctx)
		for key, val in pairs(src) do stack[key] = val end
		return len == nil and 1 or 2
	end
	if len &lt; 0 or math.floor(len) ~= len then error(modulename ..
		', ‘' .. whoami .. '’: The number of parameters to copy must be an integer greater than zero', 0) end
	copy_table_maxn(new_substack(ctx), src, len)
	return 2
end


-- Heavy lifting for `setting_by_flushing`, `combining` and `combining_values`
local function set_strings_from_substack (ctx, dest, whoami)
	if ctx.n_children &lt; 1 then error(modulename ..
		', ‘' .. whoami .. '’: There are no substacks to flush', 0) end
	local currsnap = ctx.n_children
	local stack = ctx.children[currsnap]
	for key, val in pairs(memoryslots) do
		if stack[key] ~= nil then dest[val] = stack[key] end
	end
	ctx.children[currsnap], ctx.n_children = nil, currsnap - 1
end


-- Heavy lifting for `combining` and `combining_values`
local function combine_parameters (ctx, keyval_fn, whoami)
	-- NOTE: `ctx.params` might be the original metatable! This function
	-- MUST create a copy of it before returning
	local opts = ctx.pipe
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘' .. whoami .. '’: No parameter name was provided', 0) end
	local argc
	local tbl, vars = ctx.params, {}
	local sortfn, varsarg0, do_sort, tmp = load_sort_opt(opts[2])
	if varsarg0 == 2 then tmp = opts[3] and opts[3]:match'^%s*(.-)%s*$' end
	if tmp == 'with_flushed_glue' then
		varsarg0 = varsarg0 + 1
		argc = set_strings_from_opts(vars, opts, varsarg0 + 1)
		set_strings_from_substack(ctx, vars, whoami)
	else
		if tmp == 'without_flushed_glue' then varsarg0 = varsarg0 + 1 end
		argc = set_strings_from_opts(vars, opts, varsarg0 + 1)
	end
	if argc &lt; varsarg0 then error(modulename ..
		', ‘' .. whoami .. '’: No setting directive was given', 0) end
	if next(tbl) == nil then
		if vars.ifngiven ~= nil then ctx.params =
			{ [get_parameter_name(ctx.pipe[1])] = vars.ifngiven }
		elseif tbl == ctx.oparams then ctx.params = {} end
		return argc
	end
	local cache, len
	if do_sort then
		local words
		cache, words, len, tmp = get_key_list_sorted(tbl, sortfn)
		for idx = 1, tmp do cache[len + idx] = words[idx] end
		len = len + tmp
	else
		len, cache = 0, {}
		for key in pairs(tbl) do
			len = len + 1
			cache[len] = key
		end
	end
	local pmap, nss, kvs, pps = {}, 0, vars.pairsep or '', vars.itersep or ''
	for idx = 1, len do
		tmp, pmap[nss + 1] = cache[idx], pps
		pmap[nss + 2] = keyval_fn(tmp, tbl[tmp], kvs)
		nss = nss + 2
	end
	tmp = vars.oxfordsep or vars.lastsep
	if tmp ~= nil and nss &gt; 4 then pmap[nss - 1] = tmp
	elseif nss &gt; 2 and vars.lastsep ~= nil then
		pmap[nss - 1] = vars.lastsep
	end
	pmap[1] = vars.header or ''
	if vars.footer ~= nil then pmap[nss + 1] = vars.footer end
	ctx.params = { [get_parameter_name(ctx.pipe[1])] = table.concat(pmap) }
	return argc
end


-- Concatenate the numeric keys from the table of parameters to the numeric
-- keys from the table of options; non-numeric keys from the table of options
-- will prevail over colliding non-numeric keys from the table of parameters
local function concat_params (ctx)
	local retval, tbl, nmax = {}, ctx.params, table.maxn(ctx.pipe)
	if ctx.subset == 1 then
		-- We need only the sequence
		for key, val in ipairs(tbl) do retval[key + nmax] = val end
	else
		if ctx.subset == -1 then
			for key in ipairs(tbl) do tbl[key] = nil end
		end
		for key, val in pairs(tbl) do
			if type(key) == 'number' and key &gt; 0 then
				retval[key + nmax] = val
			else retval[key] = val end
		end
	end
	for key, val in pairs(ctx.pipe) do retval[key] = val end
	return retval
end


-- Flush the parameters by calling a custom function for each value (after this
-- function has been invoked `ctx.params` will be no longer usable)
local function flush_params (ctx, fn)
	local tbl = ctx.params
	if ctx.subset == 1 then
		for key, val in ipairs(tbl) do fn(key, val) end
		return
	end
	if ctx.subset == -1 then
		for key, val in ipairs(tbl) do tbl[key] = nil end
	end
	if ctx.sorttype &gt; 0 then
		local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort)
		if ctx.sorttype == 2 then
			for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end
			for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end
			return
		end
		for idx = 1, nn do fn(nums[idx], tbl[nums[idx]]) end
		for idx = 1, nw do fn(words[idx], tbl[words[idx]]) end
		return
	end
	if ctx.subset ~= -1 then
		for key, val in ipairs(tbl) do
			fn(key, val)
			tbl[key] = nil
		end
	end
	for key, val in pairs(tbl) do fn(key, val) end
end


-- Flush the parameters by calling one of two custom functions for each value
-- (after this function has been invoked `ctx.params` will be no longer usable)
local function mixed_flush_params (ctx, fn_seq, fn_oth)
	if ctx.subset == 1 then
		for key, val in ipairs(ctx.params) do fn_seq(key, val) end
		return
	end
	if ctx.subset == -1 then
		flush_params(ctx, fn_oth)
		return
	end
	local tbl = ctx.params
	if ctx.sorttype &gt; 0 then
		local nums, words, nn, nw = get_key_list_sorted(tbl, natural_sort)
		local sequence = {}
		for key, val in ipairs(tbl) do sequence[key] = val end
		if ctx.sorttype == 2 then
			for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end
		end
		for idx = 1, nn do
			if sequence[nums[idx]] then fn_seq(nums[idx], sequence[nums[idx]])
			else fn_oth(nums[idx], tbl[nums[idx]]) end
		end
		if ctx.sorttype ~= 2 then
			for idx = 1, nw do fn_oth(words[idx], tbl[words[idx]]) end
		end
		return
	end
	for key, val in ipairs(tbl) do
		fn_seq(key, val)
		tbl[key] = nil
	end
	for key, val in pairs(tbl) do fn_oth(key, val) end
end


-- Finalize and return a concatenated list
local function finalize_and_return_concatenated_list (ctx, lst, len, modsize)
	if len &gt; 0 then
		local tmp = ctx.oxfordsep or ctx.lastsep
		if tmp ~= nil and len &gt; modsize * 2 then
			lst[len - modsize + 1] = tmp
		elseif len &gt; modsize and ctx.lastsep ~= nil then
			lst[len - modsize + 1] = ctx.lastsep
		end
		lst[1] = ctx.header or ''
		if ctx.footer ~= nil then lst[len + 1] = ctx.footer end
		ctx.text = table.concat(lst)
	else ctx.text = ctx.ifngiven or '' end
end


	---                                        ---
	---     PUBLIC ENVIRONMENT                 ---
	---    ________________________________    ---
	---                                        ---


	--[[ Modifiers ]]--
	-------------------


-- Syntax:  #invoke:params|sequential|pipe to
library.sequential = function (ctx)
	if ctx.subset == 1 then error(modulename ..
		': The ‘sequential’ directive has been provided more than once', 0) end
	if ctx.subset == -1 then error(modulename ..
		': The two directives ‘non-sequential’ and ‘sequential’ are in contradiction with each other', 0) end
	if ctx.sorttype &gt; 0 then error(modulename ..
		': The ‘all_sorted’ and ‘reassorted’ directives are redundant when followed by ‘sequential’', 0) end
	ctx.iterfunc, ctx.subset = ipairs, 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|non-sequential|pipe to
library['non-sequential'] = function (ctx)
	if ctx.subset == -1 then error(modulename ..
		': The ‘non-sequential’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The two directives ‘sequential’ and ‘non-sequential’ are in contradiction with each other', 0) end
	ctx.iterfunc, ctx.subset = pairs, -1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|all_sorted|pipe to
library.all_sorted = function (ctx)
	if ctx.sorttype == 1 then error(modulename ..
		': The ‘all_sorted’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The ‘all_sorted’ directive is redundant after ‘sequential’', 0) end
	if ctx.sorttype == 2 then error(modulename ..
		': The two directives ‘reassorted’ and ‘sequential’ are in contradiction with each other', 0) end
	ctx.sorttype = 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|reassorted|pipe to
library.reassorted = function (ctx)
	if ctx.sorttype == 2 then error(modulename ..
		': The ‘reassorted’ directive has been provided more than once', 0) end
	if ctx.subset == 1 then error(modulename ..
		': The ‘reassorted’ directive is redundant after ‘sequential’', 0) end
	if ctx.sorttype == 1 then error(modulename ..
		': The two directives ‘sequential’ and ‘reassorted’ are in contradiction with each other', 0) end
	ctx.sorttype = 2
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|setting|directives|...|pipe to
library.setting = function (ctx)
	local argc = set_strings_from_opts(ctx, ctx.pipe, 1)
	if argc &lt; 2 then error(modulename ..
		', ‘setting’: No directive was given', 0) end
	return context_iterate(ctx, argc + 1)
end


-- Syntax:  #invoke:params|scoring|new parameter name|[container]|pipe to
library.scoring = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘scoring’: No parameter name was provided', 0) end
	local tmp
	local retval, opts = 0, ctx.pipe
	for _ in pairs(ctx.params) do retval = retval + 1 end
	if opts[2] ~= nil then tmp = opts[2]:match'^%s*(.*%S)' end
	if tmp == 'in_substack' then
		new_substack(ctx)[get_parameter_name(opts[1])] = tostring(retval)
		return context_iterate(ctx, 3)
	end
	ctx.params[get_parameter_name(opts[1])] = tostring(retval)
	return context_iterate(ctx, tmp == 'here' and 3 or 2)
end


-- Syntax:  #invoke:params|squeezing|pipe to
library.squeezing = function (ctx)
	local store, indices, tbl, newlen = {}, {}, ctx.params, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			newlen = newlen + 1
			indices[newlen], store[key], tbl[key] = key, val, nil
		end
	end
	table.sort(indices)
	for idx = 1, newlen do tbl[idx] = store[indices[idx]] end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|filling_the_gaps|pipe to
library.filling_the_gaps = function (ctx)
	local newval, tbl, tmp, nmin, nmax, nnums =
		ctx.pipe[1], ctx.params, {}, 1, nil, -1
	if newval == nil then error(modulename ..
		', ‘filling_the_gaps’: No value was provided', 0) end
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			if nmax == nil then
				if key &lt; nmin then nmin = key end
				nmax = key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
			tmp[key], nnums = val, nnums + 1
		end
	end
	if nmax ~= nil and nmax - nmin &gt; nnums then
		ctx.n_available = ctx.n_available + nmin + nnums - nmax
		if ctx.n_available &lt; 0 then error(modulename ..
			', ‘filling_the_gaps’: It is possible to fill at most ' ..
			tostring(maxfill) .. ' parameters', 0) end
		for idx = nmin, nmax, 1 do tbl[idx] = newval end
		for key, val in pairs(tmp) do tbl[key] = val end
	end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|clearing|pipe to
library.clearing = function (ctx)
	local tbl, numerics = ctx.params, {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key], tbl[key] = val, nil
		end
	end
	for key, val in ipairs(numerics) do tbl[key] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|cutting|left cut|right cut|pipe to
library.cutting = function (ctx)
	local lcut = tonumber(ctx.pipe[1])
	if lcut == nil or math.floor(lcut) ~= lcut then error(modulename ..
		', ‘cutting’: Left cut must be an integer number', 0) end
	local rcut = tonumber(ctx.pipe[2])
	if rcut == nil or math.floor(rcut) ~= rcut then error(modulename ..
		', ‘cutting’: Right cut must be an integer number', 0) end
	local tbl = ctx.params
	local len = #tbl
	if lcut &lt; 0 then lcut = len + lcut end
	if rcut &lt; 0 then rcut = len + rcut end
	local tot = lcut + rcut
	if tot &gt; 0 then
		local cache = {}
		if tot &gt;= len then
			for key in ipairs(tbl) do tbl[key] = nil end
			tot = len
		else
			for idx = len - rcut + 1, len, 1 do tbl[idx] = nil end
			for idx = 1, lcut, 1 do tbl[idx] = nil end
		end
		for key, val in pairs(tbl) do
			if type(key) == 'number' and key &gt; 0 then
				if key &gt; len then cache[key - tot] = val
				else cache[key - lcut] = val end
				tbl[key] = nil
			end
		end
		for key, val in pairs(cache) do tbl[key] = val end
	end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|cropping|left crop|right crop|pipe to
library.cropping = function (ctx)
	local lcut = tonumber(ctx.pipe[1])
	if lcut == nil or math.floor(lcut) ~= lcut then error(modulename ..
		', ‘cropping’: Left crop must be an integer number', 0) end
	local rcut = tonumber(ctx.pipe[2])
	if rcut == nil or math.floor(rcut) ~= rcut then error(modulename ..
		', ‘cropping’: Right crop must be an integer number', 0) end
	local tbl = ctx.params
	local nmin, nmax
	for key in pairs(tbl) do
		if type(key) == 'number' then
			if nmin == nil then nmin, nmax = key, key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
		end
	end
	if nmin ~= nil then
		local len = nmax - nmin + 1
		if lcut &lt; 0 then lcut = len + lcut end
		if rcut &lt; 0 then rcut = len + rcut end
		if lcut + rcut - len &gt; -1 then
			for key in pairs(tbl) do
				if type(key) == 'number' then tbl[key] = nil end
			end
		elseif lcut + rcut &gt; 0 then
			for idx = nmax - rcut + 1, nmax do tbl[idx] = nil end
			for idx = nmin, nmin + lcut - 1 do tbl[idx] = nil end
			local lshift = nmin + lcut - 1
			if lshift &gt; 0 then
				for idx = lshift + 1, nmax, 1 do
					tbl[idx - lshift], tbl[idx] =
						tbl[idx], nil
				end
			end
		end
	end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|purging|start offset|length|pipe to
library.purging = function (ctx)
	local idx = tonumber(ctx.pipe[1])
	if idx == nil or math.floor(idx) ~= idx then error(modulename ..
		', ‘purging’: Start offset must be an integer number', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘purging’: Length must be an integer number', 0) end
	local tbl = ctx.params
	if len &lt; 1 then
		len = len + table.maxn(tbl)
		if idx &gt; len then return context_iterate(ctx, 3) end
		len = len - idx + 1
	end
	ctx.params = copy_table_reduced(tbl, idx, len)
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|backpurging|start offset|length|pipe to
library.backpurging = function (ctx)
	local last = tonumber(ctx.pipe[1])
	if last == nil or math.floor(last) ~= last then error(modulename ..
		', ‘backpurging’: Start offset must be an integer number', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘backpurging’: Length must be an integer number', 0) end
	local idx
	local tbl = ctx.params
	if len &gt; 0 then
		idx = last - len + 1
	else
		for key in pairs(tbl) do
			if type(key) == 'number' and (idx == nil or
				key &lt; idx) then idx = key end
		end
		if idx == nil then return context_iterate(ctx, 3) end
		idx = idx - len
		if last &lt; idx then return context_iterate(ctx, 3) end
		len = last - idx + 1
	end
	ctx.params = copy_table_reduced(ctx.params, idx, len)
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|shifting|addend|pipe to
library.shifting = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local nshift = tonumber(ctx.pipe[1])
	if nshift == nil or nshift == 0 or math.floor(nshift) ~= nshift then
		error(modulename .. ', ‘shifting’: A non-zero integer number must be provided', 0) end
	local tbl = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'number' then tbl[key + nshift] = val
		else tbl[key] = val end
	end
	ctx.params = tbl
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|reversing_numeric_names|pipe to
library.reversing_numeric_names = function (ctx)
	local tbl, numerics, nmax = ctx.params, {}, 0
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key], tbl[key] = val, nil
			if key &gt; nmax then nmax = key end
		end
	end
	for key, val in pairs(numerics) do tbl[nmax - key + 1] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|pivoting_numeric_names|pipe to
--[[
library.pivoting_numeric_names = function (ctx)
	local tbl = ctx.params
	local shift = #tbl + 1
	if shift &lt; 2 then return library.reversing_numeric_names(ctx) end
	local numerics = {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key] = val
			tbl[key] = nil
		end
	end
	for key, val in pairs(numerics) do tbl[shift - key] = val end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|mirroring_numeric_names|pipe to
--[[
library.mirroring_numeric_names = function (ctx)
	local nmax, nmin
	local tbl, numerics = ctx.params, {}
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			numerics[key] = val
			tbl[key] = nil
			if nmax == nil then nmin, nmax = key, key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
		end
	end
	for key, val in pairs(numerics) do tbl[nmax + nmin - key] = val end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|swapping_numeric_names|pipe to
--[[
library.swapping_numeric_names = function (ctx)
	local tmp
	local tbl, cache, nsize = ctx.params, {}, 0
	for key in pairs(tbl) do
		if type(key) == 'number' then
			nsize = nsize + 1
			cache[nsize] = key
		end
	end
	table.sort(cache)
	for idx = math.floor(nsize / 2), 1, -1 do
		tmp = tbl[cache[idx] ]
		tbl[cache[idx] ] = tbl[cache[nsize - idx + 1] ]
		tbl[cache[nsize - idx + 1] ] = tmp
	end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|sorting_sequential_values|[criterion]|pipe to
library.sorting_sequential_values = function (ctx)
	local sortfn
	if ctx.pipe[1] ~= nil then
		sortfn = sortfunctions[ctx.pipe[1]:match'^%s*(.-)%s*$']
	end
	if sortfn then table.sort(ctx.params, sortfn)
	else table.sort(ctx.params) end -- i.e. either `false` or `nil`
	if sortfn == nil then return context_iterate(ctx, 1) end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|splicing|[add to position]|position|increment|
--            [number of elements to write]|...|pipe to
library.splicing = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp2, argc, pos, refp
	local opts, tbl = ctx.pipe, ctx.params
	local tmp1 = opts[1]
	if tmp1 ~= nil then
		tmp2 = tonumber(tmp1)
		if tmp2 == nil or math.floor(tmp2) ~= tmp2 then
			pos, argc, tmp2 = tonumber(opts[2]), 4,
				tmp1:match'^%s*(.*%S)'
			if tmp2 ~= nil then
				refp = position_references[tmp2]
				if refp == nil then error(modulename ..
					', ‘splicing’: ‘' .. tostring(tmp2) ..
					'’ is not a valid first argument', 0) end
			else refp = 0 end
		else pos, argc, refp = tmp2, 3, 0 end
	else pos, argc, refp = tonumber(opts[2]), 4, 0 end
	if pos == nil or math.floor(pos) ~= pos then error(modulename ..
		', ‘splicing’: The position must be an integer number', 0) end
	local len = tonumber(opts[argc - 1])
	if len == nil or math.floor(len) ~= len then error(modulename ..
		', ‘splicing’: The increment must be an integer number', 0) end
	if refp == 2 then
		for _ in ipairs(tbl) do pos = pos + 1 end
		refp = 0
	end
	tmp1, tmp2 = nil, nil
	if refp ~= 0 or len ~= 0 then
		for key, val in pairs(tbl) do
			if type(key) == 'number' then
				if tmp1 == nil then tmp1, tmp2 = key, key
				elseif key &lt; tmp1 then tmp1 = key
				elseif key &gt; tmp2 then tmp2 = key end
			end
		end
	end
	if tmp2 == nil then len = 0
	elseif refp == 3 then pos = pos + tmp2
	elseif refp == 1 then pos = pos + tmp1 end
	if len &gt; 0 and pos + len &gt; tmp1 and pos &lt;= tmp2 then
		tbl = copy_table_expanded(tbl, pos, len)
	elseif len &lt; 0 and pos - len &gt; tmp1 and pos &lt;= tmp2 then
		tbl = copy_table_reduced(tbl, pos, -len)
	else tbl = copy_or_ref_table(tbl, tbl ~= ctx.oparams) end
	ctx.params = tbl
	tmp1 = tonumber(opts[argc])
	if len == 0 and (tmp1 == nil or tmp1 &lt; 1) then error(modulename ..
		', ‘splicing’: When the increment is zero the number of elements to add cannot be zero', 0) end
	if tmp1 == nil or tmp1 &lt; 0 or math.floor(tmp1) ~= tmp1 then
		return context_iterate(ctx, argc)
	end
	tmp2 = argc - pos + 1
	for key = pos, pos + tmp1 - 1 do tbl[key] = opts[key + tmp2] end
	return context_iterate(ctx, argc + tmp1 + 1)
end


-- Syntax:  #invoke:params|imposing|name|value|pipe to
library.imposing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘imposing’: Missing parameter name to impose', 0) end
	ctx.params[get_parameter_name(ctx.pipe[1])] = ctx.pipe[2]
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|providing|name|value|pipe to
library.providing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘providing’: Missing parameter name to provide', 0) end
	local key = get_parameter_name(ctx.pipe[1])
	if ctx.params[key] == nil then ctx.params[key] = ctx.pipe[2] end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|reassigning|source|destination|[mode]|pipe to
library.reassigning = function (ctx)
	local opts, tbl = ctx.pipe, ctx.params
	if opts[1] == nil then error(modulename ..
		', ‘reassigning’: Missing source parameter', 0) end
	if opts[2] == nil then error(modulename ..
		', ‘reassigning’: Missing destination parameter', 0) end
	local mode
	local src = get_parameter_name(opts[1])
	local val = tbl[src]
	if opts[3] ~= nil then mode = a_modes[opts[3]:match'^%s*(.-)%s*$'] end
	local argc = mode == nil and 3 or 4
	if val == nil then return context_iterate(ctx, argc) end
	local dest = get_parameter_name(opts[2])
	if mode == 1 or mode == 4 or (mode == 3 and tbl[dest] == nil) then
		tbl[src] = nil
	end
	if mode == nil or mode &lt; 2 or tbl[dest] == nil then tbl[dest] = val end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|discarding|name|[how many]|pipe to
library.discarding = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘discarding’: Missing parameter name to discard', 0) end
	local len = tonumber(ctx.pipe[2])
	if len == nil then
		ctx.params[get_parameter_name(ctx.pipe[1])] = nil
		return context_iterate(ctx, 2)
	end
	local key = tonumber(ctx.pipe[1])
	if key == nil or math.floor(key) ~= key then error(modulename ..
		', ‘discarding’: A range was provided, but the initial parameter name is not an integer number', 0) end
	if len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘discarding’: A range can only be an integer number greater than zero', 0) end
	for idx = key, key + len - 1 do ctx.params[idx] = nil end
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|excluding_non-numeric_names|pipe to
library['excluding_non-numeric_names'] = function (ctx)
	local tmp = ctx.params
	for key, val in pairs(tmp) do
		if type(key) ~= 'number' then tmp[key] = nil end
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|excluding_numeric_names|pipe to
library.excluding_numeric_names = function (ctx)
	local tmp = ctx.params
	for key, val in pairs(tmp) do
		if type(key) == 'number' then tmp[key] = nil end
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|with_name_matching|target 1|[plain flag 1]|[or]
--            |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag
--            N]|pipe to
library.with_name_matching = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp, ptn
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_name_matching')
	local tbl, newparams = ctx.params, {}
	for idx = 1, nptns do
		ptn = targets[idx]
		if ptn[3] then
			tmp = ptn[1]
			if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then
				tmp = tonumber(tmp)
			end
			newparams[tmp] = tbl[tmp]
		else
			for key, val in pairs(tbl) do
				if tostring(key):find(ptn[1], 1, ptn[2]) then
					newparams[key] = val
				end
			end
		end
	end
	ctx.params = newparams
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_name_not_matching|target 1|[plain flag 1]
--            |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain
--            flag N]|pipe to
library.with_name_not_matching = function (ctx)
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_name_not_matching')
	local tbl = ctx.params
	if nptns == 1 and targets[1][3] then
		local tmp = targets[1][1]
		if tmp == '0' or tmp:find'^%-?[1-9]%d*$' ~= nil then
			tbl[tonumber(tmp)] = nil
		else tbl[tmp] = nil end
		return context_iterate(ctx, argc)
	end
	local yesmatch, ptn
	for key in pairs(tbl) do
		yesmatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if tostring(key) ~= ptn[1] then
					yesmatch = false
					break
				end
			elseif not tostring(key):find(ptn[1], 1, ptn[2]) then
				yesmatch = false
				break
			end
		end
		if yesmatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_value_matching|target 1|[plain flag 1]|[or]
--            |[target 2]|[plain flag 2]|[or]|[...]|[target N]|[plain flag
--            N]|pipe to
library.with_value_matching = function (ctx)
	local nomatch, ptn
	local tbl = ctx.params
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_value_matching')
	for key, val in pairs(tbl) do
		nomatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if val == ptn[1] then
					nomatch = false
					break
				end
			elseif val:find(ptn[1], 1, ptn[2]) then
				nomatch = false
				break
			end
		end
		if nomatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|with_value_not_matching|target 1|[plain flag 1]
--            |[and]|[target 2]|[plain flag 2]|[and]|[...]|[target N]|[plain
--            flag N]|pipe to
library.with_value_not_matching = function (ctx)
	local yesmatch, ptn
	local tbl = ctx.params
	local targets, nptns, argc = load_pattern_args(ctx.pipe,
		'with_value_not_matching')
	for key, val in pairs(tbl) do
		yesmatch = true
		for idx = 1, nptns do
			ptn = targets[idx]
			if ptn[3] then
				if val ~= ptn[1] then
					yesmatch = false
					break
				end
			elseif not val:find(ptn[1], 1, ptn[2]) then
				yesmatch = false
				break
			end
		end
		if yesmatch then tbl[key] = nil end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|keeping_at_most|number of parameters to pick|pipe to
library.keeping_at_most = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local len = tonumber(ctx.pipe[1])
	if len == nil or len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘keeping_at_most’: The number of parameters to keep must be an integer greater than zero', 0) end
	ctx.params = copy_table_maxn({}, ctx.params, len)
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|trimming_values|pipe to
library.trimming_values = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:match'^%s*(.-)%s*$' end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_to_lowercase|pipe to
library.mapping_to_lowercase = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:lower() end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_to_uppercase|pipe to
library.mapping_to_uppercase = function (ctx)
	local tbl = ctx.params
	for key, val in pairs(tbl) do tbl[key] = val:upper() end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mapping_by_calling|template name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            parameters]|[parameter 1]|[parameter 2]|[...]|[parameter N]|pipe to
library.mapping_by_calling = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘mapping_by_calling’: No template name was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.values_only, ctx.params)
	local model = { title = tname, args = margs }
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return ctx.frame:expandTemplate(model)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_invoking|module name|function name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.mapping_by_invoking = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘mapping_by_invoking’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘mapping_by_invoking’: No function name was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 2, mapping_styles.values_only, ctx.params)
	local model = { title = 'Module:' .. mname, args = margs }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘mapping_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return tostring(mfunc(ctx.frame:newChild(model)))
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_magic|parser function|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.mapping_by_magic = function (ctx)
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘mapping_by_magic’: No parser function was provided', 0) end
	local margs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.values_only, ctx.params)
	value_maps[looptype](tbl, margs, karg, varg, function ()
		return ctx.frame:callParserFunction(magic, margs)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_replacing|target|replace|[count]|[plain
--            flag]|pipe to
library.mapping_by_replacing = function (ctx)
	local ptn, repl, nmax, flg, argc, die =
		load_replace_args(ctx.pipe, 'mapping_by_replacing')
	if die then return context_iterate(ctx, argc) end
	local tbl = ctx.params
	if flg == 3 then
		for key, val in pairs(tbl) do
			if val == ptn then tbl[key] = repl end
		end
	else
		if flg == 2 then
			-- Copied from Module:String's `str._escapePattern()`
			ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0')
		end
		for key, val in pairs(tbl) do
			tbl[key] = val:gsub(ptn, repl, nmax)
		end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|mapping_by_mixing|mixing string|pipe to
library.mapping_by_mixing = function (ctx)
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘mapping_by_mixing’: No mixing string was provided', 0) end
	local tbl, mix = ctx.params, ctx.pipe[1]
	if mix == '$#' then
		for key in pairs(tbl) do tbl[key] = tostring(key) end
		return context_iterate(ctx, 2)
	end
	local skel, cnv, n_parts = parse_placeholder_string(mix)
	for key, val in pairs(tbl) do
		for idx = 2, n_parts, 2 do
			if skel[idx] then cnv[idx] = val
			else cnv[idx] = tostring(key) end
		end
		tbl[key] = table.concat(cnv)
	end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|mapping_to_names|pipe to
--[[
library.mapping_to_names = function (ctx)
	local tbl = ctx.params
	for key in pairs(tbl) do tbl[key] = tostring(key) end
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|renaming_to_lowercase|pipe to
library.renaming_to_lowercase = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'string' then cache[key:lower()] = val else
		cache[key] = val end
	end
	ctx.params = cache
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|renaming_to_uppercase|pipe to
library.renaming_to_uppercase = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do
		if type(key) == 'string' then cache[key:upper()] = val else
		cache[key] = val end
	end
	ctx.params = cache
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|renaming_to_sequence|[sort order]|pipe to
library.renaming_to_sequence = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache, len
	local tbl = ctx.params
	local sortfn, argc, do_sort = load_sort_opt(ctx.pipe[1])
	if do_sort then
		local words, wl
		cache, words, len, wl = get_key_list_sorted(tbl, sortfn)
		for idx = 1, len do cache[idx] = tbl[cache[idx]] end
		for idx = 1, wl do cache[len + idx] = tbl[words[idx]] end
	else
		len, cache = 0, {}
		for _, val in pairs(tbl) do
			len = len + 1
			cache[len] = val
		end
	end
	ctx.params = cache
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_calling|template name|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            parameters]|[parameter 1]|[parameter 2]|[...]|[parameter N]|pipe to
library.renaming_by_calling = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘renaming_by_calling’: No template name was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.names_only, ctx.params)
	local model = { title = tname, args = rargs }
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return ctx.frame:expandTemplate(model)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_invoking|module name|function
--            name|[call style]|[let/use]|[...]|[let/use]|[...]|[number of
--            additional arguments]|[argument 1]|[argument 2]|[...]|[argument
--            N]|pipe to
library.renaming_by_invoking = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘renaming_by_invoking’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘renaming_by_invoking’: No function name was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 2, mapping_styles.names_only, ctx.params)
	local model = { title = 'Module:' .. mname, args = rargs }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘renaming_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return tostring(mfunc(ctx.frame:newChild(model)))
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_magic|parser function|[call
--            style]|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.renaming_by_magic = function (ctx)
	local opts = ctx.pipe
	local magic
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘renaming_by_magic’: No parser function was provided', 0) end
	local rargs, argc, looptype, karg, varg, tbl, mem =
		load_callback_opts(opts, 1, mapping_styles.names_only, ctx.params)
	map_names(tbl, rargs, karg, varg, looptype, function ()
		return ctx.frame:callParserFunction(magic, rargs)
	end)
	for key, val in pairs(mem) do tbl[key] = val end
	ctx.params = tbl
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_replacing|target|replace|[count]|[plain
--            flag]|pipe to
library.renaming_by_replacing = function (ctx)
	local ptn, repl, nmax, flg, argc, die =
		load_replace_args(ctx.pipe, 'renaming_by_replacing')
	if die then return context_iterate(ctx, argc) end
	local tbl = ctx.params
	if flg == 3 then
		ptn = get_parameter_name(ptn)
		local val = tbl[ptn]
		if val ~= nil then
			tbl[ptn], tbl[get_parameter_name(repl)] = nil, val
		end
	else
		if flg == 2 then
			-- Copied from Module:String's `str._escapePattern()`
			ptn = ptn:gsub('[%(%)%.%%%+%-%*%?%[%^%$%]]', '%%%0')
		end
		local cache = {}
		for key, val in pairs(tbl) do
			steal_if_renamed(val, tbl, key, cache,
				tostring(key):gsub(ptn, repl, nmax))
		end
		for key, val in pairs(cache) do tbl[key] = val end
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|renaming_by_mixing|mixing string|pipe to
library.renaming_by_mixing = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘renaming_by_mixing’: No mixing string was provided', 0) end
	local mix = ctx.pipe[1]:match'^%s*(.-)%s*$'
	local cache = {}
	if mix == '$@' then
		for _, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = val
		end
	else
		local skel, canvas, n_parts = parse_placeholder_string(mix)
		for key, val in pairs(ctx.params) do
			for idx = 2, n_parts, 2 do
				if skel[idx] then canvas[idx] = val
				else canvas[idx] = tostring(key) end
			end
			cache[get_parameter_name(table.concat(canvas))] = val
		end
	end
	ctx.params = cache
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|renaming_to_values|pipe to
--[[
library.renaming_to_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for _, val in pairs(ctx.params) do cache[val] = val end
	ctx.params = cache
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|grouping_by_calling|template
--            name|[let/use]|[...]|[let/use]|[...]|[number of additional
--            arguments]|[argument 1]|[argument 2]|[...]|[argument N]|pipe to
library.grouping_by_calling = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tmp, argc, tbl, mem = load_child_opts(ctx.pipe, 2, 0, ctx.params)
	local gargs = {}
	for key, val in pairs(tmp) do
		if type(key) == 'number' and key &lt; 1 then gargs[key - 1] = val
		else gargs[key] = val end
	end
	tmp = ctx.pipe[1]
	if tmp ~= nil then tmp = tmp:match'^%s*(.*%S)' end
	if tmp == nil then error(modulename ..
		', ‘grouping_by_calling’: No template name was provided', 0) end
	local model = { title = tmp }
	local groups = make_groups(tbl)
	for gid, group in pairs(groups) do
		for key, val in pairs(gargs) do group[key] = val end
		group[0], model.args = gid, group
		groups[gid] = ctx.frame:expandTemplate(model)
	end
	for key, val in pairs(mem) do groups[key] = val end
	ctx.params = groups
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|parsing|string to parse|[trim flag]|[iteration
--            delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to
library.parsing = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘parsing’: No string to parse was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '|', '=')
	parse_parameter_string(ctx.params, opts[1], isep, iplain, psep, pplain,
		trimnamed, trimunnamed)
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|reinterpreting|parameter to reinterpret|[trim
--            flag]|[iteration delimiter setter]|[...]|[key-value delimiter
--            setter]|[...]|pipe to
library.reinterpreting = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘reinterpreting’: No parameter to reinterpret was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '|', '=')
	local tbl, tmp = ctx.params, get_parameter_name(opts[1])
	local str = tbl[tmp]
	if str ~= nil then
		tbl[tmp] = nil
		parse_parameter_string(tbl, str, isep, iplain, psep, pplain,
			trimnamed, trimunnamed)
	end
	return context_iterate(ctx, argc)
end


-- Syntax:  #invoke:params|evaluating|string to parse|[trim flag]|[iteration
--            delimiter setter]|[...]|[key-value delimiter setter]|[...]|pipe to
library.evaluating = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘evaluating’: No string to parse was provided', 0) end
	local isep, iplain, psep, pplain, trimnamed, trimunnamed, argc =
		load_parse_opts(opts, 2, '!', ':')
	if opts[1]:match'^%s*(.*%S)' == nil then
		ctx.pipe = copy_or_ref_table(opts, opts ~= ctx.opipe)
		return context_iterate(ctx, argc)
	end
	local new_opts, cache = {}, {}
	local shift = parse_parameter_string(cache, opts[1], isep, iplain,
		psep, pplain, trimnamed, trimunnamed) - argc
	for key, val in pairs(opts) do
		if type(key) ~= 'number' or key &lt; 1 then new_opts[key] = val
		elseif key &gt;= argc then new_opts[key + shift] = val end
	end
	for key, val in pairs(cache) do new_opts[key] = val end
	ctx.pipe = new_opts
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|mixing_names_and_values|mixing string|pipe to
library.mixing_names_and_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	if ctx.pipe[1] == nil then error(modulename ..
		', ‘mixing_names_and_values’: No mixing string was provided for parameter names', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘mixing_names_and_values’: No mixing string was provided for parameter values', 0) end
	local tmp
	local mix_k = ctx.pipe[1]:match'^%s*(.-)%s*$'
	local cache, mix_v = {}, ctx.pipe[2]
	if mix_k == '$@' and mix_v == '$@' then
		for _, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = val
		end
	elseif mix_k == '$@' and mix_v == '$#' then
		for key, val in pairs(ctx.params) do
			cache[get_parameter_name(val)] = tostring(key)
		end
	elseif mix_k == '$#' and mix_v == '$#' then
		for key in pairs(ctx.params) do cache[key] = tostring(key) end
	else
		local skel_k, cnv_k, n_parts_k = parse_placeholder_string(mix_k)
		local skel_v, cnv_v, n_parts_v = parse_placeholder_string(mix_v)
		for key, val in pairs(ctx.params) do
			tmp = tostring(key)
			for idx = 2, n_parts_k, 2 do
				if skel_k[idx] then cnv_k[idx] = val else cnv_k[idx] = tmp end
			end
			for idx = 2, n_parts_v, 2 do
				if skel_v[idx] then cnv_v[idx] = val else cnv_v[idx] = tmp end
			end
			cache[get_parameter_name(table.concat(cnv_k))] =
				table.concat(cnv_v)
		end
	end
	ctx.params = cache
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|swapping_names_and_values|pipe to
--[[
library.swapping_names_and_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local cache = {}
	for key, val in pairs(ctx.params) do cache[val] = key end
	ctx.params = cache
	return context_iterate(ctx, 1)
end
]]--


-- Syntax:  #invoke:params|combining|new parameter name|[sort
--            order]|[with/without flushed glue]|setting directives|...|pipe to
library.combining = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return key .. kvs .. val end,
		'combining'
	) + 1)
end


-- Syntax:  #invoke:params|combining_values|new parameter name|[sort
--            order]|[with/without flushed glue]|setting directives|...|pipe to
library.combining_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return val end,
		'combining_values'
	) + 1)
end


-- Syntax:  #invoke:params|combining_by_calling|template name|new parameter
--            name|pipe to
library.combining_by_calling = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local tname = ctx.pipe[1]
	if tname ~= nil then tname = tname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_calling’: No template name was provided', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘combining_by_calling’: No parameter name was provided', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[2])] = ctx.frame:expandTemplate{
			title = tname,
			args = ctx.params
		}
	}
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|combining_by_invoking|module name|function name|new
--            parameter name|pipe to
library.combining_by_invoking = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local mname = ctx.pipe[1]
	if mname ~= nil then mname = mname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_invoking’: No module name was provided', 0) end
	local fname = ctx.pipe[2]
	if fname ~= nil then fname = fname:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_invoking’: No function name was provided', 0) end
	if ctx.pipe[3] == nil then error(modulename ..
		', ‘combining_by_invoking’: No parameter name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = ctx.params }
	local mfunc = require(model.title)[fname]
	if mfunc == nil then error(modulename ..
		', ‘mapping_by_invoking’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[3])] =
			tostring(mfunc(ctx.frame:newChild(model)))
	}
	return context_iterate(ctx, 4)
end


-- Syntax:  #invoke:params|combining_by_magic|parser function|new parameter
--            name|pipe to
library.combining_by_magic = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	local magic = ctx.pipe[1]
	if magic ~= nil then magic = magic:match'^%s*(.*%S)'
	else error(modulename ..
		', ‘combining_by_magic’: No parser function was provided', 0) end
	if ctx.pipe[2] == nil then error(modulename ..
		', ‘combining_by_magic’: No parameter name was provided', 0) end
	ctx.params = {
		[get_parameter_name(ctx.pipe[2])] =
			ctx.frame:callParserFunction(magic, ctx.params)
	}
	return context_iterate(ctx, 3)
end


-- Syntax:  #invoke:params|snapshotting|[maximum number]|pipe to
library.snapshotting = function (ctx)
	return context_iterate(ctx, make_child(ctx, ctx.params, 'snapshotting'))
end


-- Syntax:  #invoke:params|remembering|[maximum number]|pipe to
library.remembering = function (ctx)
	return context_iterate(ctx, make_child(ctx, ctx.oparams, 'remembering'))
end


-- Syntax:  #invoke:params|entering_substack|[new]|pipe to
library.entering_substack = function (ctx)
	local tbl, ncurrparent = ctx.params, ctx.n_parents + 1
	if ctx.parents == nil then ctx.parents = { tbl }
	else ctx.parents[ncurrparent] = tbl end
	ctx.n_parents = ncurrparent
	if ctx.pipe[1] ~= nil and ctx.pipe[1]:match'^%s*new%s*$' then
		ctx.params = {}
		return context_iterate(ctx, 2)
	end
	local currsnap = ctx.n_children
	if currsnap &gt; 0 then
		ctx.params, ctx.children[currsnap], ctx.n_children =
			ctx.children[currsnap], nil, currsnap - 1
	else
		local newparams = {}
		for key, val in pairs(tbl) do newparams[key] = val end
		ctx.params = newparams
	end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|pulling|parameter name|pipe to
library.pulling = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘pulling’: No parameter to pull was provided', 0) end
	local tmp = ctx.n_parents
	local parent = tmp &lt; 1 and ctx.oparams or ctx.parents[tmp]
	tmp = get_parameter_name(opts[1])
	if parent[tmp] ~= nil then ctx.params[tmp] = parent[tmp] end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|recalling|parameter name|pipe to
library.recalling = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘recalling’: No parameter to recall was provided', 0) end
	local arg = get_parameter_name(opts[1])
	if ctx.oparams[arg] ~= nil then ctx.params[arg] = ctx.oparams[arg] end
	return context_iterate(ctx, 2)
end


-- Syntax:  #invoke:params|finding|parameter name|pipe to
--[[
library.finding = function (ctx)
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘finding’: No parameter to find was provided', 0) end
	local arg = get_parameter_name(opts[1])
	local parent
	for idx = ctx.n_parents, 1, -1 do
		parent = ctx.parents[idx]
		if parent[arg] ~= nil then
			ctx.params[arg] = parent[arg]
			return context_iterate(ctx, 2)
		end
	end
	if ctx.oparams[arg] ~= nil then ctx.params[arg] = ctx.oparams[arg] end
	return context_iterate(ctx, 2)
end
]]--


-- Syntax:  #invoke:params|picking|number of parameters to pick|pipe to
--[[
library.picking = function (ctx)
	local len = tonumber(ctx.pipe[1])
	if len == nil or len &lt; 1 or math.floor(len) ~= len then error(modulename ..
		', ‘picking’: The number of parameters to pick must be an integer greater than zero', 0) end
	if ctx.n_parents &lt; 1 then copy_table_maxn(ctx.params, ctx.oparams, len)
	else copy_table_maxn(ctx.params, ctx.parents[ctx.n_parents], len) end
	return context_iterate(ctx, 2)
end
]]--


-- Syntax:  #invoke:params|detaching_substack|pipe to
library.detaching_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘detaching_substack’: No substack has been created', 0) end
	local parent = ctx.parents[ncurrparent]
	for key in pairs(ctx.params) do parent[key] = nil end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|dropping_substack|pipe to
library.dropping_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘dropping_substack’: No substack has been created', 0) end
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents =
		ctx.parents[ncurrparent], nil, ncurrparent - 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|leaving_substack|pipe to
library.leaving_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘leaving_substack’: No substack has been created', 0) end
	local currsnap = ctx.n_children + 1
	if ctx.children == nil then ctx.children = { ctx.params }
	else ctx.children[currsnap] = ctx.params end
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents, ctx.n_children =
		ctx.parents[ncurrparent], nil, ncurrparent - 1, currsnap
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|merging_substack|pipe to
library.merging_substack = function (ctx)
	local ncurrparent = ctx.n_parents
	if ncurrparent &lt; 1 then error(modulename ..
		', ‘merging_substack’: No substack has been created', 0) end
	local parent, child = ctx.parents[ncurrparent], ctx.params
	ctx.params, ctx.parents[ncurrparent], ctx.n_parents = parent, nil,
		ncurrparent - 1
	for key, val in pairs(child) do parent[key] = val end
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|flushing|pipe to
library.flushing = function (ctx)
	if ctx.n_children &lt; 1 then error(modulename ..
		', ‘flushing’: There are no substacks to flush', 0) end
	local parent, currsnap = ctx.params, ctx.n_children
	for key, val in pairs(ctx.children[currsnap]) do parent[key] = val end
	ctx.children[currsnap], ctx.n_children = nil, currsnap - 1
	return context_iterate(ctx, 1)
end


-- Syntax:  #invoke:params|setting_by_flushing|pipe to
library.setting_by_flushing = function (ctx)
	set_strings_from_substack(ctx, ctx, 'setting_by_flushing')
	return context_iterate(ctx, 1)
end


	--[[ Functions ]]--
	-----------------------------


-- Syntax:  #invoke:params|count
library.count = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local retval = 0
	for _ in ctx.iterfunc(ctx.params) do retval = retval + 1 end
	if ctx.subset == -1 then retval = retval - #ctx.params end
	ctx.text = retval
	return false
end


-- Syntax:  #invoke:args|concat_and_call|template name|[prepend 1]|[prepend 2]
--            |[...]|[item n]|[named item 1=value 1]|[...]|[named item n=value
--            n]|[...]
library.concat_and_call = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘concat_and_call’: No template name was provided', 0) end
	remove_numeric_keys(opts, 1, 1)
	ctx.text = ctx.frame:expandTemplate{
		title = tname,
		args = concat_params(ctx)
	}
	return false
end


-- Syntax:  #invoke:args|concat_and_invoke|module name|function name|[prepend
--            1]|[prepend 2]|[...]|[item n]|[named item 1=value 1]|[...]|[named
--            item n=value n]|[...]
library.concat_and_invoke = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘concat_and_invoke’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘concat_and_invoke’: No function name was provided', 0) end
	remove_numeric_keys(opts, 1, 2)
	local mfunc = require('Module:' .. mname)[fname]
	if mfunc == nil then error(modulename ..
		', ‘concat_and_invoke’: The function ‘' .. fname ..
		'’ does not exist', 0) end
	ctx.text = mfunc(ctx.frame:newChild{
		title = 'Module:' .. mname,
		args = concat_params(ctx)
	})
	return false
end


-- Syntax:  #invoke:args|concat_and_magic|parser function|[prepend 1]|[prepend
--            2]|[...]|[item n]|[named item 1=value 1]|[...]|[named item n=
--            value n]|[...]
library.concat_and_magic = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable!
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘concat_and_magic’: No parser function was provided', 0) end
	remove_numeric_keys(opts, 1, 1)
	ctx.text = ctx.frame:callParserFunction(magic, concat_params(ctx))
	return false
end


-- Syntax:  #invoke:params|value_of|parameter name
library.value_of = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local opts = ctx.pipe
	if opts[1] == nil then error(modulename ..
		', ‘value_of’: No parameter name was provided', 0) end
	local val
	local key = opts[1]:match'^%s*(.-)%s*$'
	if key == '0' or key:find'^%-?[1-9]%d*$' ~= nil then
		key = tonumber(key)
		val = ctx.params[key]
		-- No worries: #ctx.params is unused when the modifier is in
		-- first position (and therefore `ctx.params` is a metatable)
		if val ~= nil and (
			ctx.subset ~= -1 or key &gt; #ctx.params or key &lt; 1
		) and (
			ctx.subset ~= 1 or (key &lt;= #ctx.params and key &gt; 0)
		) then
			ctx.text = (ctx.header or '') .. val .. (ctx.footer or '')
		else ctx.text = ctx.ifngiven or '' end
	else
		val = ctx.params[key]
		if ctx.subset ~= 1 and val ~= nil then ctx.text = (ctx.header
			or '') .. val .. (ctx.footer or '')
		else ctx.text = ctx.ifngiven or '' end
	end
	return false
end


-- Syntax:  #invoke:params|list
library.list = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or ''
	flush_params(ctx, function (key, val)
		ret[nss + 1], ret[nss + 2], ret[nss + 3], ret[nss + 4], nss =
			pps, key, kvs, val, nss + 4
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 4)
	return false
end


-- Syntax:  #invoke:params|list_values
library.list_values = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	-- NOTE: `library.coins()` and `library.unique_coins()` rely on us
	local ret, nss, pps = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		ret[nss + 1], ret[nss + 2], nss = pps, val, nss + 2
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|list_maybe_with_names
library.list_maybe_with_names = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, kvs, pps = {}, 0, ctx.pairsep or '', ctx.itersep or ''
	mixed_flush_params(
		ctx,
		function (key, val)
			ret[nss + 1], ret[nss + 2], ret[nss + 3],
				ret[nss + 4], nss = pps, '', '', val, nss + 4
		end,
		function (key, val)
			ret[nss + 1], ret[nss + 2], ret[nss + 3],
				ret[nss + 4], nss = pps, key, kvs, val, nss + 4
		end
	)
	finalize_and_return_concatenated_list(ctx, ret, nss, 4)
	return false
end


-- Syntax:  #invoke:params|coins|[first coin = value 1]|[second coin = value
--            2]|[...]|[last coin = value N]
--[[
library.coins = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local opts, tbl = ctx.pipe, ctx.params
	for key, val in pairs(tbl) do tbl[key] = opts[get_parameter_name(val)] end
	return library.list_values(ctx)
end
]]--


-- Syntax:  #invoke:params|unique_coins|[first coin = value 1]|[second coin =
--            value 2]|[...]|[last coin = value N]
--[[
library.unique_coins = function (ctx)
	local tmp
	local opts, tbl = ctx.pipe, ctx.params
	for key, val in pairs(tbl) do
		tmp = get_parameter_name(val)
		tbl[key], opts[tmp] = opts[tmp], nil
	end
	return library.list_values(ctx)
end
]]


-- Syntax:  #invoke:params|for_each|wikitext
library.for_each = function (ctx)
	-- NOTE: `ctx.pipe` might be the original metatable!
	local ret, nss, pps, txt = {}, 0, ctx.itersep or '', ctx.pipe[1] or ''
	local skel, cnv, n_parts = parse_placeholder_string(txt)
	flush_params(ctx, function (key, val)
		for idx = 2, n_parts, 2 do
			if skel[idx] then cnv[idx] = val
			else cnv[idx] = tostring(key) end
		end
		ret[nss + 1], nss = pps, nss + 2
		ret[nss] = table.concat(cnv)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each|template name|[append 1]|[append 2]
--            |[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘call_for_each’: No template name was provided', 0) end
	local model = { title = tname, args = opts }
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	table.insert(opts, 1, true)
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|invoke_for_each|module name|module function|[append
--            1]|[append 2]|[...]|[append n]|[named param 1=value 1]|[...]
--            |[named param n=value n]|[...]
library.invoke_for_each = function (ctx)
	local mname, fname
	local opts = ctx.pipe
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘invoke_for_each’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘invoke_for_each’: No function name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = opts }
	local mfunc = require(model.title)[fname]
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = mfunc(ctx.frame:newChild(model))
		
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|magic_for_each|parser function|[append 1]|[append 2]
--            |[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.magic_for_each = function (ctx)
	local magic
	local opts = ctx.pipe
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘magic_for_each’: No parser function was provided', 0) end
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	table.insert(opts, 1, true)
	flush_params(ctx, function (key, val)
		opts[1], opts[2], ret[nss + 1], nss = key, val, ccs, nss + 2
		ret[nss] = ctx.frame:callParserFunction(magic, opts)

	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each_value|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each_value = function (ctx)
	local tname
	local opts = ctx.pipe
	if opts[1] ~= nil then tname = opts[1]:match'^%s*(.*%S)' end
	if tname == nil then error(modulename ..
		', ‘call_for_each_value’: No template name was provided', 0) end
	local model = { title = tname, args = opts }
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = ctx.frame:expandTemplate(model)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|invoke_for_each_value|module name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.invoke_for_each_value = function (ctx)
	local opts = ctx.pipe
	local mname, fname
	if opts[1] ~= nil then mname = opts[1]:match'^%s*(.*%S)' end
	if mname == nil then error(modulename ..
		', ‘invoke_for_each_value’: No module name was provided', 0) end
	if opts[2] ~= nil then fname = opts[2]:match'^%s*(.*%S)' end
	if fname == nil then error(modulename ..
		', ‘invoke_for_each_value’: No function name was provided', 0) end
	local model = { title = 'Module:' .. mname, args = opts }
	local mfunc = require(model.title)[fname]
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	remove_numeric_keys(opts, 1, 1)
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = mfunc(ctx.frame:newChild(model))
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|magic_for_each_value|parser function|[append 1]
--            |[append 2]|[...]|[append n]|[named param 1=value 1]|[...]|[named
--            param n=value n]|[...]
library.magic_for_each_value = function (ctx)
	local opts = ctx.pipe
	local magic
	if opts[1] ~= nil then magic = opts[1]:match'^%s*(.*%S)' end
	if magic == nil then error(modulename ..
		', ‘magic_for_each_value’: No parser function was provided', 0) end
	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(ctx, function (key, val)
		opts[1], ret[nss + 1], nss = val, ccs, nss + 2
		ret[nss] = ctx.frame:callParserFunction(magic, opts)
	end)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


-- Syntax:  #invoke:params|call_for_each_group|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
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  <page>
    <title>User:Ruud Loeffen/Cosmic Influx Theory(2)</title>
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      <id>2818392</id>
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      <timestamp>2026-07-16T05:25:50Z</timestamp>
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        <username>Ruud Loeffen</username>
        <id>2998353</id>
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      <comment>copied content from version (3) to (2) to avoid confusion</comment>
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      <text bytes="23000" sha1="a490v70o02qui3o39rnaxze0pch781h" xml:space="preserve">{{original research}}

[[File:CITbanner.png|center|frameless|960px|Cosmic Influx Theory]]

== Introduction ==
The '''Cosmic Influx Theory (CIT)''' explores the continuous influx of mass-energy in celestial bodies, contributing to planetary growth, geophysical activity, and gravitational effects. Beyond the macroscopic scale, CIT proposes that mass-energy influx also influences '''microscopic phenomena''' such as Van der Waals forces, the Casimir effect... [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.2.10|[8.2.10]]], and even the trajectory of falling raindrops. These phenomena may provide subtle but crucial evidence of a pervasive cosmic influx shaping both the vast and the minuscule aspects of the universe.

By delving into the '''Gravitational Constant''', we unveil compelling evidence for an '''increase in mass and heat''' for all celestial objects within an isotropic and homogenous universe as a result of the '''Lorentz Transformation of Mass- Energy''' (LTME) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.1.1|[8.1.1]]]. Traditionally, LTME has been considered relevant primarily for '''subatomic particles''' at '''high''' velocities. However, this study posits that LTME is equally applicable to '''big celestial bodies''', even at relatively '''low velocities'''.

CIT introduces the concept of a '''universal energy influx''', hypothesized as a stream of &quot;whirlings&quot; or &quot;excitations&quot; interacting with the kinetic energy of atoms, driving incremental mass increases in alignment with the Lorentz Transformation of Mass-Energy (LTME) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7.2|[8.7.2]]]

This mechanism offers a unified explanation for geological phenomena such as '''volcanic activity, seafloor spreading, and planetary expansion''' [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.15|[8.4.15]]] [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.20|[8.4.20]]], while also addressing cosmological questions such as galactic rotation curves and cosmic acceleration. Key results include calculated mass-energy growth rates consistent with geological observations as described by many researchers on '''Earth Expansion''' and '''Expansion Tectonics''' 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.20|[8.4.20]]] [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.21|[8.4.21]]], a redefinition of gravitational acceleration through the volumetric universal influx. By integrating CIT with established physics principles and observational data, this paper highlights its potential to bridge gaps in mainstream models of dark matter and dark energy [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.1.2|[8.1.2]]].

Importantly, '''CIT does not reject the occurrence of subduction zones'''. Rather, it integrates subduction as a natural consequence of localized surface adjustments during global expansion. While oceanic crust is created at mid-ocean ridges, older, '''denser crust may subduct along continental margins, often accompanied by mountain building'''. However, the net balance, according to CIT, is a continuous increase in the total mass and volume of celestial bodies. A more detailed discussion on how subduction and expansion coexist within CIT is presented in '''[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3_Geophysical_Evidence:_Plate_Tectonics_and_Planetary_Evolution|Chapter 5.3]]'''.

This pursuit contemplates the possibility of an infinitely energetic universe, where energy metamorphoses into mass through &lt;math&gt;M = \frac{E}{c^2}&lt;/math&gt;
This interpretation proposes the existence of a '''Process of Continuously Created Matter''', manifesting as an ongoing accretion, augmentation, and expansion, harmonizing with the universe's ever-expansive nature [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.7|[8.4.7]]].

CIT introduces the '''Preferred Distance (D&lt;sub&gt;pref&lt;/sub&gt;)''', derived from the '''Root Mean Square Velocity (VRMS)''' of planetary systems (see Chapter 2 for explanation)[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7.3|[8.7.3]]], as a key factor in structuring planetary orbits. This theory challenges conventional gravitational models by linking the '''gravitational constant (G)''' to the Lorentz transformation and vacuum energy properties [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7.8|[8.7.8]]].

The purpose of this Wikiversity page is to present CIT in a structured and accessible format, supported by mathematical derivations, observational data, and theoretical discussions.

= CIT–VGT: An Interdisciplinary Collaboration in Gravitational and Geometric Physics =
 
== Francesco Chiaramonte ==
In 2026, the Cosmic Influx Theory (CIT) entered a new collaborative phase through the work of Francesco Chiaramonte, developer of Vortical Geometrodynamics Theory (VGT) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.54|[8.4.54]]] . While the original CIT framework was developed by Ruud Loeffen, Chiaramonte has contributed substantially to the mathematical, geometrical, and field-theoretical interpretation of several CIT-related ideas.

The collaboration between Loeffen and Chiaramonte focuses especially on the possible complementarity between CIT and VGT. In this combined approach, CIT proposes a universal influx process related to mass-energy growth, gravitational acceleration, and preferred-distance relations, while VGT explores vortical, torsional, and geometrical structures that may provide a more formal mathematical language for such processes.

Chiaramonte’s contribution is particularly important in the development of the CIT–VGT research line, including discussions on planetary torsional coupling, gravitational lensing, vortical constraint algebra, Gaia DR3 harmonic density structures, and possible links between vacuum dynamics, angular momentum, and large-scale cosmic organization. These ideas remain exploratory and should not be presented as established physics, but they represent a serious attempt to make the CIT framework more mathematically explicit and testable.

For that reason, Francesco Chiaramonte is introduced here as co-author and theoretical collaborator for the CIT–VGT convictions, reasoning, insights, and related publications developed from 2026 onward. The aim of this collaboration is not merely to confirm CIT, but to examine, strengthen, criticize, formalize, and where necessary correct the theory through mathematical reasoning, observational comparison, and open scientific discussion.

== Professor Suresh Kumar S.== 

Professor S. Suresh Kumar contributes advanced knowledge in mathematical physics, particularly in metric-affine geometry, torsion, Palatini formulations, SU(2) structures, hypermomentum, field theory, and the geometric foundations of gravitation. These areas are closely related to the effort within CIT–VGT to describe gravity not only as an observed force, but as a deeper interaction between cosmic influx, matter, motion, and spacetime structure.

His contribution is expected to strengthen the mathematical and theoretical basis of the combined CIT–VGT framework. In particular, he can help investigate whether the proposed influx processes can be represented through torsion, affine connections, matter–geometry coupling, and microscopic degrees of freedom. This may provide a bridge between the macroscopic phenomena emphasized in Cosmic Influx Theory and the more formal geometric structures developed within VGT.

Professor Kumar may also contribute to the formulation of consistent field equations, action principles, conservation relations, and possible links between nuclear, atomic, astrophysical, and cosmological scales. His expertise is therefore especially valuable in testing the internal consistency of CIT–VGT, identifying necessary mathematical improvements, and translating its physical concepts into a more rigorous theoretical framework suitable for further scientific discussion, comparison, and development.

== RMM Loeffen ==

Ruud Loeffen is the originator and principal developer of Cosmic Influx Theory (CIT). His work begins with observable gravitational, geological, planetary, and cosmological phenomena and explores the possibility that gravity is associated with a continuous inward influx of energy and matter-forming potential.

Using accessible mathematics, numerical comparisons, and dimensional analysis, he has developed relationships involving surface gravity, planetary mass, the Lorentz transformation of mass-energy, the characteristic (V_{\mathrm{RMS}}) velocity, the gravitational constant, and possible mass-energy increase over time. His research also investigates connections between gravitational processes at planetary scales and matter formation at atomic and nuclear scales.

Within the broader CIT–VGT collaboration, Ruud provides the foundational physical concepts, numerical discoveries, observational interpretations, and cross-scale hypotheses. His aim is to encourage critical examination, mathematical development, and independent testing of CIT as an alternative framework for understanding gravity and cosmic evolution.

== Chapters ==
Below are the ten chapters explaining the Cosmic Influx Theory in detail:

* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|Chapter 1: The Foundations of Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2|Chapter 2: The Role of VRMS in Planetary Structuring]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3|Chapter 3: The Cosmic Influx and the Gravitational Constant (G)]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4|Chapter 4: Implications for Planetary and Cosmic Expansion]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5|Chapter 5: Cosmic Expansion and the Growth of Celestial Bodies]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6|Chapter 6: The Future of Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7|Chapter 7: Units, Dimensions, and Fundamental Constants in CIT]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8|Chapter 8: Supporting Research, References, and Multimedia on Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9|Chapter 9: Genesis of the Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10|Chapter 10: Feeling the Influx — A New Point of Observation]]

== Detailed Chapter and Subsection Overview ==
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|Chapter 1: The Foundations of Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.1|1.1 The Root Mean Square Velocity (VRMS)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.2|1.2 The Limitations of Traditional Gravitational Models]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.3|1.3 The Concept of an Energy Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4|1.4 Lorentz Transformation and Planck-Based Influx Concepts]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.1|1.4.1 Lorentz Transformation and Mass-Energy Increase]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.2|1.4.2 The Plinflux: Deriving the Influx Quantum from Planck Geometry]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.3|1.4.3 From Field Equations to Surface Gravity: A Practical Role for 𝜅 and Influx]]
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|Chapter 1: The Foundations of Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.1|1.1 The Root Mean Square Velocity (VRMS)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.2|1.2 The Limitations of Traditional Gravitational Models]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.3|1.3 The Concept of an Energy Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4|1.4 Lorentz Transformation and Planck-Based Influx Concepts]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.1|1.4.1 Lorentz Transformation and Mass-Energy Increase]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.2|1.4.2 The Plinflux: Deriving the Influx Quantum from Planck Geometry]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.3|1.4.3 From Field Equations to Surface Gravity: A Practical Role for 𝜅 and Influx]]
 
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.5|1.5 Understanding VRMS and Its Significance]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.6|1.6  Relating Lorentz Mass-Energy to the Gravitational Constant]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#sec_1_7|1.7 From Einstein’s Original Kappa to Vacuum Structure]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2|Chapter 2: The Role of VRMS in Planetary Structuring]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.1|2.1 How VRMS is Related to Cosmic Structuring]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.2|2.2 The Connection Between CIT and General Relativity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.3|2.3 The Preferred Distance (Dpref) and its Calculation]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.4|2.4 Empirical Confirmation from Exoplanetary Systems]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.5|2.5 Implications for Planetary Formation Models]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3|Chapter 3: The Cosmic Influx and the Gravitational Constant (G)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.1|3.1 The Traditional Definition of G]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.2|3.2 Vacuum Energy and the Gravitational Constant]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.3|3.3 The Role of Vacuum Energy in Gravity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.4|3.4 Mass, Vacuum, and the Historical Constants]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.5|3.5 A Relativistic Vacuum Model: Components A &amp; B]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.6|3.6 Observational Evidence and Implications (volcanoes etc.)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.7|3.7 Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4|Chapter 4: Implications for Planetary and Cosmic Expansion]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.1|4.1 Recap of Delta Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.2|4.2 Isostasy as Internal Pressure and Volume Stress Due to Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.3|4.3 Radius Growth: A General Response to Cosmic Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.4|4.4 Equality of Influx and Gravity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5|4.5 Implications for Planetary and Cosmic Expansion]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5.1|4.5.1 Expansion of Earth's Radius]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5.2|4.5.2 Mass Growth Across Geological Epochs]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5.3_Time_Expansion_as_a_Consequence_of_Increasing_Mass:_A_CIT_Perspective|4.5.3 Time Expansion as a Consequence of Increasing Mass: A CIT Perspective]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.6|4.6 Conclusion: Influx as the Driver of Mass-Energy Growth]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.7|4.7 Looking Back in Time]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.8|4.8 Reversing Our Perspective: Looking Back from the Primordial Energy Field]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.9|4.9 The Expanding History of the Universe]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.10|4.10 A New Perspective on the Observable Universe]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#Summary|Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5|Chapter 5: Cosmic Expansion and the Growth of Celestial Bodies]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.1|5.1 Planetary Growth Through Mass-Energy Influx Delta INFLUX]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2|5.2 The Link Between Cosmic Expansion and CIT]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.1|5.2.1 Growing Galaxies and Cosmic Redshift]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.2|5.2.2 Growing Planets Born in Protoplanetary Disks]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.3A|5.2.3A Growing Moons Born in Circumplanetary Disks]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.3B|5.2.3B Secondary Rings Created by Geological and Cryovolcanic Activity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3|5.3 Geophysical Evidence: Plate Tectonics and Planetary Evolution]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1|5.3.1 Seafloor Spreading – A Step Toward Understanding Multi-Directional Crustal Growth]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.1|5.3.1.1 Introduction]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.2|5.3.1.2 Traditional Model]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.3|5.3.1.3 Multi-Directional Seafloor Spreading]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.4|5.3.1.4 MDSS and Expansion Tectonics]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.5|5.3.1.5 Evidence on Continents]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.6|5.3.1.6 Are Some Mountain Ranges Fossil Mid-Ocean Ridges?]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.7|5.3.1.7 Fossil Spreading Ridges Preserved on Continental Crust]]
****[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.8|5.3.1.8 Isostasy in a Multi-Directional Growth Picture (MDSS)]]
****
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.4|5.4 Earth's Day Length Through Geological Time]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.5|5.5 Stellar Growth and Galactic Evolution]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.6|5.6 Bondi-Hoyle Accretion as Empirical Support]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.7|5.7 Pioneers and Contributors to Earth Expansion and Expansion Tectonics]] 
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#References|References]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#Summary|Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6|Chapter 6: The Future of Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.1|6.1 Experimental and Observational Tests for CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.2|6.2 CIT and the Unification of Physics]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.3|6.3 The Role of AI-Human Collaboration in Science]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.4|6.4 Why Local Mass Measurements Cannot Detect the Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.5|6.5 Observational Evidence for a Cosmic Influx: Accelerometer, Casimir Effect, Cloud Chamber, Van der Waals Forces, and the Human Body]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.6|6.6 The Human Sensor of Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#Summary|Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7|Chapter 7: Units, Dimensions, and Fundamental Constants in CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.1|7.1 Unit Conversions in CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.2|7.2 The Five Dimensions in CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3|7.3 Derivation of Constants in CIT]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.1|7.3.1 Gravitational Constant (G)]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.2|7.3.2 κ_CIT – Planetary Structuring Constant]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.3|7.3.3 Einsteinian Coupling Constant κ]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.4|7.3.4  Alignment Between ACT Observations and CIT Predictions]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.5|7.3.5 Updated CIT Jeans Mass Concept]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.4|7.4 Conclusion]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.5|7.5 Overview of Important Constants]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8|Chapter 8: Supporting Research, References, and Multimedia]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.1|8.1 Articles Explaining CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.2|8.2 Comments and Contributions from ChatGPT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.3|8.3 Excel Files Supporting CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4|8.4 Other Articles and Websites]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.5|8.5 Videos Supporting CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.6|8.6 Videos Related to CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7|8.7 Selected Responses from ChatGPT]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9|Chapter 9: Genesis of the Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.1|9.1 Early Insights and Thought Experiments]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.2|9.2 Connecting with Existing Theories]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.3|9.3 Mathematical Exploration and Key Discoveries]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.4|9.4 Challenges and the Scientific Landscape]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.5|9.5 The Role of AI in Theory Development]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.6|9.6 Conclusion and Future Directions]]
 
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10|Chapter 10: Feeling the Influx — A New Point of Observation]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.1|10.1 The Quiet Moment in Bed]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.2|10.2 The Accelerometer Confirms It]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.3|10.3 Falling Raindrops — The Influx Made Visible]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.4|10.4 From Concept to Realization]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.5|10.5 A Universal Gesture of Reception]]
 
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[[File:CITbanner.png|center|frameless|960px|Cosmic Influx Theory]]

= Cosmic Influx Theory (CIT) =

== Introduction ==
The '''Cosmic Influx Theory (CIT)''' explores the continuous influx of mass-energy in celestial bodies, contributing to planetary growth, geophysical activity, and gravitational effects. Beyond the macroscopic scale, CIT proposes that mass-energy influx also influences '''microscopic phenomena''' such as Van der Waals forces, the Casimir effect... [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.2.10|[8.2.10]]], and even the trajectory of falling raindrops. These phenomena may provide subtle but crucial evidence of a pervasive cosmic influx shaping both the vast and the minuscule aspects of the universe.

By delving into the '''Gravitational Constant''', we unveil compelling evidence for an '''increase in mass and heat''' for all celestial objects within an isotropic and homogenous universe as a result of the '''Lorentz Transformation of Mass- Energy''' (LTME) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.1.1|[8.1.1]]]. Traditionally, LTME has been considered relevant primarily for '''subatomic particles''' at '''high''' velocities. However, this study posits that LTME is equally applicable to '''big celestial bodies''', even at relatively '''low velocities'''.

CIT introduces the concept of a '''universal energy influx''', hypothesized as a stream of &quot;whirlings&quot; or &quot;excitations&quot; interacting with the kinetic energy of atoms, driving incremental mass increases in alignment with the Lorentz Transformation of Mass-Energy (LTME) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7.2|[8.7.2]]]

This mechanism offers a unified explanation for geological phenomena such as '''volcanic activity, seafloor spreading, and planetary expansion''' [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.15|[8.4.15]]] [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.20|[8.4.20]]], while also addressing cosmological questions such as galactic rotation curves and cosmic acceleration. Key results include calculated mass-energy growth rates consistent with geological observations as described by many researchers on '''Earth Expansion''' and '''Expansion Tectonics''' 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.20|[8.4.20]]] [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.21|[8.4.21]]], a redefinition of gravitational acceleration through the volumetric universal influx. By integrating CIT with established physics principles and observational data, this paper highlights its potential to bridge gaps in mainstream models of dark matter and dark energy [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.1.2|[8.1.2]]].

Importantly, '''CIT does not reject the occurrence of subduction zones'''. Rather, it integrates subduction as a natural consequence of localized surface adjustments during global expansion. While oceanic crust is created at mid-ocean ridges, older, '''denser crust may subduct along continental margins, often accompanied by mountain building'''. However, the net balance, according to CIT, is a continuous increase in the total mass and volume of celestial bodies. A more detailed discussion on how subduction and expansion coexist within CIT is presented in '''[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3_Geophysical_Evidence:_Plate_Tectonics_and_Planetary_Evolution|Chapter 5.3]]'''.

This pursuit contemplates the possibility of an infinitely energetic universe, where energy metamorphoses into mass through &lt;math&gt;M = \frac{E}{c^2}&lt;/math&gt;
This interpretation proposes the existence of a '''Process of Continuously Created Matter''', manifesting as an ongoing accretion, augmentation, and expansion, harmonizing with the universe's ever-expansive nature [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.7|[8.4.7]]].

CIT introduces the '''Preferred Distance (D&lt;sub&gt;pref&lt;/sub&gt;)''', derived from the '''Root Mean Square Velocity (VRMS)''' of planetary systems (see Chapter 2 for explanation)[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7.3|[8.7.3]]], as a key factor in structuring planetary orbits. This theory challenges conventional gravitational models by linking the '''gravitational constant (G)''' to the Lorentz transformation and vacuum energy properties [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7.8|[8.7.8]]].

The purpose of this Wikiversity page is to present CIT in a structured and accessible format, supported by mathematical derivations, observational data, and theoretical discussions.

= CIT–VGT: An Interdisciplinary Collaboration in Gravitational and Geometric Physics =
 
== Francesco Chiaramonte ==
In 2026, the Cosmic Influx Theory (CIT) entered a new collaborative phase through the work of Francesco Chiaramonte, developer of Vortical Geometrodynamics Theory (VGT) [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4.54|[8.4.54]]] . While the original CIT framework was developed by Ruud Loeffen, Chiaramonte has contributed substantially to the mathematical, geometrical, and field-theoretical interpretation of several CIT-related ideas.

The collaboration between Loeffen and Chiaramonte focuses especially on the possible complementarity between CIT and VGT. In this combined approach, CIT proposes a universal influx process related to mass-energy growth, gravitational acceleration, and preferred-distance relations, while VGT explores vortical, torsional, and geometrical structures that may provide a more formal mathematical language for such processes.

Chiaramonte’s contribution is particularly important in the development of the CIT–VGT research line, including discussions on planetary torsional coupling, gravitational lensing, vortical constraint algebra, Gaia DR3 harmonic density structures, and possible links between vacuum dynamics, angular momentum, and large-scale cosmic organization. These ideas remain exploratory and should not be presented as established physics, but they represent a serious attempt to make the CIT framework more mathematically explicit and testable.

For that reason, Francesco Chiaramonte is introduced here as co-author and theoretical collaborator for the CIT–VGT convictions, reasoning, insights, and related publications developed from 2026 onward. The aim of this collaboration is not merely to confirm CIT, but to examine, strengthen, criticize, formalize, and where necessary correct the theory through mathematical reasoning, observational comparison, and open scientific discussion.

== Professor Suresh Kumar S.== 

Professor S. Suresh Kumar contributes advanced knowledge in mathematical physics, particularly in metric-affine geometry, torsion, Palatini formulations, SU(2) structures, hypermomentum, field theory, and the geometric foundations of gravitation. These areas are closely related to the effort within CIT–VGT to describe gravity not only as an observed force, but as a deeper interaction between cosmic influx, matter, motion, and spacetime structure.

His contribution is expected to strengthen the mathematical and theoretical basis of the combined CIT–VGT framework. In particular, he can help investigate whether the proposed influx processes can be represented through torsion, affine connections, matter–geometry coupling, and microscopic degrees of freedom. This may provide a bridge between the macroscopic phenomena emphasized in Cosmic Influx Theory and the more formal geometric structures developed within VGT.

Professor Kumar may also contribute to the formulation of consistent field equations, action principles, conservation relations, and possible links between nuclear, atomic, astrophysical, and cosmological scales. His expertise is therefore especially valuable in testing the internal consistency of CIT–VGT, identifying necessary mathematical improvements, and translating its physical concepts into a more rigorous theoretical framework suitable for further scientific discussion, comparison, and development.

== RMM Loeffen ==

Ruud Loeffen is the originator and principal developer of Cosmic Influx Theory (CIT). His work begins with observable gravitational, geological, planetary, and cosmological phenomena and explores the possibility that gravity is associated with a continuous inward influx of energy and matter-forming potential.

Using accessible mathematics, numerical comparisons, and dimensional analysis, he has developed relationships involving surface gravity, planetary mass, the Lorentz transformation of mass-energy, the characteristic (V_{\mathrm{RMS}}) velocity, the gravitational constant, and possible mass-energy increase over time. His research also investigates connections between gravitational processes at planetary scales and matter formation at atomic and nuclear scales.

Within the broader CIT–VGT collaboration, Ruud provides the foundational physical concepts, numerical discoveries, observational interpretations, and cross-scale hypotheses. His aim is to encourage critical examination, mathematical development, and independent testing of CIT as an alternative framework for understanding gravity and cosmic evolution.

== Chapters ==
Below are the ten chapters explaining the Cosmic Influx Theory in detail:

* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|Chapter 1: The Foundations of Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2|Chapter 2: The Role of VRMS in Planetary Structuring]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3|Chapter 3: The Cosmic Influx and the Gravitational Constant (G)]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4|Chapter 4: Implications for Planetary and Cosmic Expansion]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5|Chapter 5: Cosmic Expansion and the Growth of Celestial Bodies]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6|Chapter 6: The Future of Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7|Chapter 7: Units, Dimensions, and Fundamental Constants in CIT]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8|Chapter 8: Supporting Research, References, and Multimedia on Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9|Chapter 9: Genesis of the Cosmic Influx Theory]]
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10|Chapter 10: Feeling the Influx — A New Point of Observation]]

== Detailed Chapter and Subsection Overview ==
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|Chapter 1: The Foundations of Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.1|1.1 The Root Mean Square Velocity (VRMS)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.2|1.2 The Limitations of Traditional Gravitational Models]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.3|1.3 The Concept of an Energy Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4|1.4 Lorentz Transformation and Planck-Based Influx Concepts]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.1|1.4.1 Lorentz Transformation and Mass-Energy Increase]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.2|1.4.2 The Plinflux: Deriving the Influx Quantum from Planck Geometry]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.3|1.4.3 From Field Equations to Surface Gravity: A Practical Role for 𝜅 and Influx]]
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1|Chapter 1: The Foundations of Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.1|1.1 The Root Mean Square Velocity (VRMS)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.2|1.2 The Limitations of Traditional Gravitational Models]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.3|1.3 The Concept of an Energy Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4|1.4 Lorentz Transformation and Planck-Based Influx Concepts]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.1|1.4.1 Lorentz Transformation and Mass-Energy Increase]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.2|1.4.2 The Plinflux: Deriving the Influx Quantum from Planck Geometry]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.4.3|1.4.3 From Field Equations to Surface Gravity: A Practical Role for 𝜅 and Influx]]
 
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.5|1.5 Understanding VRMS and Its Significance]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#1.6|1.6  Relating Lorentz Mass-Energy to the Gravitational Constant]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_1#sec_1_7|1.7 From Einstein’s Original Kappa to Vacuum Structure]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2|Chapter 2: The Role of VRMS in Planetary Structuring]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.1|2.1 How VRMS is Related to Cosmic Structuring]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.2|2.2 The Connection Between CIT and General Relativity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.3|2.3 The Preferred Distance (Dpref) and its Calculation]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.4|2.4 Empirical Confirmation from Exoplanetary Systems]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_2#2.5|2.5 Implications for Planetary Formation Models]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3|Chapter 3: The Cosmic Influx and the Gravitational Constant (G)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.1|3.1 The Traditional Definition of G]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.2|3.2 Vacuum Energy and the Gravitational Constant]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.3|3.3 The Role of Vacuum Energy in Gravity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.4|3.4 Mass, Vacuum, and the Historical Constants]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.5|3.5 A Relativistic Vacuum Model: Components A &amp; B]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.6|3.6 Observational Evidence and Implications (volcanoes etc.)]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_3#3.7|3.7 Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4|Chapter 4: Implications for Planetary and Cosmic Expansion]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.1|4.1 Recap of Delta Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.2|4.2 Isostasy as Internal Pressure and Volume Stress Due to Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.3|4.3 Radius Growth: A General Response to Cosmic Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.4|4.4 Equality of Influx and Gravity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5|4.5 Implications for Planetary and Cosmic Expansion]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5.1|4.5.1 Expansion of Earth's Radius]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5.2|4.5.2 Mass Growth Across Geological Epochs]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.5.3_Time_Expansion_as_a_Consequence_of_Increasing_Mass:_A_CIT_Perspective|4.5.3 Time Expansion as a Consequence of Increasing Mass: A CIT Perspective]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.6|4.6 Conclusion: Influx as the Driver of Mass-Energy Growth]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.7|4.7 Looking Back in Time]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.8|4.8 Reversing Our Perspective: Looking Back from the Primordial Energy Field]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.9|4.9 The Expanding History of the Universe]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#4.10|4.10 A New Perspective on the Observable Universe]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_4#Summary|Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5|Chapter 5: Cosmic Expansion and the Growth of Celestial Bodies]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.1|5.1 Planetary Growth Through Mass-Energy Influx Delta INFLUX]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2|5.2 The Link Between Cosmic Expansion and CIT]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.1|5.2.1 Growing Galaxies and Cosmic Redshift]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.2|5.2.2 Growing Planets Born in Protoplanetary Disks]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.3A|5.2.3A Growing Moons Born in Circumplanetary Disks]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.2.3B|5.2.3B Secondary Rings Created by Geological and Cryovolcanic Activity]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3|5.3 Geophysical Evidence: Plate Tectonics and Planetary Evolution]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1|5.3.1 Seafloor Spreading – A Step Toward Understanding Multi-Directional Crustal Growth]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.1|5.3.1.1 Introduction]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.2|5.3.1.2 Traditional Model]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.3|5.3.1.3 Multi-Directional Seafloor Spreading]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.4|5.3.1.4 MDSS and Expansion Tectonics]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.5|5.3.1.5 Evidence on Continents]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.6|5.3.1.6 Are Some Mountain Ranges Fossil Mid-Ocean Ridges?]]
**** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.7|5.3.1.7 Fossil Spreading Ridges Preserved on Continental Crust]]
****[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.3.1.8|5.3.1.8 Isostasy in a Multi-Directional Growth Picture (MDSS)]]
****
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.4|5.4 Earth's Day Length Through Geological Time]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.5|5.5 Stellar Growth and Galactic Evolution]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.6|5.6 Bondi-Hoyle Accretion as Empirical Support]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#5.7|5.7 Pioneers and Contributors to Earth Expansion and Expansion Tectonics]] 
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#References|References]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_5#Summary|Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6|Chapter 6: The Future of Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.1|6.1 Experimental and Observational Tests for CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.2|6.2 CIT and the Unification of Physics]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.3|6.3 The Role of AI-Human Collaboration in Science]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.4|6.4 Why Local Mass Measurements Cannot Detect the Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.5|6.5 Observational Evidence for a Cosmic Influx: Accelerometer, Casimir Effect, Cloud Chamber, Van der Waals Forces, and the Human Body]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#6.6|6.6 The Human Sensor of Influx]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_6#Summary|Summary]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7|Chapter 7: Units, Dimensions, and Fundamental Constants in CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.1|7.1 Unit Conversions in CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.2|7.2 The Five Dimensions in CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3|7.3 Derivation of Constants in CIT]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.1|7.3.1 Gravitational Constant (G)]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.2|7.3.2 κ_CIT – Planetary Structuring Constant]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.3|7.3.3 Einsteinian Coupling Constant κ]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.4|7.3.4  Alignment Between ACT Observations and CIT Predictions]]
*** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.3.5|7.3.5 Updated CIT Jeans Mass Concept]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.4|7.4 Conclusion]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_7#7.5|7.5 Overview of Important Constants]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8|Chapter 8: Supporting Research, References, and Multimedia]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.1|8.1 Articles Explaining CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.2|8.2 Comments and Contributions from ChatGPT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.3|8.3 Excel Files Supporting CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.4|8.4 Other Articles and Websites]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.5|8.5 Videos Supporting CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.6|8.6 Videos Related to CIT]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_8#8.7|8.7 Selected Responses from ChatGPT]]
 
[[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9|Chapter 9: Genesis of the Cosmic Influx Theory]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.1|9.1 Early Insights and Thought Experiments]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.2|9.2 Connecting with Existing Theories]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.3|9.3 Mathematical Exploration and Key Discoveries]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.4|9.4 Challenges and the Scientific Landscape]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.5|9.5 The Role of AI in Theory Development]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_9#9.6|9.6 Conclusion and Future Directions]]
 
* [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10|Chapter 10: Feeling the Influx — A New Point of Observation]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.1|10.1 The Quiet Moment in Bed]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.2|10.2 The Accelerometer Confirms It]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.3|10.3 Falling Raindrops — The Influx Made Visible]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.4|10.4 From Concept to Realization]]
** [[User:Ruud_Loeffen/Cosmic_Influx_Theory(3)/Chapter_10#10.5|10.5 A Universal Gesture of Reception]]
 
----
'''Navigation:'''  
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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        <username>Dc.samizdat</username>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="87475" sha1="hgsihc6m1ap3g9sfe3yxpxvjnauolz1" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords  form the ''edge polygon'' of the 16-cell {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the characteristic rotation of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted 3 times, then bent into a circle in the fourth dimension. Each rung is a tesseract edge.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two distinct ways we can rotate the 24-cell isoclinically in invariant planes containing 24-cell edges, called the ''characteristic left rotation'' and the ''characteristic right rotation'', respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''characteristic left rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>hgsihc6m1ap3g9sfe3yxpxvjnauolz1</sha1>
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      <id>2818374</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="87487" sha1="c0l5ftqmtz9386335x18lfwpf2rkwxq" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords  form the ''edge polygon'' of the 16-cell {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the characteristic rotation of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted 3 times, and bent into a circle in the fourth dimension. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two distinct ways we can rotate the 24-cell isoclinically in invariant planes containing 24-cell edges, called the ''characteristic left rotation'' and the ''characteristic right rotation'', respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''characteristic left rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="87489" sha1="ozlmjko3b38pwqjismrv1f29llw23v4" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords  form the ''edge polygon'' of the 16-cell {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the characteristic rotation of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted 3 times, and bent into a circle in the fourth dimension. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two distinct ways we can rotate the 24-cell isoclinically in invariant planes containing 24-cell edges, called the ''characteristic left rotation'' and the ''characteristic right rotation'', respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''characteristic left rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>ozlmjko3b38pwqjismrv1f29llw23v4</sha1>
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      <id>2818376</id>
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      <text bytes="87517" sha1="pqsrhew78rzb0j04y94yzef1qyyhtb6" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords  form the ''edge polygon'' of the 16-cell {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the characteristic rotation of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two distinct ways we can rotate the 24-cell isoclinically in invariant planes containing 24-cell edges, called the ''characteristic left rotation'' and the ''characteristic right rotation'', respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''characteristic left rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>pqsrhew78rzb0j04y94yzef1qyyhtb6</sha1>
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      <id>2818388</id>
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      <text bytes="87578" sha1="d7bcopu4xkcdfqxf8akq771zqjm3oou" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords  form the ''edge polygon'' of the 16-cell {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two chiral ways we can rotate the 24-cell isoclinically in invariant great hexagon planes containing its edges, called the ''great hexagon left rotation'' and the ''great hexagon right rotation,'' respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon left rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the  ''great hexagon right rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>d7bcopu4xkcdfqxf8akq771zqjm3oou</sha1>
    </revision>
  </page>
  <page>
    <title>Linked-Open-Exhibition-Exercise</title>
    <ns>0</ns>
    <id>329922</id>
    <revision>
      <id>2818359</id>
      <parentid>2818341</parentid>
      <timestamp>2026-07-15T15:30:20Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818359</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13042" sha1="8v40xnj6umpzj605kkorziuytvicfe1" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Exhibition: Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Graph: https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum, date, etc.

For example: https://www.wikidata.org/wiki/Q138547468 See Table 1 for minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries.  
* View the results of the exhibition record in the Wikidat Query Service results link, which shows all of your entries. 
* As a timeline: https://w.wiki/J8NJ, and 
* As a graph: https://w.wiki/J8aS.
* Review exhibition entries. Cover topics raised by creating an LOD entry: -  Wikidata basics: 

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

Step-by-step guide:

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung 
#** https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&amp;currentPosition=1
#** https://www.sprengel-museum.de/ausstellungen/archiv
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>8v40xnj6umpzj605kkorziuytvicfe1</sha1>
    </revision>
    <revision>
      <id>2818361</id>
      <parentid>2818359</parentid>
      <timestamp>2026-07-15T15:39:42Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818361</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13130" sha1="oortuop0ksy1it57ls5mzo3kh5imzd9" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: Figure 1 - https://www.wikidata.org/wiki/Q138572982  and see Table 1 for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries.  
* View the results of the exhibition record in the Wikidat Query Service results link, which shows all of your entries. 
* As a timeline: https://w.wiki/J8NJ, and 
* As a graph: https://w.wiki/J8aS.
* Review exhibition entries. Cover topics raised by creating an LOD entry: -  Wikidata basics: 

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

Step-by-step guide:

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung 
#** https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&amp;currentPosition=1
#** https://www.sprengel-museum.de/ausstellungen/archiv
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>oortuop0ksy1it57ls5mzo3kh5imzd9</sha1>
    </revision>
    <revision>
      <id>2818362</id>
      <parentid>2818361</parentid>
      <timestamp>2026-07-15T15:42:00Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818362</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13247" sha1="ma5ryh5nkosdhdb7d4k71fq2oyzb74g" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries.  
* View the results of the exhibition record in the Wikidat Query Service results link, which shows all of your entries. 
* As a timeline: https://w.wiki/J8NJ, and 
* As a graph: https://w.wiki/J8aS.
* Review exhibition entries. Cover topics raised by creating an LOD entry: -  Wikidata basics: 

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

Step-by-step guide:

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung 
#** https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&amp;currentPosition=1
#** https://www.sprengel-museum.de/ausstellungen/archiv
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>ma5ryh5nkosdhdb7d4k71fq2oyzb74g</sha1>
    </revision>
    <revision>
      <id>2818363</id>
      <parentid>2818362</parentid>
      <timestamp>2026-07-15T15:48:25Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818363</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13217" sha1="3ge8hvqtpppb942ha6z9jly4dh63z9t" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries - example Figure 1,
** as a timeline: https://w.wiki/J8NJ  - example Figure 2. and
** as a graph: https://w.wiki/J8aS - example Figure 3.
* Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics:

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

==== Step-by-step guide: ====

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung 
#** https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&amp;currentPosition=1
#** https://www.sprengel-museum.de/ausstellungen/archiv
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>3ge8hvqtpppb942ha6z9jly4dh63z9t</sha1>
    </revision>
    <revision>
      <id>2818364</id>
      <parentid>2818363</parentid>
      <timestamp>2026-07-15T15:50:06Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818364</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13259" sha1="omyj5xqtw3iy9118zdkobb7j9y976hi" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries - example Figure 1,
** as a timeline: https://w.wiki/J8NJ  - example Figure 2. and
** as a graph: https://w.wiki/J8aS - example Figure 3.
* Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics:

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

==== Step-by-step guide: ====

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung.  
#** https://portal.dnb.de/opac/showFullRecord?currentResultId=sprengel+and+museum+and+ausstellung%26any&amp;currentPosition=1
#** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026)
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised (offline July 2026)
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>omyj5xqtw3iy9118zdkobb7j9y976hi</sha1>
    </revision>
    <revision>
      <id>2818365</id>
      <parentid>2818364</parentid>
      <timestamp>2026-07-15T15:51:20Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818365</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13237" sha1="3z2rp9k2xiuvapivn0a9d8tib55nkpv" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries - example Figure 1,
** as a timeline: https://w.wiki/J8NJ  - example Figure 2. and
** as a graph: https://w.wiki/J8aS - example Figure 3.
* Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics:

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

==== Step-by-step guide: ====

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung.  
#** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&amp;cqlMode=true 
#** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026)
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised (offline July 2026)
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>3z2rp9k2xiuvapivn0a9d8tib55nkpv</sha1>
    </revision>
    <revision>
      <id>2818366</id>
      <parentid>2818365</parentid>
      <timestamp>2026-07-15T15:54:17Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818366</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13353" sha1="ogsiols0dv91b798hdybzj86ga0wq8o" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries - example Figure 1,
** as a timeline: https://w.wiki/J8NJ  - example Figure 2. and
** as a graph: https://w.wiki/J8aS - example Figure 3.
* Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics:

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

==== Step-by-step guide: ====

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records.  
#** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&amp;cqlMode=true 
#** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026)
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised (offline July 2026)
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>ogsiols0dv91b798hdybzj86ga0wq8o</sha1>
    </revision>
    <revision>
      <id>2818372</id>
      <parentid>2818366</parentid>
      <timestamp>2026-07-15T16:25:53Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818372</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13456" sha1="j8q4se66zfeaf14sa23jlym6aud45wo" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries - example Figure 1,
** as a timeline: https://w.wiki/J8NJ  - example Figure 2. and
** as a graph: https://w.wiki/J8aS - example Figure 3.
* Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics:

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

==== Step-by-step guide: ====

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records.  
#** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&amp;cqlMode=true 
#** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026)
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised (offline July 2026)
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>j8q4se66zfeaf14sa23jlym6aud45wo</sha1>
    </revision>
    <revision>
      <id>2818373</id>
      <parentid>2818372</parentid>
      <timestamp>2026-07-15T16:28:18Z</timestamp>
      <contributor>
        <username>Mrchristian</username>
        <id>281704</id>
      </contributor>
      <comment>/* 1: Complete the Wikidata entry for a Sprengel Museum exhibition */</comment>
      <origin>2818373</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13560" sha1="5y1qu5ju4by0qa843amsy2zi2aeaz5j" xml:space="preserve">
Linked Open Exhibitions (Prototype): https://nfdi4culture.github.io/linked-open-exhibition/

Back to main course: [[BIM-126-02-Data-Science-Linked-Open-Exhibition]]

DE version - see language switcher - top right.

Tasks:

# Complete the Wikidata entry for a Sprengel Museum exhibition
# Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype 
# Adding Data Model mapping to standards to forked repository
# Adding SPARQL Query network diagram to forked repository
# Adding ORCID ID to forked repository 
# AI LLMs:
## Agentic coding: VSCode Copilot exercise
## Document AI LLM use with list of use, pro and cons, and attribution
# Completion of project section of Linked Open Exhibitions 
## The three sections:
### Wikidata Exhibition entries
### DNB (Library metadata) entries sorting
### Exhibition catalogue scan - Text and Data Mining

---

== 1: Complete the Wikidata entry for a Sprengel Museum exhibition ==
[[File:Graph of exhibition 2026 06 02.png|alt=Exhibition: Das Bild ist, was es tut|none|frame|Figure 1: Exhibition - Das Bild ist, was es tut https://www.wikidata.org/wiki/Q138572982 used here in the prototype https://mrchristian.github.io/prototype/wikidata-item.html]]
[[File:Timeline 2026 06 02.jpg|alt=Timeline|left|frame|Figure 2: Timeline: https://w.wiki/J8NJ]]
[[File:Network 2026 06 02.jpg|alt=Graph|left|frame|Figure 3: Graph https://w.wiki/J8aS]]This exercise covers:

* How to record minimal information for an exhibition in Wikidata as Linked Open Data. Title, museum/venue, date, etc.

For example: '''''Figure 1: Exhibition - Das Bild ist, was es tut''''' - https://www.wikidata.org/wiki/Q138572982  and see '''''Table 1: Minimal data entries for an exhibition (Add all 9 items)''''' for a list of the nine minimal data entries for an exhibition.

* View the results collected for the exhibition record:
** In the Query Service results link that shows all of your entries - example Figure 1,
** as a timeline: https://w.wiki/J8NJ  - example Figure 2. and
** as a graph: https://w.wiki/J8aS - example Figure 3.
* Review exhibition information entries covering topics raised by creating a Linked Open Data entry and Wikidata basics:

# Wikidata good practice
# Consulting schemas
# Importance of review
# Using GitHub Issues
# Comparing available data – before and after

==== Step-by-step guide: ====

==== A. Creating the exhibition entry in Wikidata. ====
# Login to Wikidata: https://www.wikidata.org/ If you missed how to add items or edit Wikidata see: https://de.wikiversity.org/wiki/Handbuch_Wikidata use this example as a reference for what you need to create https://www.wikidata.org/wiki/Q138572982   
# Have the source information for an exhibition at hand to make a data entry, here you can find a list of exhibitions. Students we assigned exibitions and you can find your allocation [https://tib.cloud/s/fncf8W6pXs8qgiq here] (password needed - if you need an allocation or have a question contact: Simon Worthington simon.worthington@tib.eu), e.g.,
#* '''NOTE:''' Sprengel Museum website is offline - if you need more info about your exhibition use the DNB site to search your exhibition name, use keywords like Sprengel Museum Ausellung. If you have not made an exhibition entry in Wikidata use any of the exhibitions you find listed in the DNB records.  
#** https://portal.dnb.de/opac/simpleSearch?query=sprengel+and+museum+and+ausstellung&amp;cqlMode=true 
#** https://www.sprengel-museum.de/ausstellungen/archiv (offline July 2026)
#** https://www.sprengel-museum.de/besuch?view=article&amp;id=65:publikationen&amp;catid=2:uncategorised (offline July 2026)
# Check there is no existing entry for the exhibition is on Wikidata. Use the search function. 
# Create an item or edit an existing item.
#* Note: Check which language you are using. We will be adding Deutsch and English entries (starting with Deutsch).
# Create the following data entries in Wikidata, see below: Table 1: ''Minimal data entries for an exhibition.''
# Review exhibition Wikidata entries. Review is carried out by using three questions. Add comments if needed, corrections can be made. Results and notes can be added to the Discussion Page of the entry, e.g., 
#* All entries present [   ]
#* All entries correct [   ]
#* Entries are in Deutsch and English – within reason [   ]
# References can be added: Source URLs, date accessed

===== ''Table'' ''1: Minimal data entries for an exhibition (Add all 9 items)'' =====
{| class=&quot;wikitable&quot;
| colspan=&quot;7&quot; |'''Fields used to make an exhibition entry. See example: https://www.wikidata.org/wiki/Q138547468'''
|-
|A
|Label
| colspan=&quot;5&quot; |Note: Keep short. Use title from exhibition
|-
|B
|Description
| colspan=&quot;5&quot; |Note: Use to differentiate from other entries. Follow this example: Gabriela Jolowicz Holzschnitte Ausstellung im Sprengel Museum, Hannover, 2026
|-
|
|'''Property (P) and Item (Q)'''
|'''URI'''
|'''DE'''
|'''EN'''
|'''Add'''
|'''Note'''
|-
|1
|P31
|https://www.wikidata.org/wiki/Property:P31
|ist ein(e)
|instance of
|Q464980
|Add item
|-
|2
|Q464980
|https://www.wikidata.org/wiki/Q464980
|Ausstellung
|Exhibition
|
|(Used above)
|-
|3
|P1476
|https://www.wikidata.org/wiki/Property:P1476
|Titel
|Title
|Title
|Plain text
|-
|4
|P276
|https://www.wikidata.org/wiki/Property:P276
|Ort
|Location
|Sprengel Museum Hannover Q510144
|Add item
|-
|5
|P580
|https://www.wikidata.org/wiki/Property:P580
|Startzeitpunkt
|Start time
|Date
|YYYY-MM-DD
|-
|6
|P582
|https://www.wikidata.org/wiki/Property:P582
|Endzeitpunkt
|End time
|Date
|YYYY-MM-DD
|-
|7
|P1640
|https://www.wikidata.org/wiki/Property:P1640
|Kurator
|Curator
|Person
|Add item (if don't exists will need to create/can omit at present)
|-
|8
|P710
|https://www.wikidata.org/wiki/Property:P710
|Teilnehmer
|Participant
|Person (the artist)
|Add item (if don't exists will need to create/can omit at present)
|-
|9
|P856
|https://www.wikidata.org/wiki/Property:P856
|offizielle Website
|Official website
|URL
|URL
|}

Task #1 complete!

---

== 2. Completion of the GitHub task of forking repository and publishing Wikidata entry ==
[[File:Wikidata 2026 06 02.jpg|left|thumb]]
Completion of the GitHub task of forking repository and publishing Wikidata entry https://github.com/mrchristian/prototype or https://github.com/NFDI4Culture/prototype-linkedOE

Tools: Quarto, GitHub, VS Code, Jupyter Notebooks, Codespace if needed, copilot: Agentic Coding)

'''Requirements'''

# A laptop or computer where you can install VScode
# You will need 2FA on your mobile (optional)
# Create a GitHub account
# Install VScode
# Connect Github account to VScode
# Create GitHub reposoitory

'''Fork the following repository:''' https://github.com/mrchristian/prototype

Create a page for the quarto project that retrieves the data used for thie Wikidata item and renders it as  professional webpage ''&lt;Insert your exhibition here – or use this one&gt;''  https://www.wikidata.org/wiki/Q138547468 The approach should create a SPARQL query for the data and then render this as HTML using a Jupyter Notebook.

All entries: https://tib.cloud/s/fncf8W6pXs8qgiq (needs password)

===== Tasks =====

* Change exhibition - manual
* Run Jupyter Notebook 
* Run and preview Quarto 
* Publish to your GitHub Pages 

===== Step-by-step =====

====== Part one: Working environment ======
'''''NOTE: If you are having problems running locally then use the Codespace online option.'''''

# Create GitHub account - https://github.com/
# Have 2FA available - usually on mobile (Google authenticator) (optional)
# Install VSCode - https://code.visualstudio.com/download
# Install GitHub Desktop - https://desktop.github.com/download/
# Add Github account when prompted, use 2FA

====== Step two: The prototype ======
# Fork the repository:  https://github.com/mrchristian/prototype
# If working locally continue - if using Codespace - launch Codespace (see below and then continue)
# Test Quarto in the Terminal: 
## &lt;code&gt;quarto check&lt;/code&gt; 
## &lt;code&gt;quarto render&lt;/code&gt;
## &lt;code&gt;quarto preview&lt;/code&gt; (control C - to stop)
# If not working run Quarto from Agent
# Change Wikidata exhibition in Notebook
# Run notebook
# Run &lt;code&gt;quarto render&lt;/code&gt; &lt;code&gt;quarto preview&lt;/code&gt;
# Save all (or use auto save)
# Git: Message, Commit and Push
# On GitHub.com your repository
## Turn on Pages: GitHub Actions
## Code: About cog - Click use my GitHub Pages
## Actions tab: Publish Quarto Project
# ENDE - Rinse repeat  :-)

===== Codespace option: =====
Videolink: https://tib.cloud/s/LDtkN6QsdFkGGR6 (10 Minuten Zeit)

Codespace is an online Virtual Machine which can be launched from GitHub.

The repository includes a Dev Container configuration so you can work entirely in the browser without installing anything locally.

# On the repository page on GitHub, click Code → Codespaces → Create codespace on main.
# Wait for the container to build — Python packages from &lt;code&gt;requirements.txt&lt;/code&gt; are installed automatically - about 5 minu3. Adding Data Model mapping to standards to forked repositorytes.
# Once everything is installed the Codespace can be used anytime. It automatically shutsdown when left alone and can be restarted any time.
# Work done in Codespace must be pushed back to the repository.
# If Codespace is not used for 28 days the Codespace is deleted.

---

== 3. Adding Data Model mapping to standards to forked repository ==


Four data models have been made for the project. The data models have been mapped to sector data schemas: Wikidata; CIDOC CRM; and Wikibase4Research. See: https://nfdi4culture.github.io/linked-open-exhibition/

Choose data models that relate to your Wikidata entry.

Data models are:

* Artist Data Model
* Exhibition Data Model
* DNB Catalogue Data Model
* Item in Exhibition Data Model

Copy the .qmd files used over to your repository and insert them in your Quarto YAML file _quarto.yml like so:

website:

  &lt;code&gt;title: &quot;BIM Prototype 02&quot;&lt;/code&gt;

&lt;code&gt;  navbar:&lt;/code&gt;

&lt;code&gt;    left:&lt;/code&gt;

&lt;code&gt;          - href: artist-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Artist Data Model&lt;/code&gt;

&lt;code&gt;          - href: exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Exhibition Data Model&lt;/code&gt;

&lt;code&gt;          - href: dnb-catalogue-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: DNB Catalogue Data Model&lt;/code&gt;

&lt;code&gt;          - href: item-in-exhibition-datamodel.qmd&lt;/code&gt;

&lt;code&gt;            text: Item in Exhibition Data Model&lt;/code&gt;

== 4. Adding SPARQL Query network diagram to forked repository ==
'''Visualizing the Wikidata Item as a Graph'''

https://github.com/mrchristian/prototype

The following cell renders a graph visualization of the relationships for the selected Wikidata item. This helps to see how the item is connected to other entities via its properties.

In your Quarto project the Jupyter Lab Notebook will render the graph automatically&lt;blockquote&gt;wikidata-item.ipynb&lt;/blockquote&gt;

# In cell 2 input your Wikidata QID, e.g., item_id = &quot;Q138572982&quot;
# Click Run All at the top of the Jupyter Lab Notebook. The graph will then render.
# Once rendered you can preview your Quarto publication. Then render Quarto and push to GitHub.

[[File:Graph of exhibition 2026 06 02.png|alt=Graph of exhibition 2026 06 02|frame|center]]

== 5. Adding ORCID ID to forked repository ==
'''ORCID''' (Open Researcher and Contributor ID) is a free, unique, persistent digital identifier that distinguishes you from other researchers. It’s a 16-digit identifier in the format: &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;

See full details here: https://nfdi4culture.github.io/linked-open-exhibition/

==== How to Get an ORCID ====

# '''Visit''': orcid.org
# '''Click''': “Sign in” → “Register for an ORCID iD”
# '''Provide''':
#* Given name and family name
#* Email address
#* Password
#* Affiliation (optional but recommended)
# '''Verify''': Confirm your email address
# '''Complete''': Your 16-digit ORCID will be generated immediately

==== Add to Quarto ====
_quarto.yml

&lt;code&gt;project''':'''&lt;/code&gt;  

&lt;code&gt;type''':''' website&lt;/code&gt;  

&lt;code&gt;title''':''' &quot;My Project&quot;&lt;/code&gt; 

&lt;code&gt;metadata''':'''&lt;/code&gt;  

&lt;code&gt;author''':'''&lt;/code&gt;    

&lt;code&gt;'''-''' name''':''' Jane Researcher&lt;/code&gt;      

&lt;code&gt;- orcid''':''' 0000-0002-1234-5678&lt;/code&gt;

==== Add to CFF Citation File Format ====
This will make your repository citable on GitHub.

Ask Copilot to generate a CFF file in the top level of your repository and add your ORCID.

== 6. AI LLM: Agentic assistent/coding ==
For the project Copilot is used in VSCode for limited agentic coding.

A GitHub account is needed to use Copilot and the user must agree to TnCs. A free account will be used.

Once logged into VSCode, see the menu item: View &gt; Chat to access the AI on the right. Use Agent mode.

==== Exercises: ====

# Ask the agent to create a CFF file and add you ORCID ID. Promt: create a CFF file and add my ORCID ID &lt;code&gt;XXXX-XXXX-XXXX-XXXX&lt;/code&gt;
# Ask the agent to create a .QMD file describing your exhibition, give it Wikidata QID, and ask it to add the page to your Quarto project.
# Ask the agent to render and push your Auarto project to Git.

==== Request an account with KISSKI this can be used later for code and questions. ====
„KI-Servicezentrum für Sensible und Kritische Infrastrukturen“ (KISSKI) can be used for unmetered ChatGPT5 &lt;nowiki&gt;https://kisski.gwdg.de/leistungen/2-02-llm-service/&lt;/nowiki&gt; | &lt;nowiki&gt;https://chat-ai.academiccloud.de/chat&lt;/nowiki&gt;</text>
      <sha1>5y1qu5ju4by0qa843amsy2zi2aeaz5j</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Inactivity policy</title>
    <ns>4</ns>
    <id>329965</id>
    <revision>
      <id>2818360</id>
      <parentid>2812850</parentid>
      <timestamp>2026-07-15T15:36:37Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <minor />
      <comment>clarify</comment>
      <origin>2818360</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="995" sha1="qb75xbe6sbzx45jgjpfizwg1xda77rd" xml:space="preserve">{{Draft}}
Wikiversity [[Wikiversity:Support staff|support staff]] ([[Wikiversity:Curatorship|curators]], [[Wikiversity:Custodianship|custodians]] and [[Wikiversity:Bureaucratship|bureaucrats]]) are considered inactive if they have made no edits and have logged no actions within one year.

== Process ==
# A notice is sent to the [[Wikiversity:Colloquium|Colloquium]], listing the inactive support staff.
# An inactive support staff member is notified on their talk page, explaining that their rights may be removed. {{tlx|Inactive curator}} may be used for the notification.
# If no response is received from the inactive support staff member within two weeks, whether through their talk page or the Colloquium, the rights will be removed. A custodian will remove the curator permission locally, while a steward will be asked to remove bureaucrat and/or custodian permissions (per [[m:Steward requests/Permissions#Removal of access]]).

[[Category:Wikiversity administration|Inactivity policy]]</text>
      <sha1>qb75xbe6sbzx45jgjpfizwg1xda77rd</sha1>
    </revision>
  </page>
  <page>
    <title>Igbo culture</title>
    <ns>0</ns>
    <id>330007</id>
    <revision>
      <id>2818371</id>
      <parentid>2816611</parentid>
      <timestamp>2026-07-15T16:08:50Z</timestamp>
      <contributor>
        <username>Rabbi Mendl</username>
        <id>772449</id>
      </contributor>
      <minor />
      <comment>([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:Igbo woman wearing Akwete obiakwa(double wrapper) with uweobi(blouse) and Ichafu headdress.jpg]] → [[File:Igbo Woman wearing gele.jpg]] [[c:COM:FR#FR2|Criterion 2]] (meaningless or ambiguous name) · The current filename incorrectly identifies the headwear as &quot;Ichafu&quot;. Reliable published sources identify this style as gele.</comment>
      <origin>2818371</origin>
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      <text bytes="22013" sha1="5kivfjjohsh49tbr2vezkgkz65ie08w" xml:space="preserve">{{contrib-creator|Wmbata}}
{{launch}}
{{course}}
{{humanities}}
== Introduction ==
'''Igbo culture''' are the customs, practices and traditions of the Igbo people of southeastern Nigeria. It consists of ancient practices known as ''Odinala'' ''ndi'' ''igbo'' as well as new concepts added into the Igbo culture either by cultural evolution or by outside influence. These customs and traditions includes the Igbo people's visual art, music and dance forms, as well as their attire, Food, cuisine and language dialects. Because of their various subgroups, the variety of their culture is heightened further.

== Learning Objectives ==
By reviewing this material, you should be able to:

* '''Identify the major traditional musical instruments and art forms of Ndị Igbo.'''
* '''Contrast the components of traditional Ndị Igbo cosmology.'''
* '''Detail the socio-economic functions of historical practices like traditional marriage, dressing, architecture, and the apprenticeship system.'''

== Module 1: Creative Arts and Expressive Traditions ==

=== Music ===
[[File:Udu.jpg|thumb|right|95px|Udu, an Igbo instrument]]
The Igbo people have a melodic and symphonic musical style. Instruments include Ọ̀pì  otherwise known as '''Oja''' a wind instrument similar to the flute, '''igba''', and '''ichaka'''.

Another popular musical form among Igbo people is highlife, which is a fusion of jazz and traditional music and widely popular in West Africa. The modern Igbo highlife is seen in the works of Prince Nico Mbarga, Dr Sir Warrior, Oliver De Coque, Bright Chimezie, Celestine Ukwu,Chief Osita Osadebe, And many others who are some of the greatest Igbo highlife musicians of the twentieth century. There are also other notable Igbo highlife artists, like the Mike Ejeagha, Paulson Kalu, Ali Chukwuma, Ozoemena Nwa Nsugbe.

=== Art ===
Igbo art is known for various types of masquerades, masks, outfits (symbolizing people), animals and abstract conceptions. Igbo art is also known for its bronze castings found in the town of Igbo Ukwu from the 9th century.

&lt;gallery widths=&quot;200&quot; heights=&quot;200&quot; mode=&quot;packed&quot;&gt;
File:Nigeria, igbo, maschera-elmo della società mmuo, xx secolo.jpg|Helmet-mask; 20th century; Indianapolis Museum of Art (USA)
File:Nigeria, igbo, figura femminile per un tempietto, xx secolo.jpg|Female figure for a small temple, 20th century; Indianapolis Museum of Art
File:Igbo brass anklet.jpg|Anklet beaten from a solid brass bar of the type worn by Igbo women. Now in the collection of Wolverhampton Art Gallery. The leg-tube extends approximately 7&amp;nbsp;cm each side of the 35&amp;nbsp;cm disc.
File:Bronze ceremonial vessel in form of a snail shell, 9th century, Igbo-Ukwu, Nigeria.JPG|Bronze ceremonial vessel in form of a snail shell; 9th century; from Igbo-Ukwu; Nigerian National Museum (Lagos, Nigeria)
File:Eze Onyiudo (2).jpg|Eze Onyiudo Masquerade Awka-Etiti
&lt;/gallery&gt;

=== Igbo masks and masquerades ===
There are two basic types of masquerades, visible and invisible. The visible masquerades are meant for the public. They often are more entertaining. Masks used offer a visual appeal for their shapes and forms. In these visible masquerades, performances of harassment, music, dance, and parodies are acted out (Oyeneke 25).

The invisible masquerades take place at night.  Sound is the main tool for them. The masquerader uses his voice to scream so it may be heard throughout the village. The masks used are usually fierce looking and their interpretation is only fully understood by the society's members.  These invisible masquerades call upon a silent village to strike fear in the hearts of those not initiated into their society.


== Module 2: Spiritual Beliefs and Cosmological Frameworks ==

=== Mythology ===
While today many Igbo people are Christian, the traditional ancient Igbo religion is known as Odinani. In the Igbo mythology, which is part of their ancient religion, the supreme God is called Chineke (&quot;the God of creation&quot;); Chineke created the world and everything in it and is associated with all things on Earth. To the ancient Igbo, the cosmos is divided into four complex parts:
 
* OKIKE (Creation)
* ALUSI (Supernatural Forces or Deities)
* MMUO (Spirit)
* UWA (World)

==== Alusi ====
[[File:Complex sculpture Nigeria BM Af1954 23 522 img02.jpg|thumb|alt=A photo of a complex wooden carving of animals, people and spirits laid on each other to about 2 meters in height|Complex wooden carving depicting images of power and daily life, such as horsemen, imported goods, military insignia, Europeans, rifles, wild beasts and masqueraders.]]'''Alusi''', also known as '''Arusi''' or '''Arushi''', are minor deities that are worshiped and served in Igbo mythology. There are a list of many different Alusi that exists within each community and each has its own purpose. When there is no longer need for the deity, it is returned to its source, through the help of a Chief Priest or Dibia, who is aware of the procedure and ensures that its done properly.

==== Mmuo ====
Mmuo simply means spirit. It is either a good and godly spirit (mmuo oma) or it is an evil spirit (mmuo ojo). For example, the Ogbanje spirit is seen as an evil spirit (mmuo ojo) and anyone possessed by this spirit is given spiritual attention. (Spiritual attention means a way of casting out the evil spirit through deliverance (Christian way) or through African Traditional Religion&amp;nbsp; (i.e. digging out his/her '''“iyi uwa”'''. the ATR way)). Ogbanje is an Igbo (Nigeria) term that means a repeater or someone who comes and departs.  Ogbanje is not a bad spirit in Igbo Cosmology. It is a word widely used to describe a kid or teenager who is claimed to die and be born repeatedly by the same person.

=== Osu caste system ===
Osu are a group of people whose ancestors were dedicated to serving in shrines and temples for the deities of the Igbo, and therefore were deemed property of the gods. Relationships and sometimes interactions with Osu were (and to this day, still are) in many cases, forbidden.

To this day being called an ''Osu'' remains a stigma that prevents people's progress and lifestyles.


== Module 3: Social Milestones and Economic Structures ==

=== Umuada ===
The married and unmarried daughters of a particular clan or village in [[wikipedia:ala Igbo|ala Igbo]].

While Igbo society is about tracing descent through the male, Ndi Umuada serve as a vital checks and balances in the society. They represent a collective authority that balances the political power held by the males. Their words and decisions are highly respected and are often final.

=== Yam ===
The yam is very important to the Igbo as it is their staple crop. There are celebrations such as the New yam festival  which are held Every August of Every year for the harvesting of the yam.

The New Yam festival  is celebrated annually to secure a good harvest of the staple crop.  The festival is practiced primarily in Nigeria and other countries in West Africa.

=== Traditional marriage ===
Marriages in Igbo community follow a multi-step process before the bride and groom are proclaimed husband and wife in accordance with local law and tradition.
[[File:Igba nkwu ceremony 04.jpg|alt=Igbo Traditional Marriage|thumb|Traditional Igbo Marriage Attire]]
The traditional marriage is known as &quot;Igbankwu Alumdi&quot; in Igbo land, or wine carrying, since it involves the bride serving up a cup of palm wine to her fiancé. Prior to the wedding, the groom must go to the bride's compound with his father before the Igbankwu day to get the bride's father's consent to marry his daughter. If the bride's father is late, in this case, the bride's brother, uncle or male relative fills in for the bride's late father, as applies to the groom. On the second visit, when kola nuts (oji Igbo) are offered, the two fathers must arrange a price for the bride. In most cases, the bride's price is just symbolic, in addition to other requirements like kola nuts, goats, wine, fowl and so on. Normally, it takes more than one evening until the bride price is agreed upon, after which a feast is served to both parents. When the bride price is paid, another evening is set aside for the ceremony.

During the ceremony, the bride's father fills a cup with palm wine and hands it over to the daughter. Accompanied by her brides maids known as umuagbo nwunye, she then searches for the groom among the crowd of wedding guests to offer him the drink. Once the drink is offered, the bride and groom dance to the bride's father. They kneel before him and he will give them his blessings. After that, the couple dances for a while before taking their seats, then refreshment takes place followed by presentation of gifts, at times a speech from the MC, and then closing prayer and departure.

=== Apprenticeship ===
The Igbo have a unique form of apprenticeship in which either a male family member or a community member will spend time (usually in their teens to their adulthood) with another family, when they work for them. After the time spent with the family, the head of the host household, who is usually the older man who brought the apprentice into his household, will establish  the apprentice by either setting up a business for him or giving money or tools by which to make a living.

This practice was exploited by Europeans, who used this practice as a way of trading in enslaved people. Olaudah Equiano, although stolen from his home, was an Igbo person who was forced into service to an African family.  He said that he felt part of the family, unlike later, when he was shipped to North America and enslaved in the Thirteen Colonies.

The Igbo apprenticeship system is called Imu Ahia or Igba Boy in Igboland. It became more prominent among the Igbos after the Nigerian civil war, in a quest to survive the £20 policy which was proposed by Obafemi Awolowo that only £20 be given to every Biafran citizen to survive on regardless of what they had in the bank before the war and the rest of the money were held by the Nigerian government.

Petty trade was one of the only ways to build back destroyed communities as well as farming, but then, farming required time that was not readily available at that moment. Essentially, most people went into trading.

This Imu-Ahia/Igba Boy model was simple, it works in such a way that business owners would take in younger boys which can be relative, sibling or non-relative from same region, house them and have them work as apprentices in business while learning how it works and the secrets of the business. After the allotted time for the training was reached, 5–8 years’ time, a little graduation ceremony would be held for the '''Nwa Boy''' (the person that learnt the trade). He would also be paid a lump sum for their services over the years, and the money will be used to start a business for the '''Nwa Boy'''.

=== Chieftaincy Title ===
[[File:Igbo ichi marks.jpg|thumb|An Igbo man with ''Ichi'' marks, a sign of rank as an Ozo]]
Highly accomplished men and women are admitted into their noble orders for people of title such as Ndi Ozo or Ndi Nze. These people receive insignia to show their stature. Membership is highly exclusive, and to qualify an individual need to be highly regarded and well-spoken of in the community.

=== Kola nut (Ọjị) ===
[[File:Kola nut.jpg|alt=Kola nut|thumb|Kola nut]]
Kola nut  occupies a unique position in the cultural life of Igbo people. Ọjị is the first thing served to any visitor in an Igbo home. Ọjị is served before an important function begins, be it marriage ceremony, settlement of family disputes or entering into any type of agreement. Ọjị is traditionally broken into pieces by hand, and if the Kola nut breaks into 3 pieces a special celebration is arranged.


== Module 4: Material, Material Culture, Architecture, and Systems of Time ==

=== Igbo Architecture ===
Igbo architecture refers to the architectural styles and building traditions of the Igbo people. The architectural style is closely tied to the Igbo society's culture, beliefs, and social structure. While the architectural style has evolved, traditional Igbo architecture shares some common characteristics such as:

'''Compound layout'''- Igbo architectural traditions often revolve around the concept of a compound which is characterized by an enclosed area encompassing multiple family residences, open central courtyards, verandas, and auxiliary structures. These compounds are meticulously planned and sometimes paved with flat stones to foster communal living and facilitate familial engagements. Additionally, certain compounds feature unique elements like Impluvium houses, Gardens, Moats, and water wells demonstrating the diversity within Igbo architectural practices.

'''Ventilation''' - Igbo architecture integrates strategic placement of openings in buildings to promote cross-ventilation, aiding in regulating indoor temperatures. Employing expansive openings facilitates air circulation, ensuring occupant comfort. Depending on the area with high temperatures and humidity, evaporation of sweat becomes challenging; however, airflow aids this process, enhancing comfort. Moreover, construction practices involve thick walls, thatched roofs, and raised foundations to mitigate environmental challenges. The thick walls maintain cooler interiors in hot weather and warmth during rainy seasons. Thatched roofs provide insulation from direct sunlight, offering shade and contributing to thermal comfort.

'''Shrines and Sacred Spaces'''- Igbo architecture often includes designated spaces in compounds or community areas for ancestral shrines/temples and secret society meeting houses. These spaces are considered sacred and are an essential part of Igbo cultural and religious practices. These sacred structures may vary in design, ranging from simple open-air spaces to more elaborate structures with specific architectural features.

'''Decorative Elements -''' Traditional Igbo architecture often incorporates decorative elements, including painted designs on walls such as [[Uli (design)|uli]], carved wooden door frames, and intricate patterns on ceilings. These decorations may have symbolic or religious significance.

=== Traditional attire ===
Igbo traditional attire varies across regions of Southeastern and south south Nigeria with various cultural significance. 

'''&lt;big&gt;Men&lt;/big&gt;'''

For men, common garments include ''uwe mwuda'' or ''afe ntutu'' ( robe) or ''efe elu'', a basic shirt paired with underneath wrappers or skirts complemented by the ''okpu ozo'' (the feathered red cap), or ''Okpu aji'' (woolen cap), ''ofo'', ''mkpara'' (staff) and ''Akupe'' (handfans) for ceremonial or titled occasions while  loin clothes or waist wrappers were usually worn as casual wears or basic activities like hunting or farming. 

&lt;big&gt;'''Women'''&lt;/big&gt;

[[File:Igbo Woman wearing gele.jpg|thumb|Igbo women’s traditional attire showing Obiakwa(matching double wrappers) made of Akwete George, uweobi (blouse with puffed sleeves) and stiff Ichafu (headdress)]]

Traditional Igbo women's attire comprises many regional and age-based (''Ụmụagbọ'') variant, including the Obiakwa pair of matching wrappers, Uweobi (blouse), and Ịchafụ̀ (head-tie), an elaborate and voluminous headdress traditionally worn by mature women. 

[[File:Igbo_woman_wearing_Isiagu_obiakwa_maiden_attire,_aka_olu(coral_beads)_ngala(head_beads)_and_nza(horsetail).jpg|thumb|A short Obiakwa (wrapper-style) ensemble paired with a fitted blouse, complemented by nza (horsewhisk), ngala (head beads), and other beaded accessories. The textile features  Isiagu motif]]

Younger women may wear shorter Obiakwa wrappers paired with a tubular Uweobi blouse. Traditional adornments include ''aka olu'' (coral beads), ''ngala'' (head beads), and ''mgbaji'' (waist beads), often complemented by other ceremonial accessories like the ''akupe'' (hand fans) and ''nza'' (horsetail whisks). Traditional attire and adornment form many parts of Igbo cultural expressions associated with age, status, ceremony, and identity.

'''&lt;u&gt;Obiakwa&lt;/u&gt;''' 

The ''Obiakwa'' is a traditional women's double- wrapper attire unique to Igbo weaving traditions such as Akwete cloth. It consists of a pair of matching wrappers Descriptions of Akwete weaving note that such wrapper sets were engineered during the weaving process to be worn together.  It is therefore sold in matching pairs.  These wrappers are standardly paired with a blouse called uweobi and the Ichafu headdress. Younger women usually wear shorter ''obiakwa'' waist wrapper sets combined with fitted or tubular blouses or wrappers . These clothings are also complemented with ''ngala'', ''mbaji'' and ''aka'' (beaded accessories) as well ''uli'' body arts. In some regions, The Uli body art was also used to decorate both men and women in the form of lines forming patterns and shapes on the body. 

'''&lt;u&gt;Blouse&lt;/u&gt;''' 

To complete the silhouette, double wrappers(obiakwa) are paired with a Blouse (or ''uweobi''), a traditional fitted blouse. Short ''obiakwa'' styles are usually paired with tubular blouses. 


'''&lt;u&gt;Ichafu&lt;/u&gt;'''

[[File:Igbo_woman_wearing_Joojii_and_Ichafu._Igbo_regality.jpg|thumb|An Igbo woman dressed in traditional attire consisting of a white puff-sleeve Uweobi blouse, a red and gold double George wrapper, and a stiff, elaborately structured Ichafu headdress, accessorized with pearl jewelry.]]

''Ichafu'' is an elaborate head-tie or headdress worn by Igbo women, especially for church services, ceremonies and other social occasions.  It forms part of a broader clothing ensemble that may include wrappers, blouses and jewellery, and is typically tied in volumnious elevated layered styles with large folds and pleats rising above the head. Ichafu is tied with various textiles including synthetic damask, brocade, Akwete and George fabric, which gives it the stiff and highly elaborated look. In Ogadinma, published by Granta, women were described wearing colourful blouses with “expensive ichafu” tied “in layers and pleats until the scarves were piled atop their heads like large plants”. Other dialectical variations for Ichafu is ''Akwaisi'', ''ulari'', ''unari'', ''nsu n'isi'', ''ufu isi'', ''asusu isi'', ''nchafu isi,'' ''Akishi''.   

'''&lt;u&gt;Textiles&lt;/u&gt;'''

Textiles commonly used across Igbo land include ''Isiagu'' (often patterned with the tiger or Lion head motifs), ''Akwete'' and ''Akwaocha'' handwoven clothes, and richly patterned George wrappers.

[[File:Little world, Aichi prefecture - African plaza - Hat of a vassal - Ìgbo people in Nigeria - Collected in 2006.jpg|110px|thumb|left|A traditional Igbo hat made entirely from [[wool]].]]Women carried their babies on their backs with a strip of clothing binding the two with a knot at her chest. This baby carrying technique was and still is practiced by many people groups across Africa, including the Igbo. This method has been modernized in the form of the child carrier.  Both men and women wore wrappers.[[File:Igba nkwu ceremony 07.jpg|thumb|Igba nkwu, Igbo traditional marriage]]

[[File:Igbo Traditional marriage.jpg|thumb|Igbo Traditional Marriage attire]]

=== Calendar (Iguafo Igbo) ===
In the traditional Igbo calendar, a week has 4 days  (''Eke'', ''Orie'', ''Afọ'', ''Nkwọ''), seven weeks make one month, a month has 28 days and there are 13 months in a year. In the last month, an extra day is added. The names of the days have their roots in the mythology of the Kingdom of Nri. It was believed that Eri, the sky-born founder of the Nri kingdom, had gone on a journey to discover the mystery of time. On his journey he had saluted and counted the four days by the names of the spirits that governed them, and so the names of the spirits (''eke'', ''orie'', ''afọ'' and ''Nkwo'') became the days of the week.

{{col-begin}}{{col-2}}
{| class=&quot;wikitable&quot;
!No. || Months (Ọnwa) || Gregorian equivalent
|-
|1 || '''Ọnwa Mbụ''' || (3rd week of February)
|-
|2 || '''Ọnwa Abụa''' || (March)
|-
|3 || '''Ọnwa Ife Eke''' || (April)
|-
|4 || '''Ọnwa Anọ''' || (May)
|-
|5 || '''Ọnwa Agwụ''' || (June)
|-
|6 || '''Ọnwa Ifejiọkụ''' || (July)
|-
|7 || '''Ọnwa Alọm Chi''' || (August to early September)
|-
|8 || '''Ọnwa Ilo Mmụọ''' || (Late September)
|-
|9 || '''Ọnwa Ana''' || (October)
|-
|10 || '''Ọnwa Okike''' || (Early November)
|-
|11 || '''Ọnwa Ajana''' || (Late November)
|-
|12 || '''Ọnwa Ede Ajana''' || (Late November to December)
|-
|13 || '''Ọnwa Ụzọ Alụsị''' || (January to early February)&lt;ref&gt;{{cite book |last1=Onwuejeogwu |first1=M. Angulu |title=An Igbo Civilization: Nri Kingdom &amp; Hegemony |date=1981 |publisher=Ethnographica |isbn=978-978-123-105-6 }}{{page needed|date=January 2024}}&lt;/ref&gt;&lt;ref&gt;{{cite web |url=http://www.free-press-release.com/news/200802/1204305180.html |title=Eze Nri - Igu-Aro Festival - 1008th AD |publisher=Free-Press-Release Inc. |date=February 29, 2008 |access-date=2010-04-06 |archive-date=2009-07-24 |archive-url=https://web.archive.org/web/20090724025818/http://www.free-press-release.com/news/200802/1204305180.html |url-status=dead }}&lt;/ref&gt;
|}

{{col-break}}
An example of a month: '''''Ọnwa Mbụ'''''

{| class=&quot;wikitable&quot;
|-
!Eke || Orie || Afọ || Nkwọ
|-
||||| 1 || 2
|-
|3 || 4 || 5 || 6
|-
|7 || 8 || 9 || 10
|-
|11 || 12 || 13 || 14
|-
|15 || 16 || 17 || 18
|-
|19 || 20 || 21 || 22
|-
|23 || 24 || 25 || 26
|-
|27 || 28||||
|}
{{col-end}}

==== Naming after market days ====
Newborn babies were sometimes named after the day of the week when born. This is no longer the fashion.  Names such as '''Mgbeke''' (maiden [born] on the day of Eke), Mgborie (maiden [born] on the Orie day) are commonly seen among the Igbo people. For males, '''Mgbe''' is replaced by '''Nwa''' or &lt;nowiki&gt;'''&lt;/nowiki&gt;Okoro&lt;nowiki&gt;'''&lt;/nowiki&gt;(Igbo: Child [of]). Examples of this are Solomon Okoronkwo and Nwankwo Kanu, two popular footballers.</text>
      <sha1>5kivfjjohsh49tbr2vezkgkz65ie08w</sha1>
    </revision>
  </page>
  <page>
    <title>Mandelbrot CLI: Renderer with Perturbation Theory</title>
    <ns>0</ns>
    <id>330398</id>
    <revision>
      <id>2818350</id>
      <parentid>2817573</parentid>
      <timestamp>2026-07-15T14:51:41Z</timestamp>
      <contributor>
        <username>Aokoroko</username>
        <id>2811879</id>
      </contributor>
      <comment>/* C++ Source Code */</comment>
      <origin>2818350</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10213" sha1="s57mck6yuj5v9ngi36gvlicdy4mamii" xml:space="preserve">== Introduction ==
This page contains the original C++ source code used to render high-precision fragments of the Mandelbrot set using perturbation theory and 8x8 Super-Sampling Anti-Aliasing (SSAA). Created by [[User:Aokoroko]].

== Key Features ==
* '''High-Precision Reference:''' The 5000-bit reference trajectory is computed exactly once per zoom layer.
* '''Hardware-Native Performance:''' Blazing-fast math for billions of pixels utilizing hardware-native double registers.
* When using double-precision floating-point numbers (on the order of 10⁻¹⁵), perturbation theory only allows you to zoom down to the '''10⁻³⁰⁸ level—no further.'''
* '''Innovative Algorithm:''' Revolutionary *Reference Reset to Zero* implementation.
* '''True 8x8 SSAA:''' Pristine, anti-aliased image quality with 64 independent samples per pixel.
* '''OpenMP Multi-threading:''' High-speed parallel computing to maximize CPU utilization.

== C++ Source Code ==

&lt;syntaxhighlight lang=&quot;cpp&quot;&gt;
#include &lt;iostream&gt;
#include &lt;fstream&gt;
#include &lt;vector&gt;
#include &lt;cmath&gt;
#include &lt;cstdint&gt;
#include &lt;string&gt;
#include &lt;atomic&gt;
#include &lt;omp.h&gt;
#include &lt;cstdio&gt;
#include &lt;iomanip&gt;
#include &lt;gmp.h&gt;
#include &lt;mpfr.h&gt;

using namespace std;

const double PI = 3.14159265358979323846;
const mpfr_prec_t MPFR_BITS = 5000;

#pragma pack(push, 1)
struct BMPHeader {
    uint16_t type{0x4D42};
    uint32_t size{0};
    uint16_t reserved1{0};
    uint16_t reserved2{0};
    uint32_t offBits{54};
    uint32_t structSize{40};
    int32_t  width{0};
    int32_t  height{0};
    uint16_t planes{1};
    uint16_t bitCount{24};
    uint32_t compression{0};
    uint32_t sizeImage{0};
    int32_t  xpelsPerMeter{2834};
    int32_t  ypelsPerMeter{2834};
    uint32_t clrUsed{0};
    uint32_t clrImportant{0};
};
#pragma pack(pop)

struct ComplexDouble {
    double re;
    double im;
};

int main() {
    string absc_str, ordi_str, size_str;
    absc_str = &quot;-1.99999543561201124623198345433951143502785679245726844745821388800402678499411681518036306219179273434395557574279985918047221291197081186140687781560831995&quot;; 
    ordi_str = &quot;-0.00000000000000000000000026198152173811047783694060060607013913873144250985383083459221663448338433592617272786772587281530484110756597337683912309313885172&quot;; 
    size_str = &quot;1.15e-119&quot;;
    
    const int targetW = 10000;
    const int targetH = 10000;
    const int scale = 8;
    const int rawW = targetW * scale;
    const int rawH = targetH * scale;
    const int frame = 200;
    
    cout &lt;&lt; &quot;Step 1: Calculating Reference Orbit using Perturbation...&quot; &lt;&lt; endl;
    mpfr_t rx, ry, zr, zi, zr2, zi2, tmp, sz, st;
    mpfr_inits2(MPFR_BITS, rx, ry, zr, zi, zr2, zi2, tmp, sz, st, NULL);
    mpfr_set_str(rx, absc_str.c_str(), 10, MPFR_RNDN);
    mpfr_set_str(ry, ordi_str.c_str(), 10, MPFR_RNDN);
    mpfr_set_str(sz, size_str.c_str(), 10, MPFR_RNDN);
    mpfr_div_ui(st, sz, rawW, MPFR_RNDN);
    double step_d = mpfr_get_d(st, MPFR_RNDN);
    
    double ref_rec_d = mpfr_get_d(rx, MPFR_RNDN);
    double ref_imc_d = mpfr_get_d(ry, MPFR_RNDN);
    
    vector&lt;ComplexDouble&gt; ref_orbit_double(50005);
    mpfr_set_ui(zr, 0, MPFR_RNDN);
    mpfr_set_ui(zi, 0, MPFR_RNDN);
    mpfr_set_ui(zr2, 0, MPFR_RNDN);
    mpfr_set_ui(zi2, 0, MPFR_RNDN);
    uint32_t ref_i = 0;
    bool escaped = false;
    
    while (ref_i &lt; 50000) {
        ref_orbit_double[ref_i].re = mpfr_get_d(zr, MPFR_RNDN);
        ref_orbit_double[ref_i].im = mpfr_get_d(zi, MPFR_RNDN);
        mpfr_mul(tmp, zr, zi, MPFR_RNDN);
        mpfr_mul_ui(zi, tmp, 2, MPFR_RNDN);
        mpfr_add(zi, zi, ry, MPFR_RNDN);
        mpfr_sub(zr, zr2, zi2, MPFR_RNDN);
        mpfr_add(zr, zr, rx, MPFR_RNDN);
        mpfr_mul(zr2, zr, zr, MPFR_RNDN);
        mpfr_mul(zi2, zi, zi, MPFR_RNDN);
        if (escaped) {
            ref_i++;
            break;
        }
        mpfr_add(tmp, zr2, zi2, MPFR_RNDN);
        if (mpfr_cmp_d(tmp, 40000.0) &gt;= 0) {
            escaped = true;
        }
        ref_i++;
    }
    ref_orbit_double[ref_i].re = mpfr_get_d(zr, MPFR_RNDN);
    ref_orbit_double[ref_i].im = mpfr_get_d(zi, MPFR_RNDN);
    uint32_t max_valid_ref_iter = ref_i; 
    mpfr_clears(rx, ry, zr, zi, zr2, zi2, tmp, sz, st, NULL);
    
    uint8_t pal[256][3];
    for (int a = 0; a &lt; 255; ++a) {
        pal[a][0] = (uint8_t)round(127.0 + 127.0 * cos(2.0 * PI * a / 255.0));
        pal[a][1] = (uint8_t)round(127.0 + 127.0 * sin(2.0 * PI * a / 255.0));
        pal[a][2] = (uint8_t)round(127.0 + 127.0 * sin(2.0 * PI * a / 255.0));
    }
    pal[255][0] = 255; pal[255][1] = 255; pal[255][2] = 255;
    
    cout &lt;&lt; &quot;Step 2: Stream rendering Mandelbrot Set Image 112.bmp (&quot; &lt;&lt; targetW &lt;&lt; &quot;x&quot; &lt;&lt; targetH &lt;&lt; &quot;)...&quot; &lt;&lt; endl;
    int rowSize = (targetW * 3 + 3) &amp; ~3;
    BMPHeader header;
    header.width = targetW;
    header.height = targetH;
    header.sizeImage = rowSize * targetH;
    header.size = header.sizeImage + 54;
    ofstream f(&quot;Mandelbrot Set Image 112.bmp&quot;, ios::binary);
    f.write(reinterpret_cast&lt;char*&gt;(&amp;header), 54);
    vector&lt;uint8_t&gt; rowBuffer(rowSize);
    
    for (int y = 0; y &lt; targetH; ++y) {
        #pragma omp parallel for schedule(dynamic)
        for (int x = 0; x &lt; targetW; ++x) {
            uint32_t rSum = 0, gSum = 0, bSum = 0;
            const ComplexDouble* ref_ptr = ref_orbit_double.data();
            
            for (int j = 0; j &lt; scale; ++j) {
                size_t b = (size_t)y * scale + j;
                double delta_imc = (double)((long long)b - (rawH / 2)) * step_d;
                
                for (int i = 0; i &lt; scale; ++i) {
                    size_t a = (size_t)x * scale + i;
                    double delta_rec = (double)((long long)a - (rawW / 2)) * step_d;
                    
                    uint32_t index = 0;    
                    double delta_re = 0.0; 
                    double delta_im = 0.0;
                    double z_re = 0.0;     
                    double z_im = 0.0;
                    uint32_t iter = 0;
                    bool has_re_based = false;
                    
                    while (iter &lt; 50000) {
                        if ((z_re * z_re + z_im * z_im) &gt;= 40000.0) {
                            break;
                        }
                        
                        if (index &gt;= max_valid_ref_iter) {
                            if (!has_re_based) {
                                break; 
                            } else {
                                double ld_cx = ref_rec_d + delta_rec;
                                double ld_cy = ref_imc_d + delta_imc; 
                                while (iter &lt; 50000 &amp;&amp; (z_re * z_re + z_im * z_im) &lt; 40000.0) {
                                    double old_re = z_re;
                                    double old_im = z_im;
                                    z_re = old_re * old_re - old_im * old_im + ld_cx;
                                    z_im = 2.0 * old_re * old_im + ld_cy;
                                    iter++;
                                }
                                break;
                            }
                        }
                        
                        if ((z_re * z_re + z_im * z_im) &lt; (delta_re * delta_re + delta_im * delta_im)) {
                            index = 0; 
                            delta_re = z_re;
                            delta_im = z_im;
                            has_re_based = true;
                        }
                        
                        for (int k = 0; k &lt; 2; ++k) {
                            double Ur = ref_ptr[index].re;
                            double Ui = ref_ptr[index].im;
                            double next_delta_im = 2.0 * Ur * delta_im + 2.0 * Ui * delta_re + 2.0 * delta_re * delta_im + delta_imc;
                            delta_re = 2.0 * Ur * delta_re - 2.0 * Ui * delta_im + delta_re * delta_re - delta_im * delta_im + delta_rec;
                            delta_im = next_delta_im;
                            index++;
                        }
                        z_re = ref_ptr[index].re + delta_re;
                        z_im = ref_ptr[index].im + delta_im;
                        iter += 2; 
                    }
                    
                    int final_t = 50000 - iter;
                    uint8_t t = (final_t == 0) ? 255 : (uint8_t)(final_t % 254);
                    int colorIdx = (t == 255) ? 255 : (t - frame + 255) % 255;
                    bSum += pal[colorIdx][0];
                    gSum += pal[colorIdx][1];
                    rSum += pal[colorIdx][2];
                }
            }                
            int outIdx = x * 3;
            rowBuffer[outIdx + 0] = (uint8_t)(bSum &gt;&gt; 6);
            rowBuffer[outIdx + 1] = (uint8_t)(gSum &gt;&gt; 6);
            rowBuffer[outIdx + 2] = (uint8_t)(rSum &gt;&gt; 6);
        }
        f.write(reinterpret_cast&lt;const char*&gt;(rowBuffer.data()), rowSize);
        if ((y + 1) % 10 == 0 || y == targetH - 1) {
            cout &lt;&lt; &quot;Progress: &quot; &lt;&lt; (y + 1) &lt;&lt; &quot;/&quot; &lt;&lt; targetH &lt;&lt; &quot;\r&quot; &lt;&lt; flush;
        }
    }
    f.close();
    cout &lt;&lt; &quot;\nDone! Mandelbrot Set Image 112.bmp successfully saved.&quot; &lt;&lt; endl;
    return 0;
}
&lt;/syntaxhighlight&gt;

== Rendered Examples ==
&lt;gallery mode=&quot;packed&quot; heights=&quot;200&quot;&gt;
File:Mandelbrot Set Image 107.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 108.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 109.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
File:Mandelbrot Set Image 110.png|Mandelbrot set fragment using perturbation theory. Final resolution 10,000 x 10,000 pixels.
&lt;/gallery&gt;

== External Links ==
* [https://github.com/Divetoxx/Mandelbrot Official Mandelbrot CLI Repository on GitHub] — source code, documentation, and pre-compiled releases.
* [https://rosettacode.org/wiki/Mandelbrot_set#Perturbation_Theory Rosetta Code: Mandelbrot set Implementation] — C++ perturbation theory optimization showcased in the global code repository.


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      <text bytes="1363" sha1="1xx3vopascblmwawqy0cpenu1abkqin" xml:space="preserve">Audio Visual (AV) Material refers to resources that combine both  sound and visual elements, creating an engaging way to communicate ideas and information. Unlike traditional text-based content, which relies solely on words, AV materials use images, videos, sounds, and often interactive components. Audiovisual materials’ are an integral part of library and archival collections, providing users with access to a wealth of cultural, historical, and educational resources.

== Types of Audio Visual Materials ==

* '''Video and Film:''' Resources like educational documentaries, motion pictures, and recorded webinars that combine moving images and synchronized sound.

* '''Interactive Multimedia:''' Platforms that use audio, animation, and visual elements to encourage user engagement, such as educational games or virtual reality modules.

* '''Slides and Presentation Software:''' Tools like Microsoft PowerPoint or Google Slides that pair visual graphics, charts, and text with oral explanations or background narration.

== Further Readings ==
[https://www.lisedunetwork.com/audio-visual-materials/ Audio Visual Materials]

[https://eajournals.org/ijellr/wp-content/uploads/sites/56/2024/03/The-Use-of-Audio-Visual-Materials.pdf use of Audiovisual material]

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      <text bytes="5225" sha1="klm47vlxdagwmauz39rdq80nvzpk85x" xml:space="preserve">Disaster preparedness is an indispensable plan for the efficient functioning of any library. Disaster preparedness is the process of organizing a system to cope with disaster and its management either in the library or any other organization. Disaster preparedness enables the library to minimize potential damages of its resources, shorten library recovery time and provide temporary and or permanent cushion to both staff and library users. 

Disaster preparedness provides a platform to design effective, realistic and coordinated planning, reduces duplication of efforts and increase the overall effectiveness of library members disaster  preparedness and response efforts. Disaster preparedness activities embedded with risk reduction measures can prevent disaster situations and also result in saving maximum lives and livelihoods during any disaster situation, enabling the affected population to get back to normalcy within a short time period. Disaster preparedness is a continuous and integrated process resulting from a wide range of risk reduction activities and resources rather than from a distinct sectoral activity by itself. It requires the contributions of many different areas ranging from training and logistics, to health care, recovery, livelihood to institutional development.

Disaster preparedness is important for libraries, as they collect and provide access to information and knowledge of human intellectual scholarly ideas and work. The plan is prepared to minimize the impact of losses caused by disasters. In other words, the basic concept of a disaster preparedness plan is to minimize risks and maximize the efficiency of response if a disaster occurs.

== Preparedness of Library Against Disasters ==
Disaster Preparedness involves:

# Identification of a disaster response team;
# Training of an emergency action team;
# Identification of recovery work areas; and
# Ensuring supply of equipment and materials.

== '''Equipment Needed in The Library to Fight Disaster''' ==
Equipment like fire extinguishers, Audible alarm fire, smoke detectors, Break glass alarm, fire sprinklers, first aid kits, Water Detector, “You are here “Map should be made available

# '''Fire Extinguishers:''' Fire extinguishers are extremely important as they are the most commonly used for of fire protection. In many cases they are a first line of defense and often contain or extinguish a fire, preventing costly damage.
# '''Audible Fire Alarms:''' Loud sirens are a requisite part of any fire alarm system. The noise ensures that visually impaired people and those with limited hearing can still detect the need to evacuate.
# '''Smoke detectors''': Properly installed and maintained smoke alarms are considered to be one of the best and least expensive means of providing an early warning of a potentially deadly fire and could reduce by almost half the risk of dying from a fire in workplaces and homes.
# '''Break glass alarm:''' The emergency break glass alarm activates the EWIS to initiate an evacuation of the building. In some situations, you may not need to contact the Fire Brigade but do need to evacuate the building. This is where the emergency break glass alarm can help.
# '''fire sprinklers:''' A sprinkler system is designed to control or extinguish fires in the early stages. This makes it easier and safer for building occupants to exit the building, and for firefighters to extinguish any fire that remains. Sprinklers reduce the loss due to fire.
# '''First aid kits''': First-aid kits help you handle the medical emergencies as quickly as possible. In an emergency, a delay of just a single minute can cause irreconcilable damage. These kits offer basic and instant care for common medical injuries like injuries, burns, cuts etc.
# '''Water Detector:''' A water detector is an electronic device that is designed to detect the presence of water for purposes such as to provide an alert in time to allow the prevention of water leakage. Water leak detection is an expression more commonly used for larger, integrated systems installed in modern buildings or those containing valuable artifacts, materials or other critical assets where early notification of a potentially damaging leak would be beneficial. In particular, water leak detection has become a necessity in data centers, trading floors, banks, archives and other mission-critical infrastructure.
# '''“You Are Here” Map:''' One of the most important purpose and spatial task is to guide people to the nearest exits in the case of evacuation. In the case of emergency situation spatial awareness of all involved people is really critical. Therefore, You-Are-Here maps could help people to locate themselves in the place and finding the way to the nearest exit.

== Further Reading ==
[https://www.extensionjournal.com/uploads/archives/9-2-334-726.pdf Disaster management in libraries]

[https://www.ijfmr.com/special-issues/2/131.pdf Disaster Management in Libraries:]

[https://credence-publishing.com/journal/uploads/archive/202517451480090101752702.pdf Disaster preparedness and management in academic libraries]

[https://journals.journalsplace.org/index.php/CJLIL/article/download/769/663 Disaster Preparedness]</text>
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      <text bytes="5321" sha1="izvb7nbenr5dyh3or32l6o3tzw26orq" xml:space="preserve">Disaster preparedness is an indispensable plan for the efficient functioning of any library. Disaster preparedness is the process of organizing a system to cope with disaster and its management either in the library or any other organization. Disaster preparedness enables the library to minimize potential damages of its resources, shorten library recovery time and provide temporary and or permanent cushion to both staff and library users. 

Disaster preparedness provides a platform to design effective, realistic and coordinated planning, reduces duplication of efforts and increase the overall effectiveness of library members disaster  preparedness and response efforts. Disaster preparedness activities embedded with risk reduction measures can prevent disaster situations and also result in saving maximum lives and livelihoods during any disaster situation, enabling the affected population to get back to normalcy within a short time period. Disaster preparedness is a continuous and integrated process resulting from a wide range of risk reduction activities and resources rather than from a distinct sectoral activity by itself. It requires the contributions of many different areas ranging from training and logistics, to health care, recovery, livelihood to institutional development.

Disaster preparedness is important for libraries, as they collect and provide access to information and knowledge of human intellectual scholarly ideas and work. The plan is prepared to minimize the impact of losses caused by disasters. In other words, the basic concept of a disaster preparedness plan is to minimize risks and maximize the efficiency of response if a disaster occurs.

== Preparedness of Library Against Disasters ==
Disaster Preparedness involves:

# Identification of a disaster response team;
# Training of an emergency action team;
# Identification of recovery work areas; and
# Ensuring supply of equipment and materials.

== '''Equipment Needed in The Library to Fight Disaster''' ==
Equipment like fire extinguishers, Audible alarm fire, smoke detectors, Break glass alarm, fire sprinklers, first aid kits, Water Detector, “You are here “Map should be made available

# '''Fire Extinguishers:''' Fire extinguishers are extremely important as they are the most commonly used for of fire protection. In many cases they are a first line of defense and often contain or extinguish a fire, preventing costly damage.
# '''Audible Fire Alarms:''' Loud sirens are a requisite part of any fire alarm system. The noise ensures that visually impaired people and those with limited hearing can still detect the need to evacuate.
# '''Smoke detectors''': Properly installed and maintained smoke alarms are considered to be one of the best and least expensive means of providing an early warning of a potentially deadly fire and could reduce by almost half the risk of dying from a fire in workplaces and homes.
# '''Break glass alarm:''' The emergency break glass alarm activates the EWIS to initiate an evacuation of the building. In some situations, you may not need to contact the Fire Brigade but do need to evacuate the building. This is where the emergency break glass alarm can help.
# '''fire sprinklers:''' A sprinkler system is designed to control or extinguish fires in the early stages. This makes it easier and safer for building occupants to exit the building, and for firefighters to extinguish any fire that remains. Sprinklers reduce the loss due to fire.
# '''First aid kits''': First-aid kits help you handle the medical emergencies as quickly as possible. In an emergency, a delay of just a single minute can cause irreconcilable damage. These kits offer basic and instant care for common medical injuries like injuries, burns, cuts etc.
# '''Water Detector:''' A water detector is an electronic device that is designed to detect the presence of water for purposes such as to provide an alert in time to allow the prevention of water leakage. Water leak detection is an expression more commonly used for larger, integrated systems installed in modern buildings or those containing valuable artifacts, materials or other critical assets where early notification of a potentially damaging leak would be beneficial. In particular, water leak detection has become a necessity in data centers, trading floors, banks, archives and other mission-critical infrastructure.
# '''“You Are Here” Map:''' One of the most important purpose and spatial task is to guide people to the nearest exits in the case of evacuation. In the case of emergency situation spatial awareness of all involved people is really critical. Therefore, You-Are-Here maps could help people to locate themselves in the place and finding the way to the nearest exit.

== Further Reading ==
[https://www.extensionjournal.com/uploads/archives/9-2-334-726.pdf Disaster management in libraries]

[https://www.ijfmr.com/special-issues/2/131.pdf Disaster Management in Libraries:]

[https://credence-publishing.com/journal/uploads/archive/202517451480090101752702.pdf Disaster preparedness and management in academic libraries]

[https://journals.journalsplace.org/index.php/CJLIL/article/download/769/663 Disaster Preparedness]
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Disaster preparedness is an indispensable plan for the efficient functioning of any library. Disaster preparedness is the process of organizing a system to cope with disaster and its management either in the library or any other organization. Disaster preparedness enables the library to minimize potential damages of its resources, shorten library recovery time and provide temporary and or permanent cushion to both staff and library users. 

Disaster preparedness provides a platform to design effective, realistic and coordinated planning, reduces duplication of efforts and increase the overall effectiveness of library members disaster  preparedness and response efforts. Disaster preparedness activities embedded with risk reduction measures can prevent disaster situations and also result in saving maximum lives and livelihoods during any disaster situation, enabling the affected population to get back to normalcy within a short time period. Disaster preparedness is a continuous and integrated process resulting from a wide range of risk reduction activities and resources rather than from a distinct sectoral activity by itself. It requires the contributions of many different areas ranging from training and logistics, to health care, recovery, livelihood to institutional development.

Disaster preparedness is important for libraries, as they collect and provide access to information and knowledge of human intellectual scholarly ideas and work. The plan is prepared to minimize the impact of losses caused by disasters. In other words, the basic concept of a disaster preparedness plan is to minimize risks and maximize the efficiency of response if a disaster occurs.

== Preparedness of Library Against Disasters ==
Disaster Preparedness involves:

# Identification of a disaster response team;
# Training of an emergency action team;
# Identification of recovery work areas; and
# Ensuring supply of equipment and materials.

== '''Equipment Needed in The Library to Fight Disaster''' ==
Equipment like fire extinguishers, Audible alarm fire, smoke detectors, Break glass alarm, fire sprinklers, first aid kits, Water Detector, “You are here “Map should be made available

# '''Fire Extinguishers:''' Fire extinguishers are extremely important as they are the most commonly used for of fire protection. In many cases they are a first line of defense and often contain or extinguish a fire, preventing costly damage.
# '''Audible Fire Alarms:''' Loud sirens are a requisite part of any fire alarm system. The noise ensures that visually impaired people and those with limited hearing can still detect the need to evacuate.
# '''Smoke detectors''': Properly installed and maintained smoke alarms are considered to be one of the best and least expensive means of providing an early warning of a potentially deadly fire and could reduce by almost half the risk of dying from a fire in workplaces and homes.
# '''Break glass alarm:''' The emergency break glass alarm activates the EWIS to initiate an evacuation of the building. In some situations, you may not need to contact the Fire Brigade but do need to evacuate the building. This is where the emergency break glass alarm can help.
# '''fire sprinklers:''' A sprinkler system is designed to control or extinguish fires in the early stages. This makes it easier and safer for building occupants to exit the building, and for firefighters to extinguish any fire that remains. Sprinklers reduce the loss due to fire.
# '''First aid kits''': First-aid kits help you handle the medical emergencies as quickly as possible. In an emergency, a delay of just a single minute can cause irreconcilable damage. These kits offer basic and instant care for common medical injuries like injuries, burns, cuts etc.
# '''Water Detector:''' A water detector is an electronic device that is designed to detect the presence of water for purposes such as to provide an alert in time to allow the prevention of water leakage. Water leak detection is an expression more commonly used for larger, integrated systems installed in modern buildings or those containing valuable artifacts, materials or other critical assets where early notification of a potentially damaging leak would be beneficial. In particular, water leak detection has become a necessity in data centers, trading floors, banks, archives and other mission-critical infrastructure.
# '''“You Are Here” Map:''' One of the most important purpose and spatial task is to guide people to the nearest exits in the case of evacuation. In the case of emergency situation spatial awareness of all involved people is really critical. Therefore, You-Are-Here maps could help people to locate themselves in the place and finding the way to the nearest exit.

== Further Reading ==
[https://www.extensionjournal.com/uploads/archives/9-2-334-726.pdf Disaster management in libraries]

[https://www.ijfmr.com/special-issues/2/131.pdf Disaster Management in Libraries:]

[https://credence-publishing.com/journal/uploads/archive/202517451480090101752702.pdf Disaster preparedness and management in academic libraries]

[https://journals.journalsplace.org/index.php/CJLIL/article/download/769/663 Disaster Preparedness]
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      <text bytes="5321" sha1="izvb7nbenr5dyh3or32l6o3tzw26orq" xml:space="preserve">Disaster preparedness is an indispensable plan for the efficient functioning of any library. Disaster preparedness is the process of organizing a system to cope with disaster and its management either in the library or any other organization. Disaster preparedness enables the library to minimize potential damages of its resources, shorten library recovery time and provide temporary and or permanent cushion to both staff and library users. 

Disaster preparedness provides a platform to design effective, realistic and coordinated planning, reduces duplication of efforts and increase the overall effectiveness of library members disaster  preparedness and response efforts. Disaster preparedness activities embedded with risk reduction measures can prevent disaster situations and also result in saving maximum lives and livelihoods during any disaster situation, enabling the affected population to get back to normalcy within a short time period. Disaster preparedness is a continuous and integrated process resulting from a wide range of risk reduction activities and resources rather than from a distinct sectoral activity by itself. It requires the contributions of many different areas ranging from training and logistics, to health care, recovery, livelihood to institutional development.

Disaster preparedness is important for libraries, as they collect and provide access to information and knowledge of human intellectual scholarly ideas and work. The plan is prepared to minimize the impact of losses caused by disasters. In other words, the basic concept of a disaster preparedness plan is to minimize risks and maximize the efficiency of response if a disaster occurs.

== Preparedness of Library Against Disasters ==
Disaster Preparedness involves:

# Identification of a disaster response team;
# Training of an emergency action team;
# Identification of recovery work areas; and
# Ensuring supply of equipment and materials.

== '''Equipment Needed in The Library to Fight Disaster''' ==
Equipment like fire extinguishers, Audible alarm fire, smoke detectors, Break glass alarm, fire sprinklers, first aid kits, Water Detector, “You are here “Map should be made available

# '''Fire Extinguishers:''' Fire extinguishers are extremely important as they are the most commonly used for of fire protection. In many cases they are a first line of defense and often contain or extinguish a fire, preventing costly damage.
# '''Audible Fire Alarms:''' Loud sirens are a requisite part of any fire alarm system. The noise ensures that visually impaired people and those with limited hearing can still detect the need to evacuate.
# '''Smoke detectors''': Properly installed and maintained smoke alarms are considered to be one of the best and least expensive means of providing an early warning of a potentially deadly fire and could reduce by almost half the risk of dying from a fire in workplaces and homes.
# '''Break glass alarm:''' The emergency break glass alarm activates the EWIS to initiate an evacuation of the building. In some situations, you may not need to contact the Fire Brigade but do need to evacuate the building. This is where the emergency break glass alarm can help.
# '''fire sprinklers:''' A sprinkler system is designed to control or extinguish fires in the early stages. This makes it easier and safer for building occupants to exit the building, and for firefighters to extinguish any fire that remains. Sprinklers reduce the loss due to fire.
# '''First aid kits''': First-aid kits help you handle the medical emergencies as quickly as possible. In an emergency, a delay of just a single minute can cause irreconcilable damage. These kits offer basic and instant care for common medical injuries like injuries, burns, cuts etc.
# '''Water Detector:''' A water detector is an electronic device that is designed to detect the presence of water for purposes such as to provide an alert in time to allow the prevention of water leakage. Water leak detection is an expression more commonly used for larger, integrated systems installed in modern buildings or those containing valuable artifacts, materials or other critical assets where early notification of a potentially damaging leak would be beneficial. In particular, water leak detection has become a necessity in data centers, trading floors, banks, archives and other mission-critical infrastructure.
# '''“You Are Here” Map:''' One of the most important purpose and spatial task is to guide people to the nearest exits in the case of evacuation. In the case of emergency situation spatial awareness of all involved people is really critical. Therefore, You-Are-Here maps could help people to locate themselves in the place and finding the way to the nearest exit.

== Further Reading ==
[https://www.extensionjournal.com/uploads/archives/9-2-334-726.pdf Disaster management in libraries]

[https://www.ijfmr.com/special-issues/2/131.pdf Disaster Management in Libraries:]

[https://credence-publishing.com/journal/uploads/archive/202517451480090101752702.pdf Disaster preparedness and management in academic libraries]

[https://journals.journalsplace.org/index.php/CJLIL/article/download/769/663 Disaster Preparedness]
[[Category:Library and Information Science]]
[[Category:Library and Information Science stubs]]</text>
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  <page>
    <title>User talk:Nubelbariloe</title>
    <ns>3</ns>
    <id>330605</id>
    <revision>
      <id>2818353</id>
      <timestamp>2026-07-15T15:10:19Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Question */ new section</comment>
      <origin>2818353</origin>
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      <text bytes="691" sha1="e2j3gq2kn5yvpmenluu4fivfqympk87" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)</text>
      <sha1>e2j3gq2kn5yvpmenluu4fivfqympk87</sha1>
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    <revision>
      <id>2818357</id>
      <parentid>2818353</parentid>
      <timestamp>2026-07-15T15:26:53Z</timestamp>
      <contributor>
        <username>Nubelbariloe</username>
        <id>2999346</id>
      </contributor>
      <comment>/* Question */ Reply</comment>
      <origin>2818357</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="928" sha1="mfqf9i02o9k8rt0alokli1hxgzsu4rp" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)</text>
      <sha1>mfqf9i02o9k8rt0alokli1hxgzsu4rp</sha1>
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    <revision>
      <id>2818378</id>
      <parentid>2818357</parentid>
      <timestamp>2026-07-15T18:20:53Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Question */ Reply</comment>
      <origin>2818378</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1316" sha1="k52yn5mmhdeh4b84ezc52uez5ix5cni" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)
::Sure, thank you for answering. Is there a way we can perhaps integrate [[Audio visual materials]] &amp; [[Information services]] into a cohesive learning project &amp; incorporate [[Wikiversity:Learning by doing|active learning]]? See [[Wikiversity:Learning projects]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 15 July 2026 (UTC)</text>
      <sha1>k52yn5mmhdeh4b84ezc52uez5ix5cni</sha1>
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    <revision>
      <id>2818379</id>
      <parentid>2818378</parentid>
      <timestamp>2026-07-15T18:34:41Z</timestamp>
      <contributor>
        <username>Nubelbariloe</username>
        <id>2999346</id>
      </contributor>
      <comment>/* Question */ Reply</comment>
      <origin>2818379</origin>
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      <text bytes="1689" sha1="mo7zwuqkdnanu10m6yrgjtcjiqq6o87" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)
::Sure, thank you for answering. Is there a way we can perhaps integrate [[Audio visual materials]] &amp; [[Information services]] into a cohesive learning project &amp; incorporate [[Wikiversity:Learning by doing|active learning]]? See [[Wikiversity:Learning projects]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 15 July 2026 (UTC)
:::yes. this are topics which are for library and information science students. this topics helps student understand the basics of what they are and subsequently the articles will be expanded into a full learning materials [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:34, 15 July 2026 (UTC)</text>
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      <id>2818380</id>
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      <contributor>
        <username>Nubelbariloe</username>
        <id>2999346</id>
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      <text bytes="2026" sha1="gek4q8erwoiqo5uf6zsa2tx61ojxx5e" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)
::Sure, thank you for answering. Is there a way we can perhaps integrate [[Audio visual materials]] &amp; [[Information services]] into a cohesive learning project &amp; incorporate [[Wikiversity:Learning by doing|active learning]]? See [[Wikiversity:Learning projects]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 15 July 2026 (UTC)
:::yes. this are topics which are for library and information science students. this topics helps student understand the basics of what they are and subsequently the articles will be expanded into a full learning materials [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:34, 15 July 2026 (UTC)
::::if you feel they are not, you can point corrections on where i should work on but from what you tagged and from some pages i read before creating the articles, i think they are right [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:36, 15 July 2026 (UTC)</text>
      <sha1>gek4q8erwoiqo5uf6zsa2tx61ojxx5e</sha1>
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    <revision>
      <id>2818381</id>
      <parentid>2818380</parentid>
      <timestamp>2026-07-15T19:11:51Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Question */ Reply</comment>
      <origin>2818381</origin>
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      <text bytes="2515" sha1="rl6le17go654f65cecglvorvrr3ib29" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)
::Sure, thank you for answering. Is there a way we can perhaps integrate [[Audio visual materials]] &amp; [[Information services]] into a cohesive learning project &amp; incorporate [[Wikiversity:Learning by doing|active learning]]? See [[Wikiversity:Learning projects]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 15 July 2026 (UTC)
:::yes. this are topics which are for library and information science students. this topics helps student understand the basics of what they are and subsequently the articles will be expanded into a full learning materials [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:34, 15 July 2026 (UTC)
::::if you feel they are not, you can point corrections on where i should work on but from what you tagged and from some pages i read before creating the articles, i think they are right [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:36, 15 July 2026 (UTC)
:::::Yes, they are fine. I was suggesting ways so that they are more organized and easier to retrieve. Perhaps we could move both of these pages under a main &quot;course&quot;? What would you suggest? (ex, like [[History of Topics in Special Relativity]], how it has several subpages, including [[History of Topics in Special Relativity/Lorentz transformation (general)]]). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:11, 15 July 2026 (UTC)</text>
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    <revision>
      <id>2818383</id>
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      <contributor>
        <username>Nubelbariloe</username>
        <id>2999346</id>
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      <comment>/* Question */ Reply</comment>
      <origin>2818383</origin>
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      <text bytes="2741" sha1="c2w601a7tei6pal87lvehp6cyw4v8mu" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)
::Sure, thank you for answering. Is there a way we can perhaps integrate [[Audio visual materials]] &amp; [[Information services]] into a cohesive learning project &amp; incorporate [[Wikiversity:Learning by doing|active learning]]? See [[Wikiversity:Learning projects]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 15 July 2026 (UTC)
:::yes. this are topics which are for library and information science students. this topics helps student understand the basics of what they are and subsequently the articles will be expanded into a full learning materials [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:34, 15 July 2026 (UTC)
::::if you feel they are not, you can point corrections on where i should work on but from what you tagged and from some pages i read before creating the articles, i think they are right [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:36, 15 July 2026 (UTC)
:::::Yes, they are fine. I was suggesting ways so that they are more organized and easier to retrieve. Perhaps we could move both of these pages under a main &quot;course&quot;? What would you suggest? (ex, like [[History of Topics in Special Relativity]], how it has several subpages, including [[History of Topics in Special Relativity/Lorentz transformation (general)]]). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:11, 15 July 2026 (UTC)
::::::ok. 
::::::i'm new so teach me how to organize them so i can do that. [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 19:13, 15 July 2026 (UTC)</text>
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    <revision>
      <id>2818385</id>
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      <timestamp>2026-07-15T21:49:17Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
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      <comment>/* Question */ Reply</comment>
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      <text bytes="3056" sha1="pqlf4jtk920hoicle4fk4a36993gk0q" xml:space="preserve">== Question ==

Hi Nubelbariloe! Welcome to Wikiversity. I wanted to ask about your recent page creations: [[Disaster preparedness in Libraries]]. Are you using an LLM to generate these pages? If so, please adhere to the [[Wikiversity:Artificial intelligence|AI policy]] on WV. Additionally, I've left a [[Wikiversity:Proposed deletion|proposed deletion]] template on the page as it seems to be more Wikipedia-oriented, rather than Wikiversity. Please see [[Wikiversity:What is Wikiversity?]] and [[Wikiversity:Differences between Wikiversity and Wikipedia]]. Thanks! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:10, 15 July 2026 (UTC)

:no i didnt use LLM. i have references to back that up.
:i got it from a project work [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 15:26, 15 July 2026 (UTC)
::Sure, thank you for answering. Is there a way we can perhaps integrate [[Audio visual materials]] &amp; [[Information services]] into a cohesive learning project &amp; incorporate [[Wikiversity:Learning by doing|active learning]]? See [[Wikiversity:Learning projects]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 15 July 2026 (UTC)
:::yes. this are topics which are for library and information science students. this topics helps student understand the basics of what they are and subsequently the articles will be expanded into a full learning materials [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:34, 15 July 2026 (UTC)
::::if you feel they are not, you can point corrections on where i should work on but from what you tagged and from some pages i read before creating the articles, i think they are right [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 18:36, 15 July 2026 (UTC)
:::::Yes, they are fine. I was suggesting ways so that they are more organized and easier to retrieve. Perhaps we could move both of these pages under a main &quot;course&quot;? What would you suggest? (ex, like [[History of Topics in Special Relativity]], how it has several subpages, including [[History of Topics in Special Relativity/Lorentz transformation (general)]]). —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:11, 15 July 2026 (UTC)
::::::ok. 
::::::i'm new so teach me how to organize them so i can do that. [[User:Nubelbariloe|Nubelbariloe]] ([[User talk:Nubelbariloe|discuss]] • [[Special:Contributions/Nubelbariloe|contribs]]) 19:13, 15 July 2026 (UTC)
:::::::Gladly. I just need your input. What project name would you suggest putting both of these pages under? I ask because it seems that they are related since they have similar categories. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:49, 15 July 2026 (UTC)</text>
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  <page>
    <title>Information services</title>
    <ns>0</ns>
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      <contributor>
        <username>Nubelbariloe</username>
        <id>2999346</id>
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      <comment>created an article</comment>
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      <text bytes="1379" sha1="kk7lrnu6ah90gg3tymra4ib3j1o3v3n" xml:space="preserve">Information service can be define as the collection, organization, processing, and dissemination of information to meet the specific needs of users. Information services is aimed at provide relevant, accurate, and timely information that supports decision-making, problem-solving, research, and learning. These services are offered through various channels, including libraries, digital platforms, research institutions, and even business customer service departments. Information services are essential in nearly every sector, from education and healthcare to business and government, as they help individuals and organizations access the data and resources they need to function effectively.

Information services help organizations optimize internal processes and improve efficiency, which are central aspects of digital transformation. Businesses can streamline their operations and reduce the risk of human error by automating routine tasks, such as data entry, reporting, and document management. Powered by data-driven insights, workflow automation tools enable organizations to improve operational efficiency, reduce costs, and enhance productivity.

== Further Reading ==
[https://www.lisedunetwork.com/understanding-information-services-in-modern-libraries/ Information services]

[https://www.lisedunetwork.com/what-is-information-service/ What is Information Service]</text>
      <sha1>kk7lrnu6ah90gg3tymra4ib3j1o3v3n</sha1>
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        <username>Nubelbariloe</username>
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      <comment>added category</comment>
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      <text bytes="1475" sha1="67bkpfzydlaxf34so407mre020j15j4" xml:space="preserve">Information service can be define as the collection, organization, processing, and dissemination of information to meet the specific needs of users. Information services is aimed at provide relevant, accurate, and timely information that supports decision-making, problem-solving, research, and learning. These services are offered through various channels, including libraries, digital platforms, research institutions, and even business customer service departments. Information services are essential in nearly every sector, from education and healthcare to business and government, as they help individuals and organizations access the data and resources they need to function effectively.

Information services help organizations optimize internal processes and improve efficiency, which are central aspects of digital transformation. Businesses can streamline their operations and reduce the risk of human error by automating routine tasks, such as data entry, reporting, and document management. Powered by data-driven insights, workflow automation tools enable organizations to improve operational efficiency, reduce costs, and enhance productivity.

== Further Reading ==
[https://www.lisedunetwork.com/understanding-information-services-in-modern-libraries/ Information services]

[https://www.lisedunetwork.com/what-is-information-service/ What is Information Service]
[[Category:Library and Information Science]]
[[Category:Library and Information Science stubs]]</text>
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  <page>
    <title>User talk:DaraganSergio</title>
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      <comment>/* User:DaraganSergio/Social Synaptic Plasticity */ new section</comment>
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      <text bytes="350" sha1="fjun8a73s9ha0zhucc2gnc71louqamx" xml:space="preserve">== [[User:DaraganSergio/Social Synaptic Plasticity]] ==

Hi DaraganSergio, your recently created page has been moved to [[User:DaraganSergio/Social Synaptic Plasticity]], so please continue further edits in that space. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:13, 15 July 2026 (UTC)</text>
      <sha1>fjun8a73s9ha0zhucc2gnc71louqamx</sha1>
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  </page>
  <page>
    <title>User talk:Hamadbizistech01</title>
    <ns>3</ns>
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      <timestamp>2026-07-16T04:44:12Z</timestamp>
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      <comment>Created page with &quot;{{subst:welcome}} ~~~~ :Wikiversity is not a platform for advertising. If you want to contribute here, please read the above and abide by our guidelines. ~~~~&quot;</comment>
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      <text bytes="3093" sha1="cj5q4b0dxylg6hmw63sg0vxbmec9vaw" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Hamadbizistech01!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Koavf|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
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* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
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* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
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To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:44, 16 July 2026 (UTC)&lt;/div&gt;
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{{Robelbox/close}} ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:44, 16 July 2026 (UTC)
:Wikiversity is not a platform for advertising. If you want to contribute here, please read the above and abide by our guidelines. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:44, 16 July 2026 (UTC)</text>
      <sha1>cj5q4b0dxylg6hmw63sg0vxbmec9vaw</sha1>
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