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;
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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)

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‎Best regards, Dr. Obigho [[User:Royclitton|Royclitton]] ([[User talk:Royclitton|discuss]] • [[Special:Contributions/Royclitton|contribs]]) 16:47, 12 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>
      <sha1>p9vnqdyeawhkhw1jz0zp7l65lh9pdcq</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Colloquium</title>
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      <id>2818208</id>
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        <id>411839</id>
      </contributor>
      <comment>/* July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform */ new section</comment>
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      <text bytes="49722" sha1="j5bwt55720vosb4ba63clo9i2u7j421" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;

== Proposal to rehost Wikinews here ==

As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance.

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&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; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

===Comments and questions===
:Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice.
:Initial questions:
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
:* What are &quot;active editions&quot;?
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
:* Are any changes to the scope of Wikinews proposed?
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:52, 14 May 2026 (UTC)
:* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages?
::*No, not at this time.
:* What are &quot;active editions&quot;?
::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04).
:* How can Wikiversity navigate the concerns that lead to the closure of Wikinews?
::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure).
:* Are any changes to the scope of Wikinews proposed?
::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]].
:* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension?
:** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource.
::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning &lt;em&gt;about&lt;/em&gt; what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like.
::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:13, 14 May 2026 (UTC)
:::Thanks, Justin — it is food for thought.
:::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]].
:::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:49, 14 May 2026 (UTC)

:::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]].
:::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose.
:::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable.
:::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity.
:::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects.
:::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]].
:::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles.
:::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research.
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 15 May 2026 (UTC)
My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the &quot;Wikinews refugees&quot; to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote &amp; adopt this proposal independently. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 13:47, 15 May 2026 (UTC)

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

== Create an autopatrolled user group? ==

{{tracked|T428269|resolved}}
I would like to propose creating the user group &lt;code&gt;autopatrolled&lt;/code&gt; (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling.

On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC)

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&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; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&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; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&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; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&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; 07:13, 9 June 2026 (UTC)

== How much of Wikiversity’s content is LLM slop? ==

Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC)

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== Draft inactivity policy ==

I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]].

However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC)

:I support those suggestions. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:55, 4 June 2026 (UTC)
: Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC)

== Proposed user group and/or possible policy changes ==

{{tracked|T430416|fixed}}
I want to discuss about user group and possible policy changes.

# First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]].
# Second, curators. Given that curators have some sensitive custodian rights (such as &lt;code&gt;delete&lt;/code&gt; [but not &lt;code&gt;undelete&lt;/code&gt; or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and &lt;code&gt;protect&lt;/code&gt;), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions.
# Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

:#Yes, I agree.
:#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos.
:#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''&quot;II ...then you will be approved as a probationary custodian for a period of at least four weeks&quot;''.
:::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to.
:[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC)

== New user what do I do here ==
I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful.
(One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}}
:Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC)

:Thank you very much :) hope to contribute a lot.  [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC)

== Towards an Ethics policy ==

In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects.

However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC)

:One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (&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; 07:39, 9 June 2026 (UTC)
::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism.
::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted.
::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC)
:::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:06, 9 June 2026 (UTC)
::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC)
:AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC)
::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC)
:::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC)
::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC)

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

Hello community,

The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement.

We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon.

Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC)

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)

:Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (&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; 13:59, 11 June 2026 (UTC)

== New nomination template(s) ==

I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC)

== June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions.

#'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 04:00, 15 June 2026 (UTC)

== Mobile friendly main page ==

Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}}
:{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (&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; 13:30, 18 June 2026 (UTC)
::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC)
:::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (&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; 15:42, 18 June 2026 (UTC)
::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC)
:::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (&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; 20:14, 18 June 2026 (UTC)
::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC)

== Main page titles ==

Currently, the title says &quot;Wikiversity:Main Page&quot;, but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one):
# Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.)
# Option 2: Modify [[MediaWiki:Mainpage-title]] to &lt;code&gt;Welcome to Wikiversity&lt;/code&gt; (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to &lt;code&gt;&lt;nowiki&gt;Welcome to Wikiversity, $1!&lt;/nowiki&gt;&lt;/code&gt;; the latter would display to me as &lt;code&gt;Welcome to Wikiversity, Codename Noreste!&lt;/code&gt;
Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC)
: Pinging @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]] for input above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:03, 24 June 2026 (UTC)
:I'm afraid that I don't have strong feelings on this. Changing to either or staying with the status quo are all fine to me. ―[[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; 16:10, 24 June 2026 (UTC)
: I like the option of being consistent with Wikipedia and Wikibooks -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:04, 25 June 2026 (UTC)
: {{done}}. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 30 June 2026 (UTC)
: Thankyou - looks good. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:29, 2 July 2026 (UTC)

== Wiki x AI preconference day @ Wikimania ==
There will be a preconference day at Wikimania about [[meta:Artificial_intelligence/2026_Wiki_AI | Wiki AI]]. It will be mostly offline, but there will be at least one hybrid session for demos of community-developed AI tools and workflows. 

* If you've built something cool, that is a chance to show it off, list it on the gallery of tools in progress, and get feedback. 

* If you could ask the people shaping AI on the wikis (WMF, tool builders, model trainers, GLAM and policy folks) a question, what would it be? 

Cheers, &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 23:12, 20 June 2026 (UTC) and Alaexis&lt;br&gt;{{comment|1=Copied from https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FAssessment%2FUsing_generative_AI&amp;diff=2816357&amp;oldid=2807052}}

== RFC about AI-generated content in Wikimedia Commons ==

You are invited to participate in a [[c:Commons:Requests for comment/Policy update for AI content|request for comment on Wikimedia Commons about a policy update for AI content]]. This may affect files that are uploaded to Wikimedia Commons for use on this project. Thank you. [[m:User:Codename Noreste|Codename Noreste]] ([[m:User talk:Codename Noreste|discuss]]) 17:12, 23 June 2026 (UTC)
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30513860 --&gt;

== Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki ==

&lt;section begin=&quot;Message&quot;/&gt;
'''Legal &amp; Safety Contacts'''

Hello community, the Wikimedia Foundation has provided a [[wmf:Special:MyLanguage/Legal:Wikimedia Foundation Legal and Safety Contact Information|single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish and other wikis and we will deploy to your wiki soon. [[m:Special:MyLanguage/Wikimedia_Foundation_Legal_and_Safety_Contacts_FAQ|Please read more on the project page]] and leave any comments in this thread or on the [[m:Special:MyLanguage/Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|talk page]].

&lt;section end=&quot;Message&quot;/&gt;

-- [[User:Sannita (WMF)|User:Sannita (WMF)]] ([[User talk:Sannita (WMF)|talk]]) 13:31, 25 June 2026 (UTC)
&lt;!-- Message sent by User:Sannita (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Sannita_(WMF)/Mass_sending_test&amp;oldid=30731267 --&gt;

== Preparing manuscript for submission to the ''WikiJournal of Humanities'' ==

I am working on preparing an article in Wikipedia for a dual goal of submitting it for a featured article candidacy in Wikipedia and submitting it to the ''[[WikiJournal of Humanities]]''.  I have an open request for pre-submission peer review at [[en:Wikipedia:Wikipedia:Peer review/Rei Ayanami/archive2|Wikipedia:Peer review/Rei Ayanami/archive2]], and I am asking for someone experienced with submitting journals to WikiJournals.  The article is not ready for submission, and I would like to know where I can get assistance from users who submitted articles to the journal, but did not necessarily review them.  Furthermore, I said there that submitting to the ''WikiJournal of Humanities'' depends on whether the article attains featured article status in Wikipedia, as I would like to use the featured article as a manuscript for a journal article. [[User:Z. Patterson|Z. Patterson]] ([[User talk:Z. Patterson|discuss]] • [[Special:Contributions/Z. Patterson|contribs]]) 12:04, 30 June 2026 (UTC)

== I could probably save many kilobytes by compressing my LLM chat history ==

I could probably save many kilobytes by compressing my LLM chat history. Would that be ok? I like how my new &quot;method&quot; looks: [[User:ThinkingScience/All General AI Prompt History Archive]] very compressed and neat. I like how to find new ways to make people using LLMs not become &quot;secondary citizens&quot;. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 05:05, 7 July 2026 (UTC)

== July 2026 Wikimedia Café meetups regarding Wikimedia governance and options for reform ==

&lt;div class=&quot;border-box&quot; style=&quot;background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)&quot;&gt;
&lt;div class=&quot;box&quot; style=&quot;float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;&quot;&gt;[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]&lt;/div&gt;
Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities in July. Both sessions will focus on Wikimedia governance, including possible follow-ups to the [https://meta.wikimedia.org/wiki/Movement_Charter Movement Charter] and options for reform. Participants may attend either or both Café sessions. 

This month, to deconflict the Café meetups from Wikimania, the meetups will be held one day later than usual.

#'''26 July 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1785078000 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''27 July 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1785121200 timestamp converter]), at a time friendly to Asia and the Pacific

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 03:51, 13 July 2026 (UTC)</text>
      <sha1>j5bwt55720vosb4ba63clo9i2u7j421</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Sandbox</title>
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      <id>2818177</id>
      <parentid>2817863</parentid>
      <timestamp>2026-07-12T20:03:43Z</timestamp>
      <contributor>
        <username>Onlinepg</username>
        <id>3100568</id>
      </contributor>
      <minor />
      <comment>Biography of Dr. Praveen Kumar Gupta</comment>
      <origin>2818177</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2477" sha1="o5yrvege0ym89rh6zvkki3j4gvfiat7" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}

__NOTOC__
{{Learning project
| topic = Fraternal Studies
| level = Intermediate
| Subject = Social Sciences
}}

== Overview ==
Dr. Praveen Kumar Gupta is a seasoned academician and researcher with over 33 years of experience in higher education and the ICT industry. 


== Higher Education Journey ==
Started his professional career journey in 1991, with total 33 years of Teaching and industry experience. Currently serving as Professor in the School of Computer Science and Engineering at Galgotias University, he has a distinguished record in teaching, research, mentoring, and academic leadership. His expertise spans Machine Learning, Data Science, NLP, Data Mining, and Quantum Computing, with a strong focus on guiding postgraduate and doctoral research.

==Professional Qualifications ==

Ph D in Computer Science and Engineering from Maharshi Dayanand University, India https://mdu.ac.in/default.aspx

Masters in Computer Science from IETE, India https://www.iete.org/

Bachelors in Engineering from SPPU  https://www.unipune.ac.in/

== Experience and Engagement ==
Working/worked as Professor with different higher educational institutions as Professor/Associate Professor/Assistant Professor/Lecturer.

2024 - till date - Professor - https://www.galgotiasuniversity.edu.in/p/dr-praveen-kumar-gupta

2023 - 2024 - Professor - https://www.alliance.edu.in/

2015 - 2023 - Professor - https://www.jimsindia.org/faculty-achievement.aspx

and other organizations in India

== Research Credentials ==
Total Publications: more than 40 

SCI/SCIE Publications: 5

Scopus Article Publications: 10

Non Scopus/UGC Article Publications: 10

International/National Conference Publications: 12

Book Publications: 1

Patents Published: 6

Delivered lectures on IPR at PAN-India level

Workshop, FDP &amp; Training Participated and attended:

Numerous ATAL FDP attended

Membership Certification &amp; Association:
Fellow member, IETE, New Delhi

Area of Interest:
Artificial Intelligence
Deep Learning
Machine Learning
Quantum Computing


Links to Research Profiles:

https://orcid.org/0000-0002-3074-8875

https://www.scopus.com/authid/detail.uri?authorId=59895009000

https://scholar.google.com/citations?pli=1&amp;authuser=1&amp;user=Iq99FHMAAAAJ

https://www.webofscience.com/wos/author/record/JNS-4857-2023

Above are the links for orcid, scopus, google scholar, research Gate/Webof Science, for those who want to refer or connect or cite.</text>
      <sha1>o5yrvege0ym89rh6zvkki3j4gvfiat7</sha1>
    </revision>
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      <parentid>2818177</parentid>
      <timestamp>2026-07-13T01:46:34Z</timestamp>
      <contributor>
        <username>Eunice Ameh</username>
        <id>3026207</id>
      </contributor>
      <minor />
      <comment>Eunice Ameh moved page [[Wikiversity:Sandbox]] to [[Wikiversity:Cataract and types]]</comment>
      <origin>2818177</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2477" sha1="o5yrvege0ym89rh6zvkki3j4gvfiat7" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}

__NOTOC__
{{Learning project
| topic = Fraternal Studies
| level = Intermediate
| Subject = Social Sciences
}}

== Overview ==
Dr. Praveen Kumar Gupta is a seasoned academician and researcher with over 33 years of experience in higher education and the ICT industry. 


== Higher Education Journey ==
Started his professional career journey in 1991, with total 33 years of Teaching and industry experience. Currently serving as Professor in the School of Computer Science and Engineering at Galgotias University, he has a distinguished record in teaching, research, mentoring, and academic leadership. His expertise spans Machine Learning, Data Science, NLP, Data Mining, and Quantum Computing, with a strong focus on guiding postgraduate and doctoral research.

==Professional Qualifications ==

Ph D in Computer Science and Engineering from Maharshi Dayanand University, India https://mdu.ac.in/default.aspx

Masters in Computer Science from IETE, India https://www.iete.org/

Bachelors in Engineering from SPPU  https://www.unipune.ac.in/

== Experience and Engagement ==
Working/worked as Professor with different higher educational institutions as Professor/Associate Professor/Assistant Professor/Lecturer.

2024 - till date - Professor - https://www.galgotiasuniversity.edu.in/p/dr-praveen-kumar-gupta

2023 - 2024 - Professor - https://www.alliance.edu.in/

2015 - 2023 - Professor - https://www.jimsindia.org/faculty-achievement.aspx

and other organizations in India

== Research Credentials ==
Total Publications: more than 40 

SCI/SCIE Publications: 5

Scopus Article Publications: 10

Non Scopus/UGC Article Publications: 10

International/National Conference Publications: 12

Book Publications: 1

Patents Published: 6

Delivered lectures on IPR at PAN-India level

Workshop, FDP &amp; Training Participated and attended:

Numerous ATAL FDP attended

Membership Certification &amp; Association:
Fellow member, IETE, New Delhi

Area of Interest:
Artificial Intelligence
Deep Learning
Machine Learning
Quantum Computing


Links to Research Profiles:

https://orcid.org/0000-0002-3074-8875

https://www.scopus.com/authid/detail.uri?authorId=59895009000

https://scholar.google.com/citations?pli=1&amp;authuser=1&amp;user=Iq99FHMAAAAJ

https://www.webofscience.com/wos/author/record/JNS-4857-2023

Above are the links for orcid, scopus, google scholar, research Gate/Webof Science, for those who want to refer or connect or cite.</text>
      <sha1>o5yrvege0ym89rh6zvkki3j4gvfiat7</sha1>
    </revision>
    <revision>
      <id>2818190</id>
      <parentid>2818186</parentid>
      <timestamp>2026-07-13T02:15:13Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>Atcovi moved page [[Wikiversity:Cataract and types]] to [[Wikiversity:Sandbox]] over a redirect without leaving a redirect: Revert</comment>
      <origin>2818177</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2477" sha1="o5yrvege0ym89rh6zvkki3j4gvfiat7" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}

__NOTOC__
{{Learning project
| topic = Fraternal Studies
| level = Intermediate
| Subject = Social Sciences
}}

== Overview ==
Dr. Praveen Kumar Gupta is a seasoned academician and researcher with over 33 years of experience in higher education and the ICT industry. 


== Higher Education Journey ==
Started his professional career journey in 1991, with total 33 years of Teaching and industry experience. Currently serving as Professor in the School of Computer Science and Engineering at Galgotias University, he has a distinguished record in teaching, research, mentoring, and academic leadership. His expertise spans Machine Learning, Data Science, NLP, Data Mining, and Quantum Computing, with a strong focus on guiding postgraduate and doctoral research.

==Professional Qualifications ==

Ph D in Computer Science and Engineering from Maharshi Dayanand University, India https://mdu.ac.in/default.aspx

Masters in Computer Science from IETE, India https://www.iete.org/

Bachelors in Engineering from SPPU  https://www.unipune.ac.in/

== Experience and Engagement ==
Working/worked as Professor with different higher educational institutions as Professor/Associate Professor/Assistant Professor/Lecturer.

2024 - till date - Professor - https://www.galgotiasuniversity.edu.in/p/dr-praveen-kumar-gupta

2023 - 2024 - Professor - https://www.alliance.edu.in/

2015 - 2023 - Professor - https://www.jimsindia.org/faculty-achievement.aspx

and other organizations in India

== Research Credentials ==
Total Publications: more than 40 

SCI/SCIE Publications: 5

Scopus Article Publications: 10

Non Scopus/UGC Article Publications: 10

International/National Conference Publications: 12

Book Publications: 1

Patents Published: 6

Delivered lectures on IPR at PAN-India level

Workshop, FDP &amp; Training Participated and attended:

Numerous ATAL FDP attended

Membership Certification &amp; Association:
Fellow member, IETE, New Delhi

Area of Interest:
Artificial Intelligence
Deep Learning
Machine Learning
Quantum Computing


Links to Research Profiles:

https://orcid.org/0000-0002-3074-8875

https://www.scopus.com/authid/detail.uri?authorId=59895009000

https://scholar.google.com/citations?pli=1&amp;authuser=1&amp;user=Iq99FHMAAAAJ

https://www.webofscience.com/wos/author/record/JNS-4857-2023

Above are the links for orcid, scopus, google scholar, research Gate/Webof Science, for those who want to refer or connect or cite.</text>
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  <page>
    <title>Wikiversity:Requests for Deletion</title>
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      <id>2818178</id>
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      <contributor>
        <username>Koavf</username>
        <id>147</id>
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      <comment>/* Concomitant Strabismus */ Reply</comment>
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== [[Classical guitar pedagogy]] ==

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

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:18, 23 May 2026 (UTC)
: '''Delete''', the author is no longer here nor at Wikipedia, therefore it's unlikely to be developed for the foreseeable future. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:20, 10 July 2026 (UTC)

== [[Concomitant Strabismus]] ==

Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC)

:'''Delete''' per above. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:22, 12 July 2026 (UTC)</text>
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    <title>Wikiversity talk:Sandbox</title>
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      <contributor>
        <username>Eunice Ameh</username>
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      <minor />
      <comment>Eunice Ameh moved page [[Wikiversity talk:Sandbox]] to [[Wikiversity talk:Cataract and types]]</comment>
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== There should be people watching the sandbox ==
Dave and I have been watching the Sandbox constantly, is anyone else interested in volunteering? --[[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]) 15:22, 14 December 2013 (UTC)

:Yes I will, but what do I have to do? --[[Special:Contributions/71.235.232.65|71.235.232.65]] ([[User talk:71.235.232.65|discuss]]) 21:44, 28 March 2014‎ (UTC)

::Cleanup the sandbox by reverting cleanup edits! --~~{{font|face=Utah MT-SJ-NI|[[User:Goldenburg111|{{font|color=red|Golden}}]]&lt;sub&gt;[[User talk:Goldenburg111|{{font|color=orange|burg}}]][[Special:Contributions/Goldenburg111|111]]&lt;/sub&gt;}} 21:46, 28 March 2014 (UTC)

:::Yes, it just needs to be reset occasionally, but not over-zealously.  In particular, don't reset it if a user has been editing within the last hour or so.  They may still be testing something.  -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:46, 28 March 2014 (UTC)

:Keep up the good work. --[[User:Rockchalk 03|Rockchalk 03]] ([[User talk:Rockchalk 03|discuss]] • [[Special:Contributions/Rockchalk 03|contribs]]) 12:20, 31 July 2014 (UTC)

:Maybe, but it is really necessary? I can see in the page history that a robot is running sometimes (&quot;Robot: Automatically cleaned&quot;). If it was written in the information box when it was running (for example at a specified day and time every week), then nobody should be surprised if what they have written, will disappear. /[[User:PatrikN|Patrik Näsfors]] ([[User talk:PatrikN|discuss]] • [[Special:Contributions/PatrikN|contribs]]) 19:05, 12 October 2016 (UTC)

::I clean it when what is already there might discourage another editor from testing their own edits. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:13, 12 October 2016 (UTC)

:::Sorry Dave if I don't understand your answer (so clarify if you like), but if it is reset often enough, I can't see why it should be necessary to watch and do it manually. At [http://www.opensourcecms.com/scripts/details.php?scriptid=153&amp;name=MediaWiki OpenSourceCMS] (where you also can try a demo of MediaWiki, which the link points to), demos are deleted and re-installed every hour. I don't know how much a bot can be programmed to do, but as you stated above, it could be fine if the bot could see if there is more than an hour since the last edit before purging the page. Could that be done? It's just a suggestion so people don't have to check and clean manually like now. /[[User:PatrikN|Patrik Näsfors]] ([[User talk:PatrikN|discuss]] • [[Special:Contributions/PatrikN|contribs]]) 20:34, 12 October 2016 (UTC)

::::It isn't reset often enough. The bot that is active here only resets the sandbox on Saturday mornings. It's certainly possible to write a bot to do this. I don't run mine unattended, but I'd be happy to help someone else learn how to write the code to do this if they're interested. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:58, 12 October 2016 (UTC)

:::::Thanks for the clarification. I'd be happy to help with that. I also have a server running continuously if that's a requirement. I haven't learned to program in Python yet, but it's on my to-learn list and I'd like to learn what else is necessary. (I'm also a quick learner). But actually it seems the [[User:Mu301Bot|Mu301Bot]] does not run anything at the moment (and the task to clean the Sandbox have been inactive since July 2016).

:::::I can also see why it's necessary to do it manually once in a while, because the page is totally changed sometimes. That raises another question. Is it possible to protect a section at the top (the header) and prevent that it is modified, or can only whole pages be protected? Or might a solution be to create subpages for sandboxing? /[[User:PatrikN|Patrik Näsfors]] ([[User talk:PatrikN|discuss]] • [[Special:Contributions/PatrikN|contribs]]) 22:40, 12 October 2016 (UTC)

::::::I'm a bit biased, but there's a good Python course at [[Python Programming]]. There's sample bot code at [[MediaWiki API/Pywikibot]]. The [[Pywikibot]] course is very outdated, but the instructions at [[mw:Manual:Pywikibot]] were good the last time I tried them (maybe six months ago). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:28, 13 October 2016 (UTC)

== Education ==

I would like to find out more information about free learning resources on the internet that once completed and I pass the course I would earn a free certificate showing my accomplishments. [[User:ChrisREdG33|ChrisREdG33]] ([[User talk:ChrisREdG33|discuss]] • [[Special:Contributions/ChrisREdG33|contribs]]) 20:48, 20 July 2018 (UTC)

:{{At|ChrisREdG33}} In most cases, free certificates are worth what you pay for them. Free learning is great. Free proof of learning doesn't go very far. I encourage you to look at your goals and determine what proof of education is necessary for those goals. If it is a portfolio, create your own proof of experience. If certification is required, you'll likely need to pay for that testing or proof. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:07, 21 July 2018 (UTC)

== Science ==

I think we should a develop  scientific project [[User:Umeh elochukwu|Umeh elochukwu]] ([[User talk:Umeh elochukwu|discuss]] • [[Special:Contributions/Umeh elochukwu|contribs]]) 13:59, 8 November 2020 (UTC)

We should try making a drone
 [[User:Umeh elochukwu|Umeh elochukwu]] ([[User talk:Umeh elochukwu|discuss]] • [[Special:Contributions/Umeh elochukwu|contribs]]) 14:01, 8 November 2020 (UTC)

== love,live, and let go ==

I just wanted to see if anyone is actually interested in conversation on this choice of a topic that I chose. If you are interested then please say so and I will begin. [[User:Ronniewaddle79|Ronniewaddle79]] ([[User talk:Ronniewaddle79|discuss]] • [[Special:Contributions/Ronniewaddle79|contribs]]) 21:14, 15 November 2023 (UTC)

:I am interested , please proceed @[[User:Ronniewaddle79|Ronniewaddle79]] [[User:Luminary77|Sierra Seven]] ([[User talk:Luminary77|discuss]] • [[Special:Contributions/Luminary77|contribs]]) 13:17, 9 June 2024 (UTC)
::Me too!@[[User:Ronniewaddle79|Ronniewaddle79]] [[Special:Contributions/&amp;#126;2026-27795-53|&amp;#126;2026-27795-53]] ([[User talk:&amp;#126;2026-27795-53|talk]]) 09:09, 8 May 2026 (UTC)</text>
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      <comment>Atcovi moved page [[Wikiversity talk:Cataract and types]] to [[Wikiversity talk:Sandbox]] over redirect: Revert</comment>
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== There should be people watching the sandbox ==
Dave and I have been watching the Sandbox constantly, is anyone else interested in volunteering? --[[User:Goldenburg111|Goldenburg111]] ([[User talk:Goldenburg111|talk]]) 15:22, 14 December 2013 (UTC)

:Yes I will, but what do I have to do? --[[Special:Contributions/71.235.232.65|71.235.232.65]] ([[User talk:71.235.232.65|discuss]]) 21:44, 28 March 2014‎ (UTC)

::Cleanup the sandbox by reverting cleanup edits! --~~{{font|face=Utah MT-SJ-NI|[[User:Goldenburg111|{{font|color=red|Golden}}]]&lt;sub&gt;[[User talk:Goldenburg111|{{font|color=orange|burg}}]][[Special:Contributions/Goldenburg111|111]]&lt;/sub&gt;}} 21:46, 28 March 2014 (UTC)

:::Yes, it just needs to be reset occasionally, but not over-zealously.  In particular, don't reset it if a user has been editing within the last hour or so.  They may still be testing something.  -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:46, 28 March 2014 (UTC)

:Keep up the good work. --[[User:Rockchalk 03|Rockchalk 03]] ([[User talk:Rockchalk 03|discuss]] • [[Special:Contributions/Rockchalk 03|contribs]]) 12:20, 31 July 2014 (UTC)

:Maybe, but it is really necessary? I can see in the page history that a robot is running sometimes (&quot;Robot: Automatically cleaned&quot;). If it was written in the information box when it was running (for example at a specified day and time every week), then nobody should be surprised if what they have written, will disappear. /[[User:PatrikN|Patrik Näsfors]] ([[User talk:PatrikN|discuss]] • [[Special:Contributions/PatrikN|contribs]]) 19:05, 12 October 2016 (UTC)

::I clean it when what is already there might discourage another editor from testing their own edits. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:13, 12 October 2016 (UTC)

:::Sorry Dave if I don't understand your answer (so clarify if you like), but if it is reset often enough, I can't see why it should be necessary to watch and do it manually. At [http://www.opensourcecms.com/scripts/details.php?scriptid=153&amp;name=MediaWiki OpenSourceCMS] (where you also can try a demo of MediaWiki, which the link points to), demos are deleted and re-installed every hour. I don't know how much a bot can be programmed to do, but as you stated above, it could be fine if the bot could see if there is more than an hour since the last edit before purging the page. Could that be done? It's just a suggestion so people don't have to check and clean manually like now. /[[User:PatrikN|Patrik Näsfors]] ([[User talk:PatrikN|discuss]] • [[Special:Contributions/PatrikN|contribs]]) 20:34, 12 October 2016 (UTC)

::::It isn't reset often enough. The bot that is active here only resets the sandbox on Saturday mornings. It's certainly possible to write a bot to do this. I don't run mine unattended, but I'd be happy to help someone else learn how to write the code to do this if they're interested. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:58, 12 October 2016 (UTC)

:::::Thanks for the clarification. I'd be happy to help with that. I also have a server running continuously if that's a requirement. I haven't learned to program in Python yet, but it's on my to-learn list and I'd like to learn what else is necessary. (I'm also a quick learner). But actually it seems the [[User:Mu301Bot|Mu301Bot]] does not run anything at the moment (and the task to clean the Sandbox have been inactive since July 2016).

:::::I can also see why it's necessary to do it manually once in a while, because the page is totally changed sometimes. That raises another question. Is it possible to protect a section at the top (the header) and prevent that it is modified, or can only whole pages be protected? Or might a solution be to create subpages for sandboxing? /[[User:PatrikN|Patrik Näsfors]] ([[User talk:PatrikN|discuss]] • [[Special:Contributions/PatrikN|contribs]]) 22:40, 12 October 2016 (UTC)

::::::I'm a bit biased, but there's a good Python course at [[Python Programming]]. There's sample bot code at [[MediaWiki API/Pywikibot]]. The [[Pywikibot]] course is very outdated, but the instructions at [[mw:Manual:Pywikibot]] were good the last time I tried them (maybe six months ago). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:28, 13 October 2016 (UTC)

== Education ==

I would like to find out more information about free learning resources on the internet that once completed and I pass the course I would earn a free certificate showing my accomplishments. [[User:ChrisREdG33|ChrisREdG33]] ([[User talk:ChrisREdG33|discuss]] • [[Special:Contributions/ChrisREdG33|contribs]]) 20:48, 20 July 2018 (UTC)

:{{At|ChrisREdG33}} In most cases, free certificates are worth what you pay for them. Free learning is great. Free proof of learning doesn't go very far. I encourage you to look at your goals and determine what proof of education is necessary for those goals. If it is a portfolio, create your own proof of experience. If certification is required, you'll likely need to pay for that testing or proof. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:07, 21 July 2018 (UTC)

== Science ==

I think we should a develop  scientific project [[User:Umeh elochukwu|Umeh elochukwu]] ([[User talk:Umeh elochukwu|discuss]] • [[Special:Contributions/Umeh elochukwu|contribs]]) 13:59, 8 November 2020 (UTC)

We should try making a drone
 [[User:Umeh elochukwu|Umeh elochukwu]] ([[User talk:Umeh elochukwu|discuss]] • [[Special:Contributions/Umeh elochukwu|contribs]]) 14:01, 8 November 2020 (UTC)

== love,live, and let go ==

I just wanted to see if anyone is actually interested in conversation on this choice of a topic that I chose. If you are interested then please say so and I will begin. [[User:Ronniewaddle79|Ronniewaddle79]] ([[User talk:Ronniewaddle79|discuss]] • [[Special:Contributions/Ronniewaddle79|contribs]]) 21:14, 15 November 2023 (UTC)

:I am interested , please proceed @[[User:Ronniewaddle79|Ronniewaddle79]] [[User:Luminary77|Sierra Seven]] ([[User talk:Luminary77|discuss]] • [[Special:Contributions/Luminary77|contribs]]) 13:17, 9 June 2024 (UTC)
::Me too!@[[User:Ronniewaddle79|Ronniewaddle79]] [[Special:Contributions/&amp;#126;2026-27795-53|&amp;#126;2026-27795-53]] ([[User talk:&amp;#126;2026-27795-53|talk]]) 09:09, 8 May 2026 (UTC)</text>
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  <page>
    <title>Polish language/Alphabet and phonetics</title>
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      <comment>/* Vowel sounds */ I changed the example for the “a” sound in English at the one given was “tarp” which included a digraph, making the pronunciation unclear. Additionally, I removed the example for “ó” being the “u” in “put” as this not only was factually incorrect but also contradicted the previous example of the “oo” in “hoop”.</comment>
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      <text bytes="8786" sha1="h4nqf0ylus2gulz3xr9r3helejp0zen" xml:space="preserve">== The Polish Alphabet ==

The Polish alphabet is composed of 32 letters:

{{center top}}Aa Ąą Bb Cc Ćć Dd Ee Ęę Ff Gg Hh Ii Jj Kk Ll Łł Mm Nn Ńń Oo Óó Pp Rr Ss Śś Tt Uu Ww Yy Zz Źź Żż{{center bottom}}

== Phonetics ==

Unlike the English language, Polish phonetics are consistent. When understood and known well, almost any Polish word can be pronounced by a foreign reader. Pay close attention to this lesson. Picking up bad habits early on can cause problems in the future.

NB: All phonetic examples of use in the English language are used in conjunction with &quot;the Queen's English&quot; or &quot;BBC English.&quot; (See [[w:Received Pronunciation|here]].)

=== Vowel sounds ===

These are very important. The vowels in the Polish language may sound very similar to beginners, but with practice, they become easier to distinguish. The key point is that a word's meaning can change when a vowel changes. Polish speakers rely on the correct vowel sound to understand the context in which a word is used. This will become apparent later. For now, here is the full list of vowel sounds in the Polish language and an explanation of phonetics for English speakers:

All Polish vowels are short. Additionally, there is no vowel reduction in Polish, so all vowels should be pronounced as they are written.

'''Aa''' - Pronounced the same way as the word &quot;a&quot; in English, e.g., in the word &quot;f'''a'''ther&quot;. It is ''not'' used to signify the other &quot;a-sound&quot; in words like &quot;c'''a'''ke&quot; or &quot;'''a'''corn.&quot;

'''ą''' - This is nasal vowel pronounced like the French &quot;bon.&quot; However, it is only pronounced this way as the final sound of a word (do not confuse the final &quot;a&quot; and &quot;ą&quot; because they indicate completely different cases) or before f, w, s, z, sz, ż, ś, ź, or ch. When followed by &quot;l&quot; or &quot;ł,&quot; it should be pronounced like &quot;o&quot; (however, people sometimes pronounce it as a full nasal, which is hypercorrect). Before other consonants, &quot;ą&quot; splits into a diphthong that is pronounced like the combination of a short &quot;o&quot; and an &quot;m,&quot; &quot;n,&quot; or a sound similar to the English &quot;n&quot; in &quot;bang&quot; - depending on the place of articulation of the next consonant (see the exercise). No Polish word begins with &quot;ą&quot;. 

'''Ee''' - This represents the English &quot;e&quot; sound found in words like &quot;bed,&quot; &quot;wet,&quot; and &quot;pet.&quot;.

'''ę''' - This is the second nasal vowel pronounced like French &quot;vende.&quot; Its pronunciation rules are almost the same as those for &quot;ą,&quot; except that, instead of a short &quot;o,&quot; a short &quot;e&quot; is used when necessary. At the end of a word, &quot;ę&quot; is pronounced like the Polish &quot;e&quot; described above. No Polish word begins with ę.

'''Ii''' - This represents the English &quot;ee&quot; in words like &quot;week,&quot; &quot;need,&quot; and &quot;feet,&quot; but not as long.

'''Oo''' - This represents the English &quot;o&quot; sound in &quot;pot&quot; and &quot;hop.&quot;

'''Uu''' - This represents the English &quot;oo&quot; sound in &quot;boot&quot; and &quot;spoon,&quot; but it is shorter. 

'''Óó''' - The same sound as the &quot;oo&quot; sound in &quot;hoop&quot;. It is important to know whether a word should be written with &quot;ó&quot; or &quot;u&quot; because &quot;ó&quot; often marks morphological change from &quot;ó&quot; to &quot;o&quot; in some inflectional cases.  

'''Yy''' - Similar to the English &quot;i&quot; sound in words like &quot;bit,&quot; &quot;pit,&quot; and &quot;riddle.&quot;

=== Consonant Sounds ===

Polish has more consonants than English and they are quite similar, with few exceptions. However, they are all unaspirated. To pronounce Polish words correctly please note, that unlike English where neighboring consonants have the tendency to assimilate their place of articulation (word 'football' pronounced like 'fupbol') in Polish every voiced consonant (except m, n, r, l, ł)has its unvoiced counterpart and there is a strong tendency only to devoice all constituents of consonant clusters if they comprise unvoiced consonant. It happens both inside words and between two words if the final and initial part of them differ in terms of voicing. For instance word 'kwiat' ('flower') is pronounced 'kfiat'. 'Samochód Kasi'(Kate's car') is pronounced 'samochót Kasi'.Unvoicing of consonants in final position is a dialectal feature (some regions do not devoice them) but the general tendency is to devoice them.        
 
'''Bb''' - The '''same sound''' as the &quot;Bb&quot; sound in English. 

'''Cc''' - Very similar to &quot;z&quot; in &quot;zeppelin&quot;. Often described as English &quot;ts&quot; in &quot;tsar&quot; or &quot;tzatziki&quot;. The closest Germanic equivalent is 'z' in German &quot;Zeit&quot;.      

'''Ćć''' - This sound is similar to the &quot;ch&quot; sound in English words such as &quot;cheers&quot;, &quot;cheetah&quot; but is softer. Its articulation is palato-alveolar, not alveolar like English &quot;church&quot; or &quot;chair&quot;. Try to move your tongue a little bit backward pronouncing the word &quot;church&quot;.

'''Dd''' - It sounds like the &quot;d&quot; sound in English, but Polish &quot;d&quot; is truly dental, not alveolar. Try to pronounce &quot;d&quot; with the tip of your tongue touching the teeth.   

'''Ff''' - The '''same sound''' as the &quot;Ff&quot; sound in English.

'''Gg''' - The same &quot;g&quot; sound in &quot;gun&quot; and &quot;Hogwarts&quot;. 

'''Hh''' - This represents the English &quot;ch&quot; (sounding very similar to &quot;h&quot;) sound in &quot;loch&quot;.

'''Jj''' - This represents the English &quot;Yy&quot; sound in &quot;your&quot; and &quot;yearn&quot;.

'''Kk''' - The '''same sound''' as the &quot;Kk&quot; sound in English, but is never silent. An unvoiced counterpart of &quot;g&quot;

'''Ll''' - The '''same sound''' as the &quot;Ll&quot; sound in English.

'''Łł''' - The '''same sound''' as the &quot;Ww&quot; sound in English.

'''Mm''' - The '''same sound''' as the &quot;Mm&quot; sound in English.

'''Nn''' - The '''same sound''' as the &quot;Nn&quot; sound in English.

'''ń''' - This represents the &quot;ñ&quot; sound in &quot;El Niño&quot;. The best English example is &quot;n&quot; in the word &quot;new&quot; or &quot;manure&quot;. No Polish word begins with ń.

'''Pp''' - The '''same sound''' as the &quot;Pp&quot; sound in English. An unvoiced counterpart of &quot;b&quot;.

'''Rr''' - A rolled Scots-English &quot;Rr&quot;.

'''Ss''' - The '''same sound''' as the &quot;Ss&quot; sound in English. An unvoiced counterpart of &quot;z&quot;.

'''Śś''' - This sound is represented by &quot;sh&quot; in English word &quot;ship&quot;. There is a big phonemic difference between &quot;ś&quot; and &quot;sz&quot; (digraph &quot;sz&quot; stands for &quot;sh&quot; in English &quot;share&quot; or &quot;shy&quot;)

'''Tt''' - The '''same sound''' as the &quot;Tt&quot; sound in English but dental, not alveolar. An unvoiced counterpart of &quot;d&quot;.

'''Ww''' - The English &quot;Vv&quot; sound. A voiced counterpart of &quot;f&quot;.

'''Zz''' - The '''same sound''' as the &quot;Zz&quot; sound in English word &quot;zeal&quot;. A voiced counterpart of &quot;s&quot;.  

'''Źź''' - A voiced counterpart of &quot;ś&quot;. 

'''Żż''' - The &quot;j&quot; sound in the French name, &quot;Jacques&quot;. The tip of the tongue is just shy of making contact with the roof of the mouth a centimeter or so away from the teeth. A voiced counterpart of &quot;sz&quot;

=== Letter Combinations ===

Combinations of letters, just like in English, produce different sounds than one would expect at first glance. All of these combinations are listed below. Please be aware, if you come across letter combinations that do not appear in this list, you must pronounce the letters in order as described above and not the way you are used to in the English language. The list is as follows:

'''ch''' - the same sound as &quot;h&quot;  (historically it was a voiced counterpart of &quot;h&quot; but it is no longer pronounced like that except eastern dialects of Polish spoken by old people) 

'''ci''' = ć + i.

'''cz''' - the same sound as ch in English &quot;cherry&quot; or Charles&quot;.

'''dz''' - voiced counterpart of &quot;c&quot; which should be pronounced hard to distinguish from &quot;dzi&quot;

'''dź''' - The &quot;Jee&quot; sound in the English word &quot;Jeepers&quot; but pronounced very soft. Voiced counterpart of &quot;ć&quot;.

'''dzi''' = dź + i

'''dż''' - The &quot;Jj&quot; sound in English words like &quot;Jupiter&quot; and &quot;Jar&quot;. Voiced counterpart of &quot;cz&quot;.

'''rz''' = ż (there is only one-word exception: word '(z)marznąć' (to freeze) is pronounced as separate r+z cluster, not like 'mażnąć') 

'''si''' = ś + i

'''sz''' - The same sound as sh in English &quot;shy&quot;

== Exercise ==

After trawling through that exhaustive list you may feel a little disheartened. Don't be! The Polish language becomes easy to pronounce after understanding these rules. To help you analyse how much you understand already, take a look at this exercise. The list of words below are actually English words, but they have been spelt in Polish phonetics. Work through these words and write down what words they represent. If you are able to do this successfully, you understand Polish phonetics, and the course will become much more enjoyable after.

{{columns-list|colwidth=10em|
# Demokrac
# Hęp
# Fót
# Buk
# Kuszon
# Hidz
# Kug
# Jir
# Śut
# Spęd
# Ćaryti
# Łąt
# Łajd
# Chob
# Sid
# Rycz
# Chidz
# Dzins
# Ełpen
# Szód
# Słąb
# Cip
# Dżapan
# Pełland
# Chów
# Huwz
# Jeleł
# Sensybyl
# Erejrzur
# Dżawelyn
}}

{{center top}}[[Talk:Polish language/Alphabet and phonetics|'''Answers''']]{{center bottom}}

[[Category:Polish language]]</text>
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  <page>
    <title>Happiness</title>
    <ns>0</ns>
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    <revision>
      <id>2818198</id>
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      <timestamp>2026-07-13T02:36:53Z</timestamp>
      <contributor>
        <username>WaterQuark</username>
        <id>2987724</id>
      </contributor>
      <minor />
      <comment>fix typos</comment>
      <origin>2818198</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4113" sha1="phxryj04nl2lhp2q2vgc517iohekuxk" xml:space="preserve">{{Essay}}
[[File:Happy girls.jpg|thumb|300px|Happy girls]]
'''Happiness is a mental or emotional state of well-being characterized by positive or pleasant emotions ranging from contentment to intense joy. A variety of biological, psychological, religious, and philosophical approaches have striven to define happiness and identify its sources.'''

[[w:Happiness|Wikipedia Happiness:]] ''Various research groups, including Positive psychology, endeavor to apply the scientific method to answer questions about what &quot;happiness&quot; is, and how we might attain it. Philosophers and religious thinkers often define happiness in terms of living a good life, or flourishing, rather than simply as an emotion. Happiness in this sense was used to translate the Greek Eudaimonia, and is still used in virtue ethics. Happiness economics suggests that measures of public happiness should be used to supplement more traditional economic measures when evaluating the success of public policy.''

'''---&gt; [http://www.smashwords.com/books/view/166363 Download the Wikiversity Book Happiness (basic version 2012) in various formats (PDF, Epub, Kindle) for reading and give away.]'''

==Contents==
#[[Philosophy of Happiness]]
#[[Happiness research]]
#[[Positive thinking]]
#[[Everyone needs a spiritual centering]]
#[[The Five Principles of Health]]
#[[Have a Happy Relationship]]
#[[Have a Happy Family]]
#[[Be a Happy Single]]
#[[Have a Happy Old Age]]
#[[Have a Happy Job]]
#[[Happiness and Life]]
#[[Happiness and Enlightenment]]
#[[Laughter Yoga]]
#[[A World of Peace, Love and Happiness]]
#[[A Psychological Interpretation of the Tarot|Happiness oracle (Tarot)]]
#[[Optimism]]

== Jokes ==
[[File:The moment of happiness.jpg|thumb]]
[[Angel oracle#Angel_of_Humor|Angel oracle]]: Laugh at life. Who rests in his own happiness, can laugh at the chaos of life. The angel of humor. He lives the joy. He makes jokes. He sees life as a game. Be positive as you go through the day. Do not take things too seriously. Today you can be a little wild and crazy. Have fun. :-) :-) ;-) :-)

'''Men and women'''

Nils: Men and women are different. There is no doubt about it. But rather than focus on the negative characteristics of men and women, why not praise the positives? Let's start with the women: women are compassionate, caring, and socially. Women cry when they are happy. Women bring joy and laughter in this world. Women are kind and loyal. They will not give up under any circumstances to get for the best for their children. Women have the ability to smile when they are so tired that she can hardly stand up. They know how to make a simple meal into a feast. They know how to comfort a sad friend. They know how to deal with children for endless hours! They know how to give a man the feeling to be king. Women make the world to a happier place.

Sybilla: WOW. Thank you .... ;-)) That feels really good just now!

Sonja: ..... Thaaaaaaaaaank you. Dear Nils, you nearly were a man to marry. Lol :)))

Nils: Alll my ten ex-women had an other point of view!

==See also==
{{commons|Happiness}}
*[[q:Happiness|Happiness]] (Wikiquote)
*[[b:Happiness|Happiness]] (Wikibooks)
*[[:Category:Motivation and emotion/Book/Happiness|Happiness]] (Wikiversity - Motivation and emotion book chapters)
*[[Interpersonal relationships]]
*[[Positive psychology]]
*[[Wikiphilosophers/Happiness]]

==External links==
{{wikipedia|Philosophy of happiness}}{{wikipedia|Happiness}}
*[http://www.wikihow.com/Become-Positive%2C-Happy-and-Optimistic How to Become Positive, Happy and Optimistic]
*[http://www.wikihow.com/Become-Positive-Through-Forgiveness How to Become Positive Through Forgiveness]
*[http://www.wikihow.com/Live-Permanently-in-Love How to Live Permanently in Love]
*[http://www.psychologytoday.com/basics/happiness psychology today/happiness] 
*[http://www.authentichappiness.sas.upenn.edu/Default.aspx Authentic Happiness (Martin Seligman)]
*[http://www.happinesspodcast.org/ happiness podcast]
*[http://happiness-project.com/ happiness-project]


__NOTOC__
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  <page>
    <title>User talk:Atcovi</title>
    <ns>3</ns>
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      <id>2818213</id>
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      <timestamp>2026-07-13T07:27:43Z</timestamp>
      <contributor>
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      <comment>/* The Signpost: 13 July 2026 */ new section</comment>
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      <text bytes="15409" sha1="mhgtlbe1lydnzepw230ov93lqugx6fo" xml:space="preserve">[[User:Atcovi/Archive 1|/Archive 1 (September 25, 2013 - November 15, 2013)]] • [[User talk:Atcovi/Archive 2|/Archive 2 (November 15, 2013 - November 27, 2013)]] • [[User talk:Atcovi/Archive 3|/Archive 3 (December 3, 2013 - December 25, 2013)]] • [[User talk:Atcovi/Archive 4|/Archive 4 (December 24, 2013 - January 1, 2014)]] • [[User talk:Atcovi/Archive 5|/Archive 5 (January 2, 2014 - January 20, 2014)]] • [[User talk:Atcovi/Archive 6|/Archive 6 (March 24, 2014 - April 14, 2014)]] • [[User talk:Atcovi/Archive 7|/Archive 7 (April 19, 2014 - September 8, 2014)]] • [[User talk:Atcovi/Archive 8|/Archive 8 (September 12, 2014 - November 3, 2014)]] • [[User talk:Atcovi/Archive 9|/Archive 9 (November 6, 2014 - January 26, 2015)]] • [[User talk:Atcovi/Archive 10|/Archive 10 (January 28, 2015 - March 11, 2015)]] • [[User talk:Atcovi/Archive 11|/Archive 11 (March 22, 2015 - June 25, 2016)]] • [[User talk:Atcovi/Archive 12 (June 26, 2016 - January 8, 2018)|/Archive 12 (June 26, 2016 - January 8, 2018)]] • [[User talk:Atcovi/Archive 13 (January 9, 2018 - April 14, 2023)|/Archive 13 (January 9, 2018 - April 14, 2023)]] • [[User talk:Atcovi/Archive 14 (April 15, 2023 - May 5, 2026)|/Archive 14 (April 15, 2023 - May 5, 2026)]]

:''Before 2013: [https://en.wikiversity.org/w/index.php?title=User_talk:Atcovi&amp;diff=750617&amp;oldid=740650 see this]'' 
{{tmbox
|small = 
|image = [[Image:Busy desk.svg|{{#ifeq:|yes|40px|75x50px}}]]
|text = This user is busy in [http://en.wikipedia.org/wiki/Real_life Real Life] {{#if:|until&amp;nbsp;{{{end}}}&amp;nbsp;}}{{#if:|due to&amp;nbsp;{{{reason}}}&amp;nbsp;}}and may not respond swiftly to queries.{{#if:|&lt;P&gt;{{{msg}}} }} 
| style     = {{#if:|width: {{{width}}}px;}} {{#ifeq:{{{shadow}}}|yes|{{box-shadow|0px|2px|4px|rgba(0,0,0,0.2)}}|}}
}}

== Please vote ==

on Wikinews rebirth possibly on Wikiversity, thanks @[[User:Atcovi|Atcovi]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:21, 15 May 2026 (UTC)

:Hi BigKrow. I've been watching the discussion on the sidelines. Hopefully I'll have an input soon, I just have other commitments I'm catering to. Best of luck with your projects and welcome to Wikiversity! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:44, 16 May 2026 (UTC)

== ''The Signpost'': 22 May 2026 ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot; style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''&lt;/div&gt;
&lt;div style=&quot;column-count:2;&quot;&gt;
* News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/News and notes|Offline: Osama Khalid still in prison]]

* In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In the media|Indonesian editors, you shall return!]]

* Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Disinformation report|Who is a typical paid editor? Who are their typical clients?]]

* Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Recent research|WikiLambda the Ultimate]]

* Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Traffic report|This is where I'll be, so heavenly, so come and dance with me Michael!]]

* Forum: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Forum|WikiAnnotate: help us build a dataset of article quality evaluations]]

* In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/In focus|Demystifying the 2026-27 Annual Plan]]

* Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Opinion|Wikipedia isn't a battleground. So why does it feel like one?]]

* Serendipity: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Serendipity|Wikinews: Into the Wikiverse]]

* Special report: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Special report|Wikimedia Foundation closes Wikinews after 21 years]]

* Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Community view|Wikipedia's traffic drop: more on languages and freshness]]

* Gallery: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Gallery|Earth Day and Mother's Day]]

* Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-05-22/Comix|Brother, can you spare a page?]]

&lt;/div&gt;
&lt;div style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;'''[[w:en:Wikipedia:Wikipedia Signpost|Read this Signpost in full]]''' · [[w:en:Wikipedia:Signpost/Single|Single-page]] · [[m:Global message delivery/Targets/Signpost|Unsubscribe]] · [[m:Global message delivery|Global message delivery]] 05:19, 22 May 2026 (UTC)
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== Wikiversity:Candidates for Bureaucratship/Atcovi ==

RE: [[Wikiversity:Candidates for Bureaucratship/Atcovi]] I have closed this as successful. Congrats! See [https://en.wikiversity.org/w/index.php?title=Wikiversity:Candidates_for_Bureaucratship/Atcovi&amp;diff=prev&amp;oldid=2812184] and [https://en.wikiversity.org/w/index.php?title=Special:Log&amp;logid=3549048]. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 19:23, 30 May 2026 (UTC)

:Thank you Mike! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:55, 30 May 2026 (UTC)
::Congratulations @[[User:Atcovi|Atcovi]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:58, 8 June 2026 (UTC)

== Question ==
Hello don't mean to bother and Ik its a silly question, on wikipedia there is a tool that allows for the creation of user boxes does wikiversity have it? Or should I create them myself like {{Userbox
| border-c = #000000
| id = [[File:(logo) Gate jieitai kanochi nite, kaku tatakaeri.svg|100x50px]]
| id-c = #000000
| id-fc = #000000
| id-s = 14
| info = This user testified in front of the [[National Diet|national diet]]
| info-c = #000000
| info-fc = #ffffff
| info-s = 8
}}
however when i try to display them like on wikipedia i can't
{{yytop}}
{{yy|User:AUBSTRAWBS/GATE}}
{{yyend}}

Anyways sorry to bother you with somthing like this but i'm really stumped as to how to share them. Any help would be super apreciated also if you want any user boxes i can make them :). {{unsigned|AUBSTRAWBS}}
:Hello {{ping|AUBSTRAWBS}} no need to worry about bothering me, I'm always happy to help. I think for Wikiversity you'd have to manually create them, as I have done so. For example: [[Template:User Sri Lankan]] &amp; [[Template:User soccer]]. If you'd like to bring over templates from Wikipedia, then feel free to just copy them and paste them here - tho it may be better just to manually create them as it could be a lengthy and messy process. I do think the way you've created the &quot;national diet&quot; userbox is perfect and achieves the intended goal. Let me know if you have any more questions! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:38, 9 June 2026 (UTC)

:Here's one I created just now: [[Template:User university student]]. More templates are listed here: [[:Category:User templates]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:45, 9 June 2026 (UTC)
:Thank you very much [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 23:02, 9 June 2026 (UTC)

== ''The Signpost'': 21 June 2026 ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot; style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''&lt;/div&gt;
&lt;div style=&quot;column-count:2;&quot;&gt;
* From the editors: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/From the editors|Ways for beginners to support ''The Signpost'' community journalism]]

* News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/News and notes|Community Tech development team disbanded]]

* Disinformation report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Disinformation report|PR for the people?]]

* Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Recent research|Proposed tagging system for AI involvement; successful and unsuccessful AI tools for contributors]]

* In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/In the media|Who won a 14th century battle and who won the 2026 Iran war?]]

* Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Community view|Putting the Wish into the Wishlist]]

* In focus: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/In focus|A global standard for Neutral Point of View]]

* On the bright side: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/On the bright side|Flowers, blue helmets, reefs, pride, and Juneteenth]]

* Op-ed: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Op-ed|Breathe, Don’t Panic, there is a different story about Wikimedia + AI futures]]

* Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Opinion|Wikimedia Foundation staff develop union and Wikimedia user community reacts]]

* Technology report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Technology report|Community Tech team is disbanded, controversy erupts]]

* Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Traffic report|'Cause this is thriller, thriller night]]

* WikiConference report: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/WikiConference report|Report of Volunteer Supporters Network Annual Meeting 2026]]

* Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Comix|Take your turn]]

* Humour: [[w:en:Wikipedia:Wikipedia Signpost/2026-06-21/Humour|Group of banned T-shirt makers comes out of hiding to sell new Wikipedia-themed merchandise]]

&lt;/div&gt;
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== 'Bioleninism' and 'Slave, Sister, Sexborg, Sphinx' ==

Hello, I just want to give my takes on the removal of my pages 'Bioleninism' and 'Slave, Sister, Sexborg, Sphinx'

Both of those terms are widely used online by adherents of a political movement known as '[[w:Dark Enlightenment|Dark Enlightenment]]'- the source I used for Bioleninism is the manifesto that coined the term, the same source is used on the Wikitionary entry for Bioleninism (https://en.wiktionary.org/wiki/Bioleninism) - whereas for 'Slave, Sister, Sexborg, Sphinx' I used three sources from [[w:Nick Land|Nick Land]], a Philosopher associated with the movement. 

The Wikipedia page for Nick Land's book [[w:Fanged Noumena|Fanged Noumena]] briefly mentions 'Slave, Sister, Sexborg, Sphinx' but doesn't get into deep detail.

Both of those articles are for people learning about The Dark Enlightenment movement, Wikiveristy has a page for [[Neocameralism|neocamerialism]] - which is another important Dark Enlightenment term - and I believe these two pages are necessarily as well for analysis.

Thank you! [[User:Apallo334|Apallo334]] ([[User talk:Apallo334|discuss]] • [[Special:Contributions/Apallo334|contribs]]) 04:28, 27 June 2026 (UTC)

:Hello {{ping|Apallo334}} and welcome to Wikiversity! Please note that the issue is not notability (I did a few quick web searches and was able to figure out that this is a notable topic), the issue is that these seem to be loner pages that are not integrated into a [[Wikiversity:Learning projects|learning project]]. Ideally, we want these pages to be under the same project (similar to [[Neocameralism/Definitions]]). So for example:
:*[[The Cracker Factory]] --&gt; [[Neocameralism/The Cracker Factory]]
:*[[Slave, Sister, Sexborg, Sphinx]] --&gt; [[Neocameralism/Slave, Sister, Sexborg, Sphinx]]
:Example projects which you can model your pages off of would be:
:#[[Introduction to Computers]]
:#[[Philosophy of science]]
:Here are some extra useful resources:
:*[[Wikiversity:Learning by doing]]
:*[[Wikiversity:What is Wikiversity?]]
:*[[Wikiversity:Scope]]
:Please let me know if you have any questions. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:51, 27 June 2026 (UTC)

== ''The Signpost'': 13 July 2026 ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot; style=&quot;margin-top:10px; font-size:90%; padding-left:5px; font-family:Georgia, Palatino, Palatino Linotype, Times, Times New Roman, serif;&quot;&gt;[[File:WikipediaSignpostIcon.svg|40px|right]] ''News, reports and features from the English Wikipedia's newspaper''&lt;/div&gt;
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* News and notes: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/News and notes|An exclusive club]]

* In the media: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/In the media|Battle for a soul - who won?]]

* Opinion: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Opinion|We need to innovate with Wikimedia decision-making]]

* Recent research: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Recent research|LLMs and NPOV, 20 years of user blocks, Esperanto and Volapük Wikipedias]]

* News from Diff: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/News from Diff|How to host Wikicurious in your own community]]

* Community view: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Community view|CUNY Newmark Wikimedian-in-Residence Quarterly Brief – April to June 2026]]

* Special report: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Special report|Wikipedia escapes Category 1 designation under the UK Online Safety Act - for now]]

* On the bright side: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/On the bright side|Fatherhood, weather, and diplomacy]]

* Op-ed: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Op-ed|A Layup Easy Proposal for Wikimedia at 25: Spend 25% of Donations on the Community]]

* Traffic report: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Traffic report|The grass was greener, the light was brighter]]

* Comix: [[w:en:Wikipedia:Wikipedia Signpost/2026-07-13/Comix|schnozzed]]

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      <sha1>mhgtlbe1lydnzepw230ov93lqugx6fo</sha1>
    </revision>
  </page>
  <page>
    <title>Computer Skills/Basic/Internet</title>
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      <id>2818195</id>
      <parentid>2811076</parentid>
      <timestamp>2026-07-13T02:29:16Z</timestamp>
      <contributor>
        <username>~2026-39316-68</username>
        <id>3100648</id>
      </contributor>
      <comment>/* Note on Artificial Intelligence */ Added a note on AI warning learners about the dangers of blindly trusting AI assistants.</comment>
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Basic Internet concepts include browser navigation, bookmarks, search terms, web addresses and hyperlinks.

== Objectives and Skills ==
Objectives and skills for Internet concepts include:&lt;ref&gt;[https://pdfhost.io/v/kZr9QRh6MH_University_of_New_South_Wales__Computer_Skills_Assessment_Framework University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Understand how to navigate using a browser
* Use bookmark/favorites
* Understand simple search terms/conventions
* Locate web address
* Understand concept of hyperlink

== Multimedia ==
# [https://www.youtube.com/watch?v=FxirRVJWUTs YouTube: Browser Basics]

== Activities ==
# Complete the tutorial [https://www.learnfree.org/series/internet-basics Internet Basics].
# Identify the name of your browser application (Google Chrome, Microsoft Internet Explorer, Mozilla Firefox, Apple Safari, etc.).
# Identify parts of the user interface such as the address bar, navigation buttons, the search bar, and links.
# Add a bookmark in your browser for the [[Computer Skills/Basic]] page.

== Note on Artificial Intelligence (AI) Assistants ==
AI assistants are starting to be built into the most popular search engines, showing answers to queries before human made websites. It's recommended to take these answers with a grain of salt, since as of October 22nd 2025, there have been studies&lt;ref&gt;{{Cite web|url=https://www.bbc.co.uk/mediacentre/2025/new-ebu-research-ai-assistants-news-content/|title=Largest study of its kind shows AI assistants misrepresent news content 45% of the time – regardless of language or territory|website=www.bbc.co.uk|language=en|access-date=2026-07-13}}&lt;/ref&gt; showing that AI summaries are often inaccurate. As time goes on AI may get more accurate but generally be careful with the answers produced by these assistants.

== See Also ==
* [[../../Intermediate/Internet]]
* [[../../Advanced/Internet]]
* [[../../Proficient/Internet]]
* [[Internet Fundamentals/Web Browsers]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Internet]]
[[Category:Web browsers]]
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  <page>
    <title>Octave programming in plain view</title>
    <ns>0</ns>
    <id>150328</id>
    <revision>
      <id>2818166</id>
      <parentid>2460042</parentid>
      <timestamp>2026-07-12T16:18:40Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <minor />
      <comment>added [[Category:GNU Octave]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="2363" sha1="myrluw3j93zysud5n9lilb3y91u1v5v" xml:space="preserve">==''' Part I '''==

&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;
===   Handling Repetition ===
 
* Control ([[Media:Octave.Repeat1.Control.1.A.20150817.pdf |pdf]])

* Loop ([[Media:Octave.Repeat2.Loop.1.A.20150822.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Octave.Work1.Function.1.A.20151023.pdf |pdf]])
* Scripts ([[Media:Octave.Work2.Script.1.A.20151009.pdf |pdf]])
* Scope ([[Media:Octave.Work3.Scope.1.A.20151008.pdf |pdf]])
 
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Octave.Series1.Array.1.A.20150915.pdf |pdf]])
* Structures ([[Media:Octave.Series2.Structure.1.A.20150828.pdf |pdf]])
* Cell Arrays ([[Media:Octave.Series3.CellArray.1.A.20151016.pdf |pdf]])
* Comma Separated Lists ([[Media:Octave.Series4.List.1.A.20150908.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Typecasts ([[Media:Octave.Data2.Typecast.1.A.20150918.pdf |pdf]])
* Operators ([[Media:Octave.Data3.Operators.1.A.20151217.pdf |pdf]])
* Files ([[Media:Octave.Data4.File.1.A.20150917.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Class 
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

==''' Part II '''==

&lt;!----------------------------------------------------------------------&gt;

=== Octave in Complex Analysis ===
* [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf |Complex Functions]] 
* [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf |Complex Integrals ]]
* [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf |Complex Series]]
&lt;/br&gt;

=== Octave in Control Systems ===
* [[Media:CCAS.2.A.FreqAnal.20141001.pdf |Frequency Domain Analysis]]
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

===External links===
* [http://octave.sourceforge.net/control/ Octave control package] 
* [http://octave.sourceforge.net/control/ Octave signal processing package]

[[Category:Computer programming]]
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  <page>
    <title>Thailand</title>
    <ns>0</ns>
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      <timestamp>2026-07-13T07:30:49Z</timestamp>
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        <id>3100681</id>
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      <text bytes="1138" sha1="7je4wogm2vke1okhb7caxb0dsaoi6jn" xml:space="preserve">Thailand, officially the Kingdom of Thailand, formerly known as Siam, is a country at the centre of the Indochina peninsula in Southeast Asia. It is bordered to the north by Burma and Laos, to the east by Laos and Cambodia, to the south by the Gulf of Thailand and Malaysia, and to the west by the Andaman Sea and the southern extremity of Burma. Its maritime boundaries include Vietnam in the Gulf of Thailand to the southeast, and Indonesia and India on the Andaman Sea to the southwest.&lt;ref&gt;[[Wikipedia: Thailand]]&lt;/ref&gt;

Thailand is also popular travel destination because of its nightlife. In Bangkok and Pattaya, there are a variety of nightlife venues&lt;ref&gt;{{Cite web|url=https://thaiolddriver.com/|title=泰國老司機 Thai Old Driver｜泰國夜生活與成人娛樂目錄指南|website=泰國老司機 Thai Old Driver｜泰國夜生活目錄指南|language=zh-TW|access-date=2026-07-13}}&lt;/ref&gt;, such as nightclubs, bars, soapy massage parlor.

== Resources ==
* [[/National Elephant Day/]]

== See also ==
* [[Comparative law and justice/Thailand]]
* [[Cooking/Recipes/Thai]]

== References ==
{{Reflist}}

[[Category:Thailand]]</text>
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  </page>
  <page>
    <title>Computer Skills/Basic/Hardware</title>
    <ns>0</ns>
    <id>158061</id>
    <revision>
      <id>2818199</id>
      <parentid>2811078</parentid>
      <timestamp>2026-07-13T02:38:31Z</timestamp>
      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <minor />
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Computer hardware covered in this section includes the computer case, the display, the keyboard, mouse, computer parts, cables, and connectors.

== Objectives and Skills ==
Objectives and skills for basic hardware include:&lt;ref&gt;[https://pdfhost.io/v/kZr9QRh6MH_University_of_New_South_Wales__Computer_Skills_Assessment_Framework University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Identify component parts: cables, etc

== Multimedia ==
# [https://www.youtube.com/watch?v=mLgTnkw558w Computer Basics - Basic Parts of a Computer]

== Activities ==
# Complete the tutorial [https://www.learnfree.org/series/computer-basics Getting to Know Computers].  Make sure to complete the quiz at the end of the course.
# Learn more about [https://red-dot-geek.com/basic-computer-parts-functions/ Basic Computer Components]. A Computers internal architectural design comes in different types and sizes, but the basic structure remains same of all computer systems.
#Learn about the [https://artoftesting.com/block-diagram-of-computer Block Diagram of Computer]. A computer can perform major computer operations or functions irrespective of their size and make.
#Learn about other [https://www.geeksforgeeks.org/computer-science-fundamentals/types-of-computers/ Types of Computers]. There are several other kinds of computers that may interest you.

== See Also ==
* [[../../Intermediate/Hardware]]
* [[../../Advanced/Hardware]]
* [[../../Proficient/Hardware]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Computer hardware]]
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  </page>
  <page>
    <title>Computer Skills/Basic/Software Concepts</title>
    <ns>0</ns>
    <id>158068</id>
    <revision>
      <id>2818197</id>
      <parentid>2811079</parentid>
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      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <minor />
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Basic software concepts include operating systems and applications.

== Objectives and Skills ==
Objectives and skills for basic software concepts include:&lt;ref&gt;[https://pdfhost.io/v/kZr9QRh6MH_University_of_New_South_Wales__Computer_Skills_Assessment_Framework University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Distinguish what is software / hardware
* Understand menu bars; etc
* Understand basic terminology: ﬁle; application
* Understand purpose of backup

== Multimedia ==
# [https://www.youtube.com/watch?v=3gMOYZoMtEs YouTube: Computer Basics: Understanding Applications]

== Activities ==
# Complete the tutorial [https://www.learnfree.org/series/basic-computer-skills Basic Computer Skills]. This tutorial will teach you how to set up, use, and customize software, while reviewing previously learned concepts.
# Learn how to [https://www.pcmag.com/explainers/your-future-self-will-thank-you-beginners-guide-to-backing-up-your-pc Back Up Your Files].  Create a cloud backup or a physical backup of your important files.
:

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Software]]
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    <revision>
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        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <minor />
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Basic software concepts include operating systems and applications.

== Objectives and Skills ==
Objectives and skills for basic software concepts include:&lt;ref&gt;[https://pdfhost.io/v/kZr9QRh6MH_University_of_New_South_Wales__Computer_Skills_Assessment_Framework University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Distinguish what is software / hardware
* Understand menu bars; etc
* Understand basic terminology: ﬁle; application
* Understand purpose of backup

== Multimedia ==
# [https://www.youtube.com/watch?v=3gMOYZoMtEs Computer Basics: Understanding Applications]

== Activities ==
# Complete the tutorial [https://www.learnfree.org/series/basic-computer-skills Basic Computer Skills]. This tutorial will teach you how to set up, use, and customize software, while reviewing previously learned concepts.
# Learn how to [https://www.pcmag.com/explainers/your-future-self-will-thank-you-beginners-guide-to-backing-up-your-pc Back Up Your Files].  Create a cloud backup or a physical backup of your important files.
:

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Software]]
[[Category:Completed resources]]</text>
      <sha1>gazwpm4ykaogikgqn1sb5ncp00ztz9i</sha1>
    </revision>
  </page>
  <page>
    <title>Computer Skills/Basic/Word Processing</title>
    <ns>0</ns>
    <id>158766</id>
    <revision>
      <id>2818201</id>
      <parentid>2811085</parentid>
      <timestamp>2026-07-13T02:40:44Z</timestamp>
      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <comment>/* Activities */ Moved LearnFree tutorial to activites</comment>
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Basic word processing operations include editing and saving files, using the clipboard, recognizing user interface features, spell check, and printing files.

== Objectives and Skills ==
Objectives and skills for basic word processing concepts include:&lt;ref&gt;[https://pdfhost.io/v/kZr9QRh6MH_University_of_New_South_Wales__Computer_Skills_Assessment_Framework University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Create a new document: enter text; delete; save; save as
* Use basic edit features: cut; copy; paste 
* Identify features: tool bar; icons; cursor 
* Use dictionary; spell check
* Operate print; print preview

== Multimedia ==
# [https://www.youtube.com/watch?v=j-ZAVHk5SaU Word: Getting Started]
# [https://www.youtube.com/watch?v=PafCMUVH_OA Word: Creating and Opening Documents]
# [https://www.youtube.com/watch?v=iHuFzz7Wvt4 Word: Saving and Sharing Documents]
# [https://www.youtube.com/watch?v=vmEzxQfVj5c Word: Text Basics]
# [https://www.youtube.com/watch?v=jgNpoksYOLE Word: Page Layout]
# [https://www.youtube.com/watch?v=uzrpa-gwN1A Word: Check Spelling and Grammar]
# [https://www.youtube.com/watch?v=7bLQFTCsH8Y Word: Printing]

== Activities ==
Complete the tutorial [https://www.learnfree.org/series/word-2016 Word 2016] by LearnFree.org.

These activities may be completed using any word processing application (Microsoft Word, LibreOffice Writer, Google Document, Apple Pages, etc.):
# Start your word processing application.  It should automatically open with a new, blank document.
# Enter two paragraphs of text in your new document.  You can write about the user interface features you see (toolbar, icons, menus, status bar, scroll bars, etc.) and where they are located.
# Save your document.
# Copy the first paragraph and paste it at the end of the document, creating a third paragraph.
# Delete the duplicate third paragraph.
# Use Spell Check to verify and correct spelling for your document.
# Use Print Preview to see the page layout for your document.
# If a printer is available, print your document.
# Use Save As to save your document with a new name.

== See Also ==
* [[Computer Skills/Intermediate/Word Processing]]
* [[Computer Skills/Advanced/Word Processing]]
* [[Computer Skills/Proficient/Word Processing]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Word processing]]
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  </page>
  <page>
    <title>Computer Skills/Basic/Formatting</title>
    <ns>0</ns>
    <id>158767</id>
    <revision>
      <id>2818202</id>
      <parentid>2805570</parentid>
      <timestamp>2026-07-13T02:41:09Z</timestamp>
      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <minor />
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      <text bytes="1888" sha1="1xy6edb498y35r6q5gmkrmaga9yd7kq" xml:space="preserve">{{:{{BASEPAGENAME}}/Sidebar}}
Simple formatting includes fonts, font size, font style, font color, and text alignment.

== Objectives and Skills ==
Objectives and skills for simple formatting include:&lt;ref&gt;[http://www.eaa.unsw.edu.au/forms/pdf/icas/subjects/computer-skills-framework.pdf University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Apply basic formatting: font; font size; font style; color; align text

== Multimedia ==
# [https://www.youtube.com/watch?v=4RSrSNQZj9A Word 2010: Text Formatting]

== Activities ==
These activities may be completed using any word processing application (Microsoft Word, LibreOffice Writer, Google Docs, Apple Pages, etc.):
# Start your word processing application and open a new, blank document.
# Enter two paragraphs of text in your new document.  You can write about the formatting features you see (font, font size, font style; color, text alignment, etc.) and where they are located.
# Save your document.
# Add a heading at the top of your document labeled 'Simple Formatting'.
# Change the font for the heading to a sans-serif font (Calibri, Arial, Helvetica, Sans, etc.). 
# Change the font for the body paragraphs to a serif font (Cambria, Times, Times New Roman, Serif, etc.).
# Change the font size for the heading to 18 point.
# Change the font size for the body paragraphs to 12 point.
# Change the heading font style to bold and italic.
# Change the heading font color to blue.
# Change the heading text alignment to centered.
# Change the body paragraphs text alignment to justified.
# Save your document.

== See Also ==
* [[Computer Skills/Intermediate/Formatting]]
* [[Computer Skills/Advanced/Formatting]]
* [[Computer Skills/Proficient/Formatting]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Word processing]]
[[Category:Completed resources]]</text>
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    </revision>
    <revision>
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      <timestamp>2026-07-13T02:42:12Z</timestamp>
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        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
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Simple formatting includes fonts, font size, font style, font color, and text alignment.

== Objectives and Skills ==
Objectives and skills for simple formatting include:&lt;ref&gt;[http://www.eaa.unsw.edu.au/forms/pdf/icas/subjects/computer-skills-framework.pdf University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Apply basic formatting: font; font size; font style; color; align text

== Multimedia ==
# [https://www.youtube.com/watch?v=ViGf0RKbCyA Word: Formatting Text]

== Activities ==
These activities may be completed using any word processing application (Microsoft Word, LibreOffice Writer, Google Docs, Apple Pages, etc.):
# Start your word processing application and open a new, blank document.
# Enter two paragraphs of text in your new document.  You can write about the formatting features you see (font, font size, font style; color, text alignment, etc.) and where they are located.
# Save your document.
# Add a heading at the top of your document labeled 'Simple Formatting'.
# Change the font for the heading to a sans-serif font (Calibri, Arial, Helvetica, Sans, etc.). 
# Change the font for the body paragraphs to a serif font (Cambria, Times, Times New Roman, Serif, etc.).
# Change the font size for the heading to 18 point.
# Change the font size for the body paragraphs to 12 point.
# Change the heading font style to bold and italic.
# Change the heading font color to blue.
# Change the heading text alignment to centered.
# Change the body paragraphs text alignment to justified.
# Save your document.

== See Also ==
* [[Computer Skills/Intermediate/Formatting]]
* [[Computer Skills/Advanced/Formatting]]
* [[Computer Skills/Proficient/Formatting]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Word processing]]
[[Category:Completed resources]]</text>
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    </revision>
  </page>
  <page>
    <title>Computer Skills/Basic/Graphics</title>
    <ns>0</ns>
    <id>158777</id>
    <revision>
      <id>2818206</id>
      <parentid>2807141</parentid>
      <timestamp>2026-07-13T02:46:35Z</timestamp>
      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <minor />
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Basic graphics includes graphics manipulation and drawing tools.

== Objectives and Skills ==
Objectives and skills for simple formatting include:&lt;ref&gt;[http://www.eaa.unsw.edu.au/forms/pdf/icas/subjects/computer-skills-framework.pdf University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Manipulate graphics: resize; rotate; flip; order; color
* Understand function of drawing tools

== Multimedia ==
# [https://www.youtube.com/watch?v=hlnodEYZ71w How to Use: Microsoft Paint]

== Activities ==
These activities may be completed using any drawing or graphics application (Microsoft Paint, Paintbrush, LibreOffice Draw, Google Drawings, GIMP, etc.).
# Start your graphics application. It should automatically open with a new, blank drawing.  If it does not, add a new blank drawing.
# Either insert an image into your drawing or paste an image from the clipboard.
# Select the image.  Drag the border to resize the image.
# Rotate the selected image.  You should be able to rotate it 90 or 180 degrees.  Some graphics applications will allow more precise rotations.
# Flip the selected image.  You should be able to flip the image both horizontally and vertically.
# Insert or paste a second image.  Move the image to layer it on top of the first image.  Some graphics programs allow you to have multiple layers of images and change the order (which image is on top as viewed).
# Use the pen or color fill feature to add color to your drawing.
# Use the text feature to add a title and your name to the drawing.
# Save your drawing.
# Print your drawing.

== See Also ==
* [[Computer Skills/Intermediate/Graphics]]
* [[Computer Skills/Advanced/Graphics]]
* [[Computer Skills/Proficient/Graphics]]

== References ==
{{Reflist}}
{{subpage navbar}}

[[Category:Computer Skills]]
[[Category:Graphics]]
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  <page>
    <title>Computer Skills/Basic/Presentations</title>
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      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <comment>/* Multimedia */ Removed unnecessary videos. /* Activities */ Added Introduction to Microsoft PowerPoint course by LearnFree.</comment>
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      <model>wikitext</model>
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      <text bytes="2238" sha1="ljo3e18ukza8l23ckc73lixlvx76z79" xml:space="preserve">{{:{{BASEPAGENAME}}/Sidebar}}
Simple presentations includes creating a simple slideshow.

== Objectives and Skills ==
Objectives and skills for simple formatting include:&lt;ref&gt;[http://www.eaa.unsw.edu.au/forms/pdf/icas/subjects/computer-skills-framework.pdf University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Create simple PowerPoint / Google Slides presentation. 

== Multimedia ==
# [https://www.youtube.com/watch?v=k6pg4nZS6fA PowerPoint: Getting Started]

== Activities ==
Complete [https://www.learnfree.org/series/introduction-to-microsoft-powerpoint Introduction to Microsoft PowerPoint].

These activities may be completed using any presentation application (Microsoft PowerPoint, LibreOffice Impress, Google Slides, Apple Keynote, etc.):
# Start your presentation application.  It should automatically open with a new, blank presentation.
# Add a title and your name.  Title the presentation 'Presentations'.
# Add a new slide.  Title the slide 'Text'.  Add text that describes the text options available in your presentation application.
# Add a new slide.  Title the slide 'Images'.  Add an image to the slide.
# Select a theme for your presentation.  Apply the theme to all slides.
# Check spelling for your presentation.
# Save your presentation.
# Practice presenting your presentation.
# Print your presentation.
If you would like to learn more about how to create a more technical PowerPoint presentation that stands out, LearnFree has an extensive [https://www.learnfree.org/series/powerpoint-2016 PowerPoint 2016 tutorial].

== Opensource Alternatives ==
* [https://www.youtube.com/watch?v=MOv2Pg1rJKU&amp;list=PLy7Kah3WzqrHun-wvFV7hGO-Tyeh2w686 LibreOffice Impress]&lt;ref&gt;LibreOffice Impress (01) – Introduction to Impress (2016) by TheFrugalComputerGuy - Youtube Video - URL: https://www.youtube.com/watch?v=MOv2Pg1rJKU&amp;list=PLy7Kah3WzqrHun-wvFV7hGO-Tyeh2w686 (accessed 2020/03/10)&lt;/ref&gt;

== See Also ==
* [[Computer Skills/Intermediate/Presentations]]
* [[Computer Skills/Advanced/Presentations]]
* [[Computer Skills/Proficient/Presentations]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Presentations]]
[[Category:Completed resources]]</text>
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      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <comment>Removed open source alternative section.</comment>
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Simple presentations includes creating a simple slideshow.

== Objectives and Skills ==
Objectives and skills for simple formatting include:&lt;ref&gt;[http://www.eaa.unsw.edu.au/forms/pdf/icas/subjects/computer-skills-framework.pdf University of New South Wales: Computer Skills Assessment Framework]&lt;/ref&gt;
* Create simple PowerPoint / Google Slides presentation. 

== Multimedia ==
# [https://www.youtube.com/watch?v=k6pg4nZS6fA PowerPoint: Getting Started]

== Activities ==
Complete [https://www.learnfree.org/series/introduction-to-microsoft-powerpoint Introduction to Microsoft PowerPoint].

These activities may be completed using any presentation application (Microsoft PowerPoint, LibreOffice Impress, Google Slides, Apple Keynote, etc.):
# Start your presentation application.  It should automatically open with a new, blank presentation.
# Add a title and your name.  Title the presentation 'Presentations'.
# Add a new slide.  Title the slide 'Text'.  Add text that describes the text options available in your presentation application.
# Add a new slide.  Title the slide 'Images'.  Add an image to the slide.
# Select a theme for your presentation.  Apply the theme to all slides.
# Check spelling for your presentation.
# Save your presentation.
# Practice presenting your presentation.
# Print your presentation.
If you would like to learn more about how to create a more technical PowerPoint presentation that stands out, LearnFree has an extensive [https://www.learnfree.org/series/powerpoint-2016 PowerPoint 2016 tutorial].

== See Also ==
* [[Computer Skills/Intermediate/Presentations]]
* [[Computer Skills/Advanced/Presentations]]
* [[Computer Skills/Proficient/Presentations]]

== References ==
{{Reflist}}
{{subpage navbar}}

{{CourseCat}}
[[Category:Computer Skills]]
[[Category:Presentations]]
[[Category:Completed resources]]</text>
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    <title>MATLAB/Octave</title>
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        <username>Mu301</username>
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      </contributor>
      <minor />
      <comment>added [[Category:GNU Octave]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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{{under construction|'''WARNING:''' This procedure for downloading Octave and Notepad++ needs to be tested.  Please fix the inevitable errors made in constructing this procedure!}}

==Installing GNU Octave 3.8 (GUI version)==
As of 9/1/2014 we were unable install the latest version of Octave, which is a 3.8 version with  a Matlab-like GUI.  
==Installing GNU Octave 3.6.4==
This runs in batch mode, which will at first seem alien to those accustomed to the menu of a GUI.  But after a while, typing in the command mode will seem easier.  The only drawback is that you need to be confident of your MATLAB coding skills because you won't want to run the code every time you write a new line.

===Command window===
* It is important to learn these commands
(For more information see https://www.gnu.org/software/octave/doc/interpreter/System-Utilities.html)
# '''help *command*''' gives help for a command.
# '''cd *directory*''' moves into the named directory
# '''dir''' shows the contents of the current directory
# '''cd ..''' moves up to the parent directory
# '''mkdir *directory*''' creates the named directory in the current directory
# '''rmdir *diretory*''' removes the named directory
# '''rename *oldfile* *newfile*''' renames a file
#  '''copyfile *oldfile* *newfile*''' changes from the old filename to the new.
# '''pwd''' shows the path to the current working directory (useful for knowing how to find the folder using Windows)
# '''q''' stops the output (but does not display if one attempts to learn about it using '''help q''')
# '''path''' prints out a lot of something and is why you need to learn the '''q''' command.
# '''diary *filename*''' creates a file with a record of command window statements
# '''quit''' quits the program (if you close without using it you get an error message)

===Organizing files===
Here is one procedure for organizing your files
# Make a folder in a convenient place.  It can either be your desktop, or if you are using campus computers, a flash drive or the &quot;campus cloud&quot;.  For our purposes this folder will be called ''OctaveConvenient''.  You will install shortcuts to Octave and Notepad++ into ''OctaveConvenient''.  You might want to store internet to some of the '''[[MATLAB#Links|links]]''' already mentioned.

===Installing===
# Install Octave 3.6.4 from http://www.gnu.org/software/octave/ Verify that it works by opening it and typing ''2+3'' (enter).  You should get 5.  Then type ''quit''
# Install Notepad++ from http://notepad-plus-plus.org/. 
# Make a bookmark to our &quot;textbook&quot; at https://www.gnu.org/software/octave/doc/interpreter/
## Another resource ishttp://en.flossmanuals.net/command-line/gnu-octave/
# If you are using Windows, make a file that we shall call ''Octave_convenient'' that will contain
## shortcuts to these programs
## a journal written in either Notepad++ or your favorite word processor
## shortcuts to the files that Octave uses to store files and programs (you will create these later)
# From Octave, use '''pwd''' to deduce  where Octave resides and put a shortcut to that place in your folder.  Use '''dir''' to see what folders you have and '''cd''' ''newname'' where ''newname'' is a folder that you will use to store your programs.  Write the as ''newprogram.m'' files but call them from Octave by typing ''newname'' without the .m extension.

===First batch program===
Open Octave and type '''dir''' to verify that no folder called ''myfiles'' exists.  Then type

&lt;pre&gt;mkdir myfiles&lt;/pre&gt;

Verify that you have created ''myfiles'' by typing '''dir'''.  Then type

&lt;pre&gt;pwd&lt;/pre&gt;

to find the location ''myfiles'' on your computer.  Use that information to create a shortcut to ''myfiles'' in ''Octave_convenient''.

Open Notepad++ and write the following:
&lt;pre&gt;
close all; clear all;clc;
s='hello world'
&lt;/pre&gt;
Do NOT put a semicolon ; after the last line. Save the code as a ''hello.m'' (MATrix Laboratory) file, and save it in your myfiles directory that you see when you type '''dir''' from the command window in Octave.  Use '''cd myfiles''' to go into that folder and verify that ''hello.m'' is in your active directory.  Now type into your command window:

&lt;pre&gt;hello&lt;/pre&gt;

You should see &quot;hello world&quot;.  Now you can type 
 quit
Congratulations, you have just run your first batch program!


==Footnotes==
&lt;references/&gt;

==See also==
* [[datamash]] command-line program which performs basic numeric and statistical operations

[[Category:GNU Octave]]</text>
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{{under construction|'''WARNING:''' This procedure for downloading Octave and Notepad++ needs to be tested.  Please fix the inevitable errors made in constructing this procedure!}}
{{note|This resource describes versions of Octave that were released before 2015.}}

==Installing GNU Octave 3.8 (GUI version)==
As of 9/1/2014 we were unable install the latest version of Octave, which is a 3.8 version with  a Matlab-like GUI.  
==Installing GNU Octave 3.6.4==
This runs in batch mode, which will at first seem alien to those accustomed to the menu of a GUI.  But after a while, typing in the command mode will seem easier.  The only drawback is that you need to be confident of your MATLAB coding skills because you won't want to run the code every time you write a new line.

===Command window===
* It is important to learn these commands
(For more information see https://www.gnu.org/software/octave/doc/interpreter/System-Utilities.html)
# '''help *command*''' gives help for a command.
# '''cd *directory*''' moves into the named directory
# '''dir''' shows the contents of the current directory
# '''cd ..''' moves up to the parent directory
# '''mkdir *directory*''' creates the named directory in the current directory
# '''rmdir *diretory*''' removes the named directory
# '''rename *oldfile* *newfile*''' renames a file
#  '''copyfile *oldfile* *newfile*''' changes from the old filename to the new.
# '''pwd''' shows the path to the current working directory (useful for knowing how to find the folder using Windows)
# '''q''' stops the output (but does not display if one attempts to learn about it using '''help q''')
# '''path''' prints out a lot of something and is why you need to learn the '''q''' command.
# '''diary *filename*''' creates a file with a record of command window statements
# '''quit''' quits the program (if you close without using it you get an error message)

===Organizing files===
Here is one procedure for organizing your files
# Make a folder in a convenient place.  It can either be your desktop, or if you are using campus computers, a flash drive or the &quot;campus cloud&quot;.  For our purposes this folder will be called ''OctaveConvenient''.  You will install shortcuts to Octave and Notepad++ into ''OctaveConvenient''.  You might want to store internet to some of the '''[[MATLAB#Links|links]]''' already mentioned.

===Installing===
# Install Octave 3.6.4 from http://www.gnu.org/software/octave/ Verify that it works by opening it and typing ''2+3'' (enter).  You should get 5.  Then type ''quit''
# Install Notepad++ from http://notepad-plus-plus.org/. 
# Make a bookmark to our &quot;textbook&quot; at https://www.gnu.org/software/octave/doc/interpreter/
## Another resource ishttp://en.flossmanuals.net/command-line/gnu-octave/
# If you are using Windows, make a file that we shall call ''Octave_convenient'' that will contain
## shortcuts to these programs
## a journal written in either Notepad++ or your favorite word processor
## shortcuts to the files that Octave uses to store files and programs (you will create these later)
# From Octave, use '''pwd''' to deduce  where Octave resides and put a shortcut to that place in your folder.  Use '''dir''' to see what folders you have and '''cd''' ''newname'' where ''newname'' is a folder that you will use to store your programs.  Write the as ''newprogram.m'' files but call them from Octave by typing ''newname'' without the .m extension.

===First batch program===
Open Octave and type '''dir''' to verify that no folder called ''myfiles'' exists.  Then type

&lt;pre&gt;mkdir myfiles&lt;/pre&gt;

Verify that you have created ''myfiles'' by typing '''dir'''.  Then type

&lt;pre&gt;pwd&lt;/pre&gt;

to find the location ''myfiles'' on your computer.  Use that information to create a shortcut to ''myfiles'' in ''Octave_convenient''.

Open Notepad++ and write the following:
&lt;pre&gt;
close all; clear all;clc;
s='hello world'
&lt;/pre&gt;
Do NOT put a semicolon ; after the last line. Save the code as a ''hello.m'' (MATrix Laboratory) file, and save it in your myfiles directory that you see when you type '''dir''' from the command window in Octave.  Use '''cd myfiles''' to go into that folder and verify that ''hello.m'' is in your active directory.  Now type into your command window:

&lt;pre&gt;hello&lt;/pre&gt;

You should see &quot;hello world&quot;.  Now you can type 
 quit
Congratulations, you have just run your first batch program!


==Footnotes==
&lt;references/&gt;

==See also==
* [[datamash]] command-line program which performs basic numeric and statistical operations

[[Category:GNU Octave]]</text>
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        <id>3705</id>
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==This quiz is different because it also runs on GNU Octave==
This quiz was one of the first wikiquizzes, created so inartfully using Excel that it was impossible to add to the initial three versions that were used at the time.

The reader may note that this quiz uses scientific notation in exponential form using three digits ''E001'' instead of MatLabs customary ''E01''.  This testbank was the first to created by GNU Octave (months after a quiz-making program was written for MatLab).  Octave is the open source alternative to MatLab that is more difficult to use because its GUI is not user-friendly.  For that reason, as of November 2014, there are no immediate plans to use Octave to develop more wikiquizzes.  However at some point, it would be nice if open source tests were written using open source software.  (Also, it might be better to use another open source language such as Python or Ruby).

To run this code in Octave, put these two files into one folder and run ''main.m''.  For more information visit '''[[MATLAB/Octave]]'''.

==testeditor.m==
&lt;pre&gt;
function [ wikiName, wikiHeadABC, wikiHeadD_Z,wikiFirst, wikiCell, wikiMatrix, wikiType] = testeditor(  ) %NO EDIT
%% Change the following lines as needed:
wikiName='pe07-Cart1'; %Name of wikiquiz
wikiHeadABC='[[/testbank/]]   ' ; %What follows is long, so we concatenate:
wikiHeadABC=[wikiHeadABC '[http://www.wright.edu/~guy.vandegrift'];
wikiHeadABC=[wikiHeadABC '/openstaxphysics/quiz_solutions/quiz_solutions.htm'];
wikiHeadABC=[wikiHeadABC ' Solutions to many wikiquiz questions (pdf)]'] ; 
wikiHeadD_Z='Testbank';%'[[Category:Physics_equations/Testbanks]]';
wikiFirst='noIntroLine';% (If 'noIntroLine' there is no introductory line)
wikiType =1;%Set wikiType to 1 or 2
%   If WikiType=1, wrong answers are 
%       multiplied by 10^(n/N) where n is a small integer and N=wikiDR
%   If WikiType=2, wrong answers are 
%       larger by n*N where n is a small integer and N=wikiDR
%               
wikiQI=0; %NO EDIT  (counts the questions)
%
% Copy and paste from %%BEGIN QUESTION%% to %%END QUESTION%% as needed
%% BEGIN QUESTION 1%%
wikiS='If the initial velocity after leaving the spring is '; % S0 DONE
wikiV=randi([21,99])/10;  v=wikiV ; % V1 DONE (inserted earlier to test: wikiV=5;)
wikiS= [wikiS num2str(wikiV)];  %V2 DONE
wikiS=[wikiS,' m/s, how high does it reach before coming to rest?']; % S1 DONE
wikiS=[wikiS,'[[File:Roller coaster energy conservation.jpg|right|280px]]']; % S1 DONE
answer=v^2/2/9.8; 
wikiA =answer; 
wikiU =' m';
wikiDR = 12; 
wikiSigfig=2;
wikiQI=wikiQI+1;wikiCell{wikiQI,1}=wikiS;wikiCell{wikiQI,2}=wikiU; 
wikiMatrix(wikiQI,:)=[wikiA wikiDR wikiSigfig]; 
%% BEGIN QUESTION 2%%
wikiS='The mass of the cart is '; % S0 DONE
wikiV=randi([21,99])/10;  m=wikiV ; % V1 DONE (Inserted earlier to test: wikiV= 2;
wikiS= [wikiS num2str(wikiV)]; % V2  DONE
wikiS=[wikiS,'kg, and the spring constant is ']; % S1 DONE
wikiV=randi([4001,9999]);  k=wikiV; % V1  DONE  (Inserted earlier to test: wikiV=5447;
wikiS= [wikiS num2str(wikiV)]; % V2  DONE  (Inserted earlier to test: 
wikiS=[wikiS 'N/m. If the initial compression of the spring is ']; % S1 DONE
wikiV=randi([1,7])/10;  L=wikiV; % V1    DONE (Inserted earlier to test: wikiV=0.1;
wikiS= [wikiS num2str(wikiV)]; % V2 DONE
wikiS=[wikiS 'm, how high does it reach before coming to rest? ' ];  % S1 DONE
wikiS=[wikiS,'[[File:Roller coaster energy conservation.jpg|right|280px]]']; % S1 DONE
answer=k*L^2/2/m/9.8; %  use as many lines as you need to obtain wikiA:
wikiA =answer; % wikiA =  answer (reserved)
wikiU =' m';% wikiU = units (reserved)
wikiDR = 12; % wikiDR = detractor ratio is wikType=1 (reserved) DONE
wikiSigfig=2;% wikiSigfig = sig figs (reserved)
%Housekeeping  (do not edit)
wikiQI=wikiQI+1;wikiCell{wikiQI,1}=wikiS;wikiCell{wikiQI,2}=wikiU; %Do not edit
wikiMatrix(wikiQI,:)=[wikiA wikiDR wikiSigfig]; %Do not edit D
%%% BEGIN QUESTION 3%%
wikiS='What is the highest point the cart reaches if the speed was '; % S0 
wikiV=randi([65,85])/10;  v1=wikiV ; % V1 DONE  (Inserted earlier to test: wikiV=.14;
wikiS= [wikiS num2str(wikiV)]; % V2 
wikiS=[wikiS,'m/s when the cart was situated at a height of ']; % S1
wikiV=randi([11,31])/10;  h1=wikiV; % V1  DONE  (Inserted earlier to test:  wikiV=2.2;
wikiS= [wikiS num2str(wikiV)]; % V2 
wikiS=[wikiS 'm? ']; % S1
wikiS=[wikiS,'[[File:Roller coaster energy conservation.jpg|right|280px]]']; % S1 DONE
answer=h1+v1^2/2/9.8; %  use as many lines as you need to obtain wikiA:
%h1overSum=h1/answer
wikiA =answer; % wikiA =  answer (reserved)
wikiU =' m';% wikiU = units (reserved)
wikiDR = 14; % wikiDR = detractor ratio is wikType=1 (reserved)
wikiSigfig=2;% wikiSigfig = sig figs (reserved)
%Housekeeping  (do not edit)
wikiQI=wikiQI+1;wikiCell{wikiQI,1}=wikiS;wikiCell{wikiQI,2}=wikiU; %Do not edit
wikiMatrix(wikiQI,:)=[wikiA wikiDR wikiSigfig]; %Do not edit
%
end % NO EDIT (must follow last question)

&lt;/pre&gt;

==main.m==
&lt;pre&gt;
function [void]=main(void) %NO EDIT
%MAIN is the program you run to create the wikiquizzes after
%numericWiki.m has been suitably modified.  But you do not edit MAIN.
% variable list (may be outdated) %NO EDIT
% wikiMatrix(answer,wikiDR,wikiSigFig) %NO EDIT
% wikiType = 1 most of the time.%NO EDIT
% WikiSigFig is called sf in bookeeping.%NO EDIT
%
[ wikiName, wikiHeadABC,wikiHeadD_Z, wikiFirst, wikiCell, wikiMatrix, wikiType] = testeditor(  );%NO EDIT
cellsofar{1,1}=wikiHeadABC;%%NO EDIT First line of a wikiquiz
for index=65:67%NO EDIT
[ wikiName, wikiHeadABC,wikiHeadD_Z, wikiFirst, wikiCell, wikiMatrix, wikiType] = testeditor(  );%NO EDIT
cellsofar=startonequiz(wikiName,char(index) , wikiFirst, cellsofar);% ... &lt;quiz display=simple&gt;
cellsofar=finishthequiz(wikiMatrix,wikiCell,1, cellsofar);% ... &lt;/quiz&gt;
printafile([wikiName 'ABC'],cellsofar);%NO EDIT
end%NO EDIT
clear cellsofar;%NO EDIT
cellsofar{1,1}=wikiHeadD_Z;%First line of a wikiquiz
%NO EDIT
for index=68:90%NO EDIT
[ wikiName, wikiHeadABC,wikiHeadD_Z, wikiFirst, wikiCell, wikiMatrix, wikiType] = testeditor(  );%NO EDIT
cellsofar=startonequiz(wikiName,char(index) , wikiFirst, cellsofar);% ... &lt;quiz display=simple&gt;
cellsofar=finishthequiz(wikiMatrix,wikiCell,1, cellsofar);% ... &lt;/quiz&gt;
printafile([wikiName 'D_Z'],cellsofar);%NO EDIT
end%NO EDIT
%NO EDIT
%%% functions %%%%%NO EDIT
    function [cellsofar]=finishthequiz(wikiMatrix,wikiCell,wikiType, cellsofar)%NO EDIT
        % FINISHTHEQUIZ finishes a wikiquiz with &lt;/quiz&gt;
        nQues=size(wikiMatrix,1);%NO EDIT
        for iQues=1:nQues%NO EDIT
            cellsofar{size(cellsofar,1)+1,1}=['{',char(wikiCell(iQues,1)),'}'];%NO EDIT
            anspos = randi([1,5]);%NO EDIT
            for iAns=1:5%NO EDIT
                lett=char(96+iAns);%NO EDIT
                if iAns==anspos%NO EDIT
                    key = '+';%NO EDIT
                else%NO EDIT
                    key = '-';%NO EDIT
                end%NO EDIT
                s=[key,lett,') '];%NO EDIT
                %%%%%%%%The default is wikiType =1%%%%%%%%%NO EDIT
%This default multiplies the answer by 10^(n/N) where N=wikiDR=wikiMatrix(:,2)                 
                if wikiType ==1%NO EDIT
                ef=10^((iAns-anspos)/wikiMatrix(iQues,2));%NO EDIT
                sf=num2str(wikiMatrix(iQues,3)); % = wikiSigFig in testeditor
                form=['%.' sf 'E'];%NO EDIT
                num = num2str(ef*wikiMatrix(iQues,1),form);%NO EDIT
                end%NO EDIT
                %%%%%%%%%%%%%% wikiType not yet incorporated%%%%%%%%
              if wikiType==2%NO EDIT
                ef=iAns-anspos; %ef is now what is added or subtracted 
                sf=num2str(WikiMatrix(iQues,3));%NO EDIT
                form=['%6.' sf 'f'];  % now floating point
                num = num2str(ef+WikiMatrix(iQues,1),form);%NO EDIT
              end %NO EDIT
                %end copy to below
                %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                s=[s num ' ' char(wikiCell(iQues,2))];%NO EDIT
                cellsofar{size(cellsofar,1)+1,1}=s;%NO EDIT
            end%NO EDIT
            cellsofar{size(cellsofar,1)+1,1}= '  ' ;%NO EDIT
        end%NO EDIT
            cellsofar{size(cellsofar,1)+1,1}= '&lt;/quiz&gt;  ' ;%NO EDIT
        end%NO EDIT
    end%NO EDIT
%NO EDIT
    function [cellsofar]=startonequiz(wikiName, letter ,wikiFirst,cellsofar)%NO EDIT
        %STARTONEQUIZ adds: ==wikiName letter==    wikifirst   &lt;quiz...
        % cellsofar is appended to original cellsofar
        string = ['==' wikiName ' ' letter '=='];%NO EDIT
        cellsofar{size(cellsofar,1)+1,1}=string;%NO EDIT
        if ~strcmp(wikiFirst,'noIntroLine');%NO EDIT
            cellsofar{size(cellsofar,1)+1,1}=wikiFirst;%NO EDIT
        end%NO EDIT
        cellsofar{size(cellsofar,1)+1,1}= '&lt;quiz display=simple&gt;';%NO EDIT
    end%NO EDIT
    function []=printafile(filename,cellsofar)%NO EDIT
        %PRINTAFILE makes a txt file with the name filename that contains the cell array cellsofar
        fout=fopen([filename '.txt'], 'w+');%NO EDIT
        for count=1:size(cellsofar,1)%NO EDIT
            string=char(cellsofar(count,1));%NO EDIT
            fprintf(fout,'%s\n',char(string));%NO EDIT
        end%NO EDIT
        fclose(fout);%NO EDIT
    end%NO EDIT
&lt;/pre&gt;

{{CourseCat}}</text>
      <sha1>e80vk4xllahh8pczudtbl3213lqzeu9</sha1>
    </revision>
  </page>
  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
    <id>212733</id>
    <revision>
      <id>2818209</id>
      <parentid>2817001</parentid>
      <timestamp>2026-07-13T06:41:54Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2818209</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="2q7d2a84cvq3t43trcma6al5i0s7spx" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260629.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>2q7d2a84cvq3t43trcma6al5i0s7spx</sha1>
    </revision>
    <revision>
      <id>2818211</id>
      <parentid>2818209</parentid>
      <timestamp>2026-07-13T06:43:06Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2818211</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="bysxlpv0nkwru89izh9daven8cd7861" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260630.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>bysxlpv0nkwru89izh9daven8cd7861</sha1>
    </revision>
    <revision>
      <id>2818220</id>
      <parentid>2818211</parentid>
      <timestamp>2026-07-13T09:53:41Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2818220</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="hp5gziafjxbk6pub4ux27f70at6eixt" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260706.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>hp5gziafjxbk6pub4ux27f70at6eixt</sha1>
    </revision>
    <revision>
      <id>2818222</id>
      <parentid>2818220</parentid>
      <timestamp>2026-07-13T09:54:57Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2818222</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="lowwlfgu81m3qyojwqdivr8lpigkh4h" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260707.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>lowwlfgu81m3qyojwqdivr8lpigkh4h</sha1>
    </revision>
    <revision>
      <id>2818224</id>
      <parentid>2818222</parentid>
      <timestamp>2026-07-13T09:55:55Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2818224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="9hlwm6l61pakjv4uyqcufh6rqt58fvu" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20260713.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
      <sha1>9hlwm6l61pakjv4uyqcufh6rqt58fvu</sha1>
    </revision>
  </page>
  <page>
    <title>Internet Fundamentals/Web Browsers</title>
    <ns>0</ns>
    <id>216126</id>
    <revision>
      <id>2818196</id>
      <parentid>2814542</parentid>
      <timestamp>2026-07-13T02:33:52Z</timestamp>
      <contributor>
        <username>SunKissedMocha</username>
        <id>3067192</id>
      </contributor>
      <comment>Added Note from internet basics on AI to web browsers page.</comment>
      <origin>2818196</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16701" sha1="duwe2ujjoptxtvrpp63op4dx9dnbuju" xml:space="preserve">{{:{{BASEPAGENAME}}/Sidebar}}
[[File:Google Chrome.svg|right|100px|Chrome logo]]
[[File:Firefox logo, 2019.svg|right|100px|Firefox logo]]
[[File:Opera 2015 icon.svg|right|100px|Opera logo]]

A web browser (commonly referred to as a browser) is a software application for retrieving, presenting and traversing information resources on the World Wide Web.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt; This lesson introduces web browsers.

__TOC__

== Objectives and Skills ==
Objectives and skills for this lesson include:&lt;ref&gt;[https://www.ciwcertified.com/resources/documents/objectives/1D0-61a.pdf CIW: Internet Business Associate Exam Objectives]&lt;/ref&gt;&lt;ref&gt;[https://www.ciwcertified.com/resources/documents/course-descriptions/CCN02CFIBFNKL1208.pdf CIW: Internet Business Associate Course Description]&lt;/ref&gt;&lt;ref&gt;[http://training.gov.au/Training/Details/ICTICT103 Training.gov.au: ICTICT103 - Use, communicate and search securely on the Internet]&lt;/ref&gt;
* Identify the functions of Web browsers, and use them to access the World Wide Web and other computer resources.
* Identify and configure user customization features in Web browsers, including preferences, caching, cookies.
* Connect to and access the Internet
** Connect to Internet via existing Internet connection and confirm functionality
** Open Internet browser and set home page of personal choice by setting Internet options
** Ensure Internet browser software security
** Adjust display of the Internet browser to suit personal requirements
** Modify toolbar to meet user and Internet browser needs
** Access a particular website, note privacy and other conditions of use, and retrieve data
** Use socially responsible behaviour when sharing information on the Internet
** Enter uniform resource locator (URL) in address line of Internet browser
* Access and use consumer specific sites on the Internet
** Identify, access and review information specific sites to gain consumer information
** Identify and use Internet application sites to lodge details and gain access and information
** Access and use online forms on the Internet
* Undertake online transactions
** Access online transaction site
** Ensure security of transaction site
** Enter required information into fields on merchant's website
** Ensure pop-up dialog boxes, prompts or feedback mechanisms are completed
** Enter, check and make changes to preferred transaction options
** Complete online transaction
** Record and archive receipts according to business processes
** Close down and leave transaction process

== Readings ==
# [[Wikipedia: Web browser]]
# [[Wikipedia: Browser extension]]

== Multimedia ==
# [https://www.youtube.com/watch?v=BrXPcaRlBqo YouTube: What is a browser?]
# [https://www.youtube.com/watch?v=FxirRVJWUTs YouTube: Browser Basics]
# [https://www.youtube.com/watch?v=ntmhwxCqHzI YouTube: Browsing in Chrome]
# [https://www.youtube.com/watch?v=ZeuWjUp0PnQ YouTube: Getting Started with Firefox]
# [https://www.youtube.com/watch?v=Xbl5qd3mfKc YouTube: Customizing Chrome]
# [https://www.youtube.com/watch?v=94tAqUObEfc YouTube: Customize Firefox controls, buttons and toolbars]

== Student Presentations ==
# [https://www.youtube.com/watch?v=Os49B_fMkrc YouTube: Chrome Features]

== Activities ==
# Complete the following tutorials:
#* [https://www.tutorialspoint.com/internet_technologies/web_browsers.htm TutorialsPoint: Web Browser]
#* [https://www.learnfree.org/series/internet-tips LearnFree: Internet Tips]
# Install multiple web browsers to compare the different programs:
#* Brave
#*# Review [https://brave.com/index/ Brave: Download and Install Brave]
#*# Download and install Brave.
#* Chrome
#*# Review [https://support.google.com/chrome/answer/95346?co=GENIE.Platform%3DDesktop&amp;hl=en-GB Google: Download and Install Google Chrome].
#*# Download and install Chrome.
#* Edge
#*# Review [https://www.microsoft.com/en-us/edge Microsoft: Download Microsoft Edge Web Browser]
#*# Download and install Edge
#* Firefox
#*# Review [https://www.mozilla.org/en-US/firefox/products/ Mozilla: Firefox]
#*# Download and install Firefox.
#* Opera
#*# Review [http://www.opera.com/ Opera: Browser]
#*# Download and install Opera.
# Practice using web browsers:
#* All platforms:
#*# Complete the tutorial ̺[https://www.learnfree.org/episode/internet-basics/?id=16422 LearnFree: Chrome]
#*# Complete the tutorial [https://www.learnfree.org/episode/internet-basics/?id=16422 LearnFree: Edge]
#*# Complete the tutorial [https://www.learnfree.org/episode/internet-basics/?id=16422 LearnFree: Firefox]
#*# Complete the tutorial [https://help.opera.com/en/get-started/ Opera: Getting started with Opera]
#* MacOS
#*# Complete the tutorial [https://support.apple.com/guide/safari/get-started-ibrw6fde4ee8/mac  Getting started with Safari on Mac]
#* All
#*# After completing the tutorials, search the Internet for tips on using your preferred web browser(s).
# Configure browser settings.
#* Brave
#** Review available resources in the [https://brave.com/features/ Brave: Brave Help Center].
#** Check Brave settings and advanced settings and modify any settings that would improve your Internet experience.
#* Chrome
#** Review available resources in the [https://support.google.com/chrome Google: Chrome Help Center].
#** Check Chrome settings and advanced settings and modify any settings that would improve your Internet experience.
#* Edge
#** Review available resources under [https://support.microsoft.com/en-us/help/4027776/microsoft-edge-find-settings-tools-internet-options Edge: Settings and Tools].
#** Check Edge settings and modify any settings that would improve your Internet experience.
#* Firefox
#** Review available resources under [https://support.mozilla.org/en-US/products/firefox Mozilla: Firefox Help Topics].
#** Check Firefox preferences and modify any settings that would improve your Internet experience.
#* Opera
#** Review available resources under [http://www.opera.com/help/tutorials/personalize/ Opera: Personalize Opera].
#** Check Opera preference settings and modify and settings that would improve your Internet experience.
#* Safari
#** Review available resources under [https://support.apple.com/safari Apple: Safari Support].
#** Check Safari preferences and modify any settings that would improve your Internet experience.
# Configure browser extensions and add-ons.
#* Review [https://websitebuilders.com/how-to/learn-to-download/install-plug-in/ Websitebuilders: Installing Plug-ins] and [https://www.computerhope.com/issues/ch001411.htm ComputerHope: How to Disable or Remove Installed Extensions in Any Browser].
#* View add-ons installed in your browser(s). Disable any add-ons you don't use.
#* Search the Internet for the most popular add-ons for your browser(s). Consider adding and testing a password manager, ad blocker, autoplay blocker, grammar checker, or other add-on.
# Compare browser HTML5 compatibility.
#* Review [https://www.html5accessibility.com/ HTML Accessibility]. Note the version numbers of the browsers being compared and how well those versions met the HTML5 standard. All browsers have made improvements since the comparisons were made, but the results can be an indicator of how important standards are to a given browser's development team.

== Lesson Summary ==
* A web browser (commonly referred to as a browser) is a software application for retrieving, presenting and traversing information resources on the World Wide Web.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* The most popular web browsers are Google Chrome, Microsoft Edge (preceded by Internet Explorer), Firefox, Safari, and Opera.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;&lt;ref&gt;[https://www.netmarketshare.com/browser-market-share.aspx?qprid=2&amp;qpcustomd=0 NetMarketShare: Browser market share]&lt;/ref&gt;
* The primary purpose of a web browser is to bring information resources to the user (&quot;retrieval&quot; or &quot;fetching&quot;), allowing them to view the information (&quot;display&quot;, &quot;rendering&quot;), and then access other information (&quot;navigation&quot;, &quot;following links&quot;).&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* The prefix of the URL, the Uniform Resource Identifier or URI, determines how the URL will be interpreted.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* The most commonly used kind of URI starts with http: and identifies a resource to be retrieved over the Hypertext Transfer Protocol (HTTP).&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Many browsers also support a variety of other prefixes, such as https: for HTTPS, ftp: for the File Transfer Protocol, and file: for local files.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Prefixes that the web browser cannot directly handle are often handed off to another application entirely. For example, mailto: URIs are usually passed to the user's default e-mail application, and news: URIs are passed to the user's default newsgroup reader.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* HTML and associated content (image files, formatting information such as CSS, etc.) is passed to the browser's layout engine to be transformed from markup to an interactive document, a process known as &quot;rendering&quot;.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Most browsers can display images, audio, video, and XML files, and often have plug-ins to support Flash applications and Java applets.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* All major web browsers allow the user to open multiple information resources at the same time, either in different browser windows or in different tabs of the same window.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Major browsers also include pop-up blockers to prevent unwanted windows from &quot;popping up&quot; without the user's consent.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Most web browsers can display a list of web pages that the user has bookmarked so that the user can quickly return to them. Bookmarks are also called &quot;Favorites&quot; in Internet Explorer.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* In addition, all major web browsers have some form of built-in web feed aggregator.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Most browsers can be extended via plug-ins, downloadable components that provide additional features.&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
* Most major web browsers have common user interface elements:&lt;ref&gt;[[Wikipedia: Web browser]]&lt;/ref&gt;
** Back and forward buttons to go back to the previous resource and forward respectively.
** A refresh or reload button to reload the current resource.
** A stop button to cancel loading the resource. In some browsers, the stop button is merged with the reload button.
** A home button to return to the user's home page.
** An address bar to input the Uniform Resource Identifier (URI) of the desired resource and display it.
** A search bar to input terms into a web search engine. In some browsers, the search bar is merged with the address bar.
** A status bar to display progress in loading the resource and also the URI of links when the cursor hovers over them, and page zooming capability.
** The viewport, the visible area of the webpage within the browser window.
** The ability to view the HTML source for a page.
** An incremental find features to search within a web page.
* A browser extension is a plug-in that extends the functionality of a web browser.&lt;ref&gt;[[Wikipedia: Browser extension]]&lt;/ref&gt;
* Many browsers have an online store that allows users to find extensions and see lists of popular extensions.&lt;ref&gt;[[Wikipedia: Browser extension]]&lt;/ref&gt;
* Browser extensions are used for improving a browser's user interface, security or accessibility, blocking advertisements, and various other features to make browsing the internet easier and more pleasant.&lt;ref&gt;[[Wikipedia: Browser extension]]&lt;/ref&gt;
* Browser extensions have access to everything done by the browser, and can do things like inject ads into web pages, or make &quot;background&quot; HTTP requests to third-party servers. As a result, a malicious browser extension may take action against the interest of the user that installed it.&lt;ref&gt;[[Wikipedia: Browser extension]]&lt;/ref&gt;

== Key Terms ==
;aggregator
:A web site or computer software that aggregates a specific type of information from multiple online sources.&lt;ref&gt;[[Wikipedia: Aggregator]]&lt;/ref&gt;
;ActiveX
:An early software framework developed by Microsoft to enhance browser functionality, which was supported only by x86-based computers using Internet Explorer.&lt;ref&gt;[[Wikipedia: ActiveX]]&lt;/ref&gt;
;bookmark
:A Uniform Resource Identifier (URI) that is stored for later retrieval in any of various storage formats.&lt;ref&gt;[[Wikipedia: Bookmark (World Wide Web)]]&lt;/ref&gt;
;browser cache
:An information technology for the temporary storage of web documents, such as HTML pages and images, to reduce bandwidth usage, server load, and perceived lag.&lt;ref&gt;[[Wikipedia: Web cache]]&lt;/ref&gt;
;CAPTCHA
:A type of challenge-response test used in computing to determine whether or not the user is human.&lt;ref&gt;[[Wikipedia: CAPTCHA]]&lt;/ref&gt;
;cookie
:A small piece of data sent from a website and stored on the user's computer by the user's web browser while the user is browsing.&lt;ref&gt;[[Wikipedia: HTTP cookie]]&lt;/ref&gt;
;favorite
:See bookmark.
;Flash
:A multimedia software platform developed by Adobe which used for production of animations, rich Internet applications, desktop applications, mobile applications and mobile games.&lt;ref&gt;[[Wikipedia: Adobe Flash]]&lt;/ref&gt;
;history
:The list of web pages a user has visited recently.&lt;ref&gt;[[Wikipedia: Web browsing history]]&lt;/ref&gt;
;home page
:The initial or main web page of a website or a browser.&lt;ref&gt;[[Wikipedia: Home page]]&lt;/ref&gt;
;hyperlink
:A reference to data that the reader can directly follow either by clicking, tapping, or hovering.&lt;ref&gt;[[Wikipedia: Hyperlink]]&lt;/ref&gt;
;Java applet
:A small application which is written in Java or another programming language that compiles to Java bytecode and delivered to users in the form of that bytecode.&lt;ref&gt;[[Wikipedia: Java applet]]&lt;/ref&gt;
;mashup
:A web page, or web application, that uses content from more than one source to create a single new service displayed in a single graphical interface.&lt;ref&gt;[[Wikipedia: Mashup (web application hybrid)]]&lt;/ref&gt;
;plug-in
:A software component that adds a specific feature to an existing computer program.&lt;ref&gt;[[Wikipedia: Plug-in (computing)]]&lt;/ref&gt;
;pop-up ad
:Online advertising using a new web browser window for display.&lt;ref&gt;[[Wikipedia: Pop-up ad]]&lt;/ref&gt;
;proxy server
:A computer system or an application that acts as an intermediary for requests from clients seeking resources from other servers.&lt;ref&gt;[[Wikipedia: Proxy server]]&lt;/ref&gt;
;RSS
:A type of web feed which allows users to access updates to online content in a standardized, computer-readable format.&lt;ref&gt;[[Wikipedia: RSS]]&lt;/ref&gt;
;Silverlight
:A deprecated application framework developed by Microsoft for writing and running rich Internet applications.&lt;ref&gt;[[Wikipedia: Microsoft Silverlight]]&lt;/ref&gt;
;Uniform Resource Identifier (URI)
:A string of characters used to identify a resource.&lt;ref&gt;[[Wikipedia: Uniform Resource Identifier]]&lt;/ref&gt;
;Uniform Resource Locator (URL)
:A reference to a web resource that specifies its location on a computer network and a mechanism for retrieving it.&lt;ref&gt;[[Wikipedia: URL]]&lt;/ref&gt;
;web feed
:A subscription-supporting data format used for providing users with frequently updated content.&lt;ref&gt;[[Wikipedia: Web feed]]&lt;/ref&gt;

== Assessments ==
* Flashcards: [https://quizlet.com/dj/648152511/web-browsers-flash-cards/ Quizlet: Web Browsers]
* Quiz: [https://quizlet.com/648152511/test?answerTermSides=2&amp;promptTermSides=6&amp;questionCount=10&amp;questionTypes=4&amp;showImages=true Quizlet: Web Browsers]

== Note on Artificial Intelligence (AI) Assistants ==
AI assistants are starting to be built into the most popular search engines, showing answers to queries before human made websites. It's recommended to take these answers with a grain of salt, since as of October 22nd 2025, there have been studies&lt;ref&gt;{{Cite web|url=https://www.bbc.co.uk/mediacentre/2025/new-ebu-research-ai-assistants-news-content/|title=Largest study of its kind shows AI assistants misrepresent news content 45% of the time – regardless of language or territory|website=www.bbc.co.uk|language=en|access-date=2026-07-13}}&lt;/ref&gt; showing that AI summaries are often inaccurate. As time goes on AI may get more accurate but generally be careful with the answers produced by these assistants.

== See Also ==
* [[Computer Skills/Basic/Internet]]
* [[IC3/Internet Fundamentals]]
* [https://dev.to/vip3rousmango/battle-of-the-modern-browsers-14c3 Web Designer Depot: Best Browsers for Developers]

== References ==
{{reflist}}

{{DEFAULTSORT:{{SUBPAGENAME}}}}
{{CourseCat}}
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[[Category:Web browsers]]
[[Category:Completed resources]]</text>
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'''Simulation universe modelling at the Planck scale'''

The model is constructed around geometrical Planck objects, the following is a handy reference guide.
{{main|User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)}}


Key (mathematical electron model):
:&lt;math&gt;a = 137.035996368786&lt;/math&gt;, (θ = 0)
:&lt;math&gt;\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)}  = 2.0071349543... &lt;/math&gt;, (θ = 0)
:&lt;math&gt;v = 11843707.84994 ...,\; units = \frac{m}{s}&lt;/math&gt;, (θ = 17)
:&lt;math&gt;r = 0.712562517313 ...,\; units = (\frac{kg.m}{s})^{1/4}&lt;/math&gt;, (θ = 8)
:&lt;math&gt;\psi_{electron} = \frac{(2^7 3 \pi^3 a \Omega^5)^3}{2\pi}&lt;/math&gt;, (θ = 0)
{| class=&quot;wikitable&quot;
|+Geometrical objects (reference table)
! attribute
! object
! θ
! scalars
! value
! units
|-
| mass 
| &lt;math&gt;M = (1)&lt;/math&gt; fundamental
| 15
| &lt;math&gt;k = \frac{r^4}{v}&lt;/math&gt;
| 0.217672822274 x 10&lt;sup&gt;-7&lt;/sup&gt;
| &lt;math&gt;kg&lt;/math&gt;
|-
| time 
| &lt;math&gt;T = (\pi)&lt;/math&gt; fundamental
|  -30 
| &lt;math&gt;t = \frac{r^9}{v^6}&lt;/math&gt;
| 0.539051822204 x 10&lt;sup&gt;-43&lt;/sup&gt;
| &lt;math&gt;s&lt;/math&gt;
|-
| sqrt(momentum)  
| &lt;math&gt;P = (\Omega) \times (M^{(4/5)} \;(\pi/T)^{(2/15)}) = \Omega&lt;/math&gt;
| 16
| &lt;math&gt;r^2&lt;/math&gt;
| 1.019113421977
| &lt;math&gt;\sqrt{\frac{kg m}{s}}&lt;/math&gt;
|-
| velocity  
| &lt;math&gt;V = \frac{2\pi P^2}{M} = (2\pi\Omega^2)&lt;/math&gt;
| 17
| &lt;math&gt;v&lt;/math&gt;
| 299792458 
| &lt;math&gt;\frac{m}{s}&lt;/math&gt;
|-
| length  
| &lt;math&gt;L = VT = (2\pi^2\Omega^2)&lt;/math&gt; 
| -13
| &lt;math&gt;\frac{r^9}{v^5}&lt;/math&gt;
| 0.1616036707678 x 10&lt;sup&gt;-34&lt;/sup&gt;
| &lt;math&gt;m&lt;/math&gt;
|-
| ampere 
| &lt;math&gt;A = \frac{2^4 V^3}{a P^3} = (\frac{2^7 \pi^3 \Omega^3}{a})&lt;/math&gt; 
| 3
| &lt;math&gt;\frac{v^3}{r^6}&lt;/math&gt;
| 0.2972212562267 x 10&lt;sup&gt;25&lt;/sup&gt;
| &lt;math&gt;\sqrt{\frac{m^3}{kg^3 s^3}}&lt;/math&gt;
|-
| [[w:Vacuum permeability | Vacuum permeability]] 
| &lt;math&gt;\mu_0^* = \frac{4 \pi V^2 M}{a L A^2}&lt;/math&gt; 
| 56
| &lt;math&gt;r^7&lt;/math&gt;
| 4π/10^7
| &lt;math&gt;\frac{kg m}{A^2 s^2}&lt;/math&gt;
|-
| [[w:Electron mass | Electron mass]]
| &lt;math&gt;m_e^* = \frac{M}{\psi_{electron}}&lt;/math&gt; 
| 15
| &lt;math&gt;\frac{r^4}{v}&lt;/math&gt;
| 9.10938231256 x 10&lt;sup&gt;-31&lt;/sup&gt;
| &lt;math&gt;kg&lt;/math&gt;
|-
| [[w:Rydberg constant | Rydberg constant]] 
| &lt;math&gt;R^* = (\frac{m_e^*}{4 \pi L a^2 M})&lt;/math&gt; 
| 13
| &lt;math&gt;\frac{v^5}{r^9}&lt;/math&gt;
| 10973731.568508
| &lt;math&gt;\frac{1}{m}&lt;/math&gt;
|-
| [[w:Elementary charge | Elementary charge]] 
| &lt;math&gt;e^* = A T&lt;/math&gt;
| -27
| &lt;math&gt;\frac{r^3}{v^3}&lt;/math&gt;
| 1.60217651130 x 10&lt;sup&gt;-19&lt;/sup&gt;
| &lt;math&gt;A s&lt;/math&gt;
|-
| [[w:Planck constant | Planck constant]] 
| &lt;math&gt;h^* = 2 \pi M V L&lt;/math&gt; 
| 19
| &lt;math&gt;\frac{r^{13}}{v^5}&lt;/math&gt;
| 6.626069134 x 10&lt;sup&gt;-34&lt;/sup&gt;
| &lt;math&gt;\frac{kg m^2}{s}&lt;/math&gt;
|}

 
==Article series==
* [[https://simulationuniverse.org/ simulationuniverse.org]]: Home page

===Wiki series===
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_units_(geometrical)]]: Planck units MLTPA as geometrical objects
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Physical_constant_(anomaly)]]: Anomalies within the physical constants 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Gravity_via_Atomic_orbitals]]: Gravity as a function of atomic orbitals 
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Relativity]]: Relativity as a translation between 2 co-ordinate systems
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Planck_unit_scaffolding]]: CMB and a Planck unit universe scaffolding
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/Sqrt_Planck_momentum]]: Link between charge and mass
* [[User:Platos_Cave_(physics)/Simulation_Hypothesis/God_(programmer)]]: Introduction to a Planck scale Programmer God Simulation Hypothesis model

===General articles===
Articles have been transcribed to HTML format for ease of reference.
* [[https://simulationuniverse.org/1-Planck-unit-CMB.html 1-Planck-unit-CMB.html]]: Constructs the universe frame from Planck units 
* [[https://simulationuniverse.org/2-Relativity-hypersphere.html 2-Relativity-hypersphere.html]]: Relativity as the mathematics of perspective 
* [[https://simulationuniverse.org/3-Gravitational-orbitals.html 3-Gravitational-orbitals.html]]: Gravity as sum of n-body rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/4-Atomic-orbitals.html 4-Atomic-orbitals.html]]: Atomic orbitals as single rotating orbital particle-particle pairs
* [[https://simulationuniverse.org/5-w_axis.html 5-w_axis.html]]: Imaginary number axis (radiation domain)
* [[https://simulationuniverse.org/6-Physical-constant-anomalies.html 6-Physical-constant-anomalies.html]]: Statistical analysis of physical constant anomalies
* [[https://simulationuniverse.org/7-Monopole-quarks.html 7-Monopole-quarks.html]]: Quarks as monopoles
* [[https://simulationuniverse.org/8-Holographic-universe.html 8-Holographic-universe.html]]: Hypersphere surface as 2-D analogue


==References==
{{Reflist}}


[[Category:Physics| ]]
[[Category:Philosophy of science| ]]</text>
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      <text bytes="3749" sha1="lbju6jlxcrkozjqh3updrex6majbkx0" xml:space="preserve">[[File:Octave-v4.0.0-gui-german.png|thumb|Use Octave in KnitR]]
In general KnitR was designed to combine a report document with R code (&quot;Knit&quot; a document with &quot;R&quot; code&lt;math&gt;\Rightarrow
&lt;/math&gt; &quot;KnitR&quot;). The follow learning modul show how to integrate Octave (Open Source software numeric calculations) in KnitR code for [[Dynamic Document Generation]].

== Octave Code in KnitR Documents ==
The following section explains how to perform calculation with Octave and generation of figure in R with KnitR package in R. The &lt;kbd&gt;knitr4octave.Rmd&lt;/kbd&gt;&lt;ref&gt;knitr4octave.Rmd (2024) R-Markdown document in the knitr4education package on GitHub - URL: https://github.com/niebert/knitr4education/tree/main/en &lt;/ref&gt; is an example that can be used in the learning resource. Keep in mind to adapt the path to Octave into the head of the Octave code chunk so that Octave can be executed in R.
 
=== Working Octave Code ===
Assume we have the following Octave code, that plots a function in Octave and saves the generated code in a JPG file &lt;kbd&gt;myfigure.jpg&lt;/kbd&gt;.

&lt;syntaxhighlight lang=&quot;octave&quot;&gt;
  x = -10:0.1:10;
  plot (x, sin (x)); 
  print -djpg ./myfigure.jpg 
&lt;/syntaxhighlight&gt;

=== Embed the Octave Code in the KnitR document ===
The code chunk in R-Markdown looks a bit different than the code chunk in the statistical language R. 
* '''(Path to Octave)''' The header must contain the location/path to the Octave interpreter that executes the code. Here Octave is located at &lt;kbd&gt;/usr/local/bin/octave/&lt;/kbd&gt; on a Linux machine. You can identify the path to Octave on Linux and MacOSX by
:&lt;syntaxhighlight lang=&quot;bash&quot;&gt;
which octave
&lt;/syntaxhighlight&gt;
The which command provides the path for starting &lt;kbd&gt;octave&lt;/kbd&gt;, e.g.
:&lt;syntaxhighlight lang=&quot;bash&quot;&gt;
/usr/bin/octave
&lt;/syntaxhighlight&gt;
: Use this path as engine path in the code chunk for KnitR.
: On Windows add the path to the &lt;kbd&gt;octave.exe&lt;/kbd&gt; to your PATH variable.
* '''(Echo-Boolean)'''  &lt;kbd&gt;echo=TRUE&lt;/kbd&gt; prints the code in the document. This is a standard feature of code chunks that can be used e.g. for code chunks in R. With &lt;kbd&gt;echo=FALSE&lt;/kbd&gt; the code will disappear in the generated document by KnitR. The code will be executed even if the code does not appear in the document. So figures based on updated data will be included in the [[KnitR]] generated output after a KnitR run.
* '''(Figures)''' Octave generates figures as usual similar to calls in Graphical User Interface of Octave. Save figures on the harddisk an import the generated figures with the corresponding filename in the R-Markdown document. and executes the code (i.e. generates the figure). 
Now we will use the Octave-Code in the R-Markdown document with the 3 backticks. The next code chunk is integrated in KnitR markdown with preceeding text and an import in the document for the generated figure.
&lt;syntaxhighlight lang=&quot;octave&quot;&gt;
This is pure text before code chunk in Octave.
```{octave,engine.path='/usr/bin/octave',results='asis',echo=TRUE}
  x = -10:0.1:10;
  plot (x, sin (x)); 
  print -djpg myfigure.jpg 
```
More text and then import the generated figure from Octave.
![My Plot of the sin-function with Octave](./myfigure.jpg) 
&lt;/syntaxhighlight&gt;

=== Path to Octave ===
The engine path defines, where 
The import of the generated figure is not necessary with R code. With an R code chunk a plot command in the code chunk will be visible in the output document. After the lass3 backticks there is some more text in the R-Markdown document. The generate figure will be imported with a standard image import syntax in R-Markdown.

== References ==
&lt;references/&gt;

&lt;noinclude&gt;[[de:KnitR/Octave]]&lt;/noinclude&gt;

[[Category:Document Management]]
[[Category:Computer Algebra]]
[[Category:GNU Octave]]</text>
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GNU Octave is a free-as-in-freedom mathematical software package and a programming language, with considerable compatibility with MATLAB, a proprietary package. This page complements Wikipedia and Wikibooks. It should help readers who want to learn GNU Octave rather than just read about it. For a start, the page collects links to learning materials outside of Wikiversity.

==Getting started==
To get started, one can use octave-online.net, avoiding the need to install Octave. Using that environment, one can peruse [https://wiki.octave.org/Using_Octave Using Octave] from octave.org, try the examples and modify them. One can then move to the Octave Programming Tutorial in Wikibooks and solve the exercises that it has.

A quick glance, to be expanded:
* sin(3)
** Evaluates an expression and outputs the result.
* x = 0:0.1:7; plot (x, sin (x));
** Plots sine function on the given range, with the given step of 0.1.
* plot(x, sin(x), x, cos(x))
** Building on the above, plots multiple functions.
* m = [1, 1, 2; 3, 5, 8; 13, 21, 34]
** Creates a matrix.

Links:
* [https://octave-online.net/ Try Octave online], octave-online.net

==Compatibility with MATLAB==
As per octave.org, &quot;The Octave syntax is largely compatible with Matlab.&quot;&lt;ref&gt;[https://octave.org/ octave.org]&lt;/ref&gt; Moreover, differences between Octave and MATLAB are usually considered to be bugs. &lt;ref&gt;[https://wiki.octave.org/Differences_between_Octave_and_Matlab Differences between Octave and Matlab], wiki.octave.org&lt;/ref&gt;

Links:
* {{W|GNU Octave#MATLAB compatibility}}, wikipedia.org

==References==
&lt;references/&gt;

==Further reading==
Wikipedia and Wikibooks:
* {{W|GNU Octave}}, wikipedia.org
* [[B:Octave Programming Tutorial|Octave Programming Tutorial]], wikibooks.org

Official pages and documentation from octave.org:
* [https://www.octave.org/ GNU Octave], octave.org
* [https://wiki.octave.org/GNU_Octave_Wiki Octave], wiki.octave.org -- has links to different kinds of Octave documentation
* [https://wiki.octave.org/Using_Octave Using Octave], wiki.octave.org -- a quick introduction
* [https://wiki.octave.org/Octave_Basics Octave Basics], wiki.octave.org -- something like a reference card
* [https://docs.octave.org/latest/ Top (GNU Octave (version 8.4.0))], docs.octave.org -- the complete documentation

YouTube:
* [https://www.youtube.com/watch?v=woiU5PRVm7M GNU Octave - Full Tutorial For Beginners], MCC Py Tutorials, youtube.com
* [https://www.youtube.com/watch?v=LhPZwdhutgU Octave/MATLAB® for Beginners, Part 1: Starting from Scratch], MIT OpenCourseWare, youtube.com

Other:
* [https://octave-online.net/ Try Octave online], octave-online.net
* [http://www-h.eng.cam.ac.uk/help/programs/octave/tutorial/ CUED - Introduction to Octave], www-h.eng.cam.ac.uk -- a single-page introduction with about 15 000 words
* [https://learnxinyminutes.com/docs/matlab/ Learn MATLAB in Y Minutes], learnxinyminutes.com -- although for MATLAB, will work to a considerable extent for Octave

[[Category:Software]]
[[Category:GNU Octave]]</text>
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        <username>Mu301</username>
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      </contributor>
      <minor />
      <comment>removed [[Category:Software]] using [[Help:Gadget-HotCat|HotCat]]</comment>
      <origin>2818163</origin>
      <model>wikitext</model>
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      <text bytes="3021" sha1="qjxkonifherqpkepj77ntlmhcexovz1" xml:space="preserve">{{Original research}}
GNU Octave is a free-as-in-freedom mathematical software package and a programming language, with considerable compatibility with MATLAB, a proprietary package. This page complements Wikipedia and Wikibooks. It should help readers who want to learn GNU Octave rather than just read about it. For a start, the page collects links to learning materials outside of Wikiversity.

==Getting started==
To get started, one can use octave-online.net, avoiding the need to install Octave. Using that environment, one can peruse [https://wiki.octave.org/Using_Octave Using Octave] from octave.org, try the examples and modify them. One can then move to the Octave Programming Tutorial in Wikibooks and solve the exercises that it has.

A quick glance, to be expanded:
* sin(3)
** Evaluates an expression and outputs the result.
* x = 0:0.1:7; plot (x, sin (x));
** Plots sine function on the given range, with the given step of 0.1.
* plot(x, sin(x), x, cos(x))
** Building on the above, plots multiple functions.
* m = [1, 1, 2; 3, 5, 8; 13, 21, 34]
** Creates a matrix.

Links:
* [https://octave-online.net/ Try Octave online], octave-online.net

==Compatibility with MATLAB==
As per octave.org, &quot;The Octave syntax is largely compatible with Matlab.&quot;&lt;ref&gt;[https://octave.org/ octave.org]&lt;/ref&gt; Moreover, differences between Octave and MATLAB are usually considered to be bugs. &lt;ref&gt;[https://wiki.octave.org/Differences_between_Octave_and_Matlab Differences between Octave and Matlab], wiki.octave.org&lt;/ref&gt;

Links:
* {{W|GNU Octave#MATLAB compatibility}}, wikipedia.org

==References==
&lt;references/&gt;

==Further reading==
Wikipedia and Wikibooks:
* {{W|GNU Octave}}, wikipedia.org
* [[B:Octave Programming Tutorial|Octave Programming Tutorial]], wikibooks.org

Official pages and documentation from octave.org:
* [https://www.octave.org/ GNU Octave], octave.org
* [https://wiki.octave.org/GNU_Octave_Wiki Octave], wiki.octave.org -- has links to different kinds of Octave documentation
* [https://wiki.octave.org/Using_Octave Using Octave], wiki.octave.org -- a quick introduction
* [https://wiki.octave.org/Octave_Basics Octave Basics], wiki.octave.org -- something like a reference card
* [https://docs.octave.org/latest/ Top (GNU Octave (version 8.4.0))], docs.octave.org -- the complete documentation

YouTube:
* [https://www.youtube.com/watch?v=woiU5PRVm7M GNU Octave - Full Tutorial For Beginners], MCC Py Tutorials, youtube.com
* [https://www.youtube.com/watch?v=LhPZwdhutgU Octave/MATLAB® for Beginners, Part 1: Starting from Scratch], MIT OpenCourseWare, youtube.com

Other:
* [https://octave-online.net/ Try Octave online], octave-online.net
* [http://www-h.eng.cam.ac.uk/help/programs/octave/tutorial/ CUED - Introduction to Octave], www-h.eng.cam.ac.uk -- a single-page introduction with about 15 000 words
* [https://learnxinyminutes.com/docs/matlab/ Learn MATLAB in Y Minutes], learnxinyminutes.com -- although for MATLAB, will work to a considerable extent for Octave

[[Category:GNU Octave]]</text>
      <sha1>qjxkonifherqpkepj77ntlmhcexovz1</sha1>
    </revision>
    <revision>
      <id>2818167</id>
      <parentid>2818163</parentid>
      <timestamp>2026-07-12T16:20:16Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <minor />
      <comment>sort key</comment>
      <origin>2818167</origin>
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      <format>text/x-wiki</format>
      <text bytes="3023" sha1="7ov9tgvci4mdk9vg7itekldtee7cgjf" xml:space="preserve">{{Original research}}
GNU Octave is a free-as-in-freedom mathematical software package and a programming language, with considerable compatibility with MATLAB, a proprietary package. This page complements Wikipedia and Wikibooks. It should help readers who want to learn GNU Octave rather than just read about it. For a start, the page collects links to learning materials outside of Wikiversity.

==Getting started==
To get started, one can use octave-online.net, avoiding the need to install Octave. Using that environment, one can peruse [https://wiki.octave.org/Using_Octave Using Octave] from octave.org, try the examples and modify them. One can then move to the Octave Programming Tutorial in Wikibooks and solve the exercises that it has.

A quick glance, to be expanded:
* sin(3)
** Evaluates an expression and outputs the result.
* x = 0:0.1:7; plot (x, sin (x));
** Plots sine function on the given range, with the given step of 0.1.
* plot(x, sin(x), x, cos(x))
** Building on the above, plots multiple functions.
* m = [1, 1, 2; 3, 5, 8; 13, 21, 34]
** Creates a matrix.

Links:
* [https://octave-online.net/ Try Octave online], octave-online.net

==Compatibility with MATLAB==
As per octave.org, &quot;The Octave syntax is largely compatible with Matlab.&quot;&lt;ref&gt;[https://octave.org/ octave.org]&lt;/ref&gt; Moreover, differences between Octave and MATLAB are usually considered to be bugs. &lt;ref&gt;[https://wiki.octave.org/Differences_between_Octave_and_Matlab Differences between Octave and Matlab], wiki.octave.org&lt;/ref&gt;

Links:
* {{W|GNU Octave#MATLAB compatibility}}, wikipedia.org

==References==
&lt;references/&gt;

==Further reading==
Wikipedia and Wikibooks:
* {{W|GNU Octave}}, wikipedia.org
* [[B:Octave Programming Tutorial|Octave Programming Tutorial]], wikibooks.org

Official pages and documentation from octave.org:
* [https://www.octave.org/ GNU Octave], octave.org
* [https://wiki.octave.org/GNU_Octave_Wiki Octave], wiki.octave.org -- has links to different kinds of Octave documentation
* [https://wiki.octave.org/Using_Octave Using Octave], wiki.octave.org -- a quick introduction
* [https://wiki.octave.org/Octave_Basics Octave Basics], wiki.octave.org -- something like a reference card
* [https://docs.octave.org/latest/ Top (GNU Octave (version 8.4.0))], docs.octave.org -- the complete documentation

YouTube:
* [https://www.youtube.com/watch?v=woiU5PRVm7M GNU Octave - Full Tutorial For Beginners], MCC Py Tutorials, youtube.com
* [https://www.youtube.com/watch?v=LhPZwdhutgU Octave/MATLAB® for Beginners, Part 1: Starting from Scratch], MIT OpenCourseWare, youtube.com

Other:
* [https://octave-online.net/ Try Octave online], octave-online.net
* [http://www-h.eng.cam.ac.uk/help/programs/octave/tutorial/ CUED - Introduction to Octave], www-h.eng.cam.ac.uk -- a single-page introduction with about 15 000 words
* [https://learnxinyminutes.com/docs/matlab/ Learn MATLAB in Y Minutes], learnxinyminutes.com -- although for MATLAB, will work to a considerable extent for Octave

[[Category:GNU Octave|*]]</text>
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    <title>User:Tommy Kronkvist</title>
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        <username>Tommy Kronkvist</username>
        <id>31941</id>
      </contributor>
      <comment>User statistics.</comment>
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[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (July 13, 2026), I've made '''just over 396,500 edits''' to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found '''[[meta:Special:CentralAuth/Tommy Kronkvist|here]]'''.

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
{{User committed identity|a6edd6d2fdbf82621f0cda4e5525c71f8da9b5dfd308242c3c63365e998c32c5406b75448380903265a5403edffd1a0435b61ac943f3c65870db9250f8b884a9|SHA-512|background=#e0e8ff|border=e0e8ff}}</text>
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  <page>
    <title>Just sustainability transitions: a living review</title>
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      <comment>/* Discussion */ adding future research possibilities inspired by conference discussion with colleagues</comment>
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      <text bytes="108463" sha1="kf5obo64kokh7ckpkkihj6vwuylrgk9" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!Name
!Affiliation
!ORCID
!Contribution
|-
|Adélie Ranville
|IAE de Grenoble, CERAG lab (https://ror.org/0509qp208)
|https://orcid.org/0000-0002-3993-6135
|Research design, database search, article screening, knowledge modelling, article writing
|-
|Amélie E. Pereira
|Laboratoire DICEN IDF
|https://orcid.org/0009-0005-5928-5586
|Meta-data enrichement, article writing
|-
|Finn Nielsen
|Technical University of Denmark
|https://orcid.org/0000-0001-6128-3356
|Data visualisation
|}
Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review

== Introduction ==
Just sustainability transition refers to the process of shifting towards sustainable practices in a way that is equitable and inclusive. It includes dimensions of procedural, recognition, distributive and reparative justice and the concept is related to climate justice, environmental justice and energy justice&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. The study of sustainability transitions in social sciences requires dynamic and adaptive research synthesis methods. Sustainability transitions involve complex, multi-level processes influenced by technological, economic, social, and policy factors&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. Given the rapidly evolving nature of sustainability-related research, static literature reviews often become outdated, limiting their usefulness for policymakers, scholars, and practitioners. A living literature review – continuously updated with new findings – ensures that emerging insights, case studies, and theoretical developments are integrated cumulatively into the knowledge base. Developing such review will answer the call for more evidence-based practices in management sciences&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. Our project assesses the potential of Wikidata to build living review workflow on sustainability transition. We address three issues encountered by scientists: information overload, knowledge synthesis and results dissemination.

=== The problem of academic information overload ===
Global scientific output doubles every nine years&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, pushed by the “publish or perish” model incentivizing researchers to increase the quantity of research outputs. Researchers are subject to information overload as the number of publications to read is beyond what a human brain can handle, they are expected to produce high-quality research under an increasing time pressure. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “Wikifying science” may in this context contribute to facilitating researcher’s work while preserving scientific quality. That is why in this project, we aim to build a searchable academic publication database with enriched meta-data that will allow scholars to navigate the existing publications corpus related to just sustainability transition more easily.

=== The problem of knowledge synthesis ===
The volume of academic production is rendering knowledge synthesis difficult. Scholars have thus called for making literature reviews cumulative and updatable&lt;ref&gt;{{Citation|title=Day 2 {{!}} Arnaud Vaganay: Reproducible Literature Reviews|url=https://www.youtube.com/watch?v=Nspd_1cx9kc|date=2017-10-19|accessdate=2026-06-23|last=Berkeley Initiative for Transparency in the Social Sciences (BITSS)}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). While such reviews method exist for quantitative research producing standardized results, they are not adapted to synthetize social science studies on sustainability transitions that involve diverse methodologies and various disciplinary perspectives. The goal of the project is to propose a demonstration of a living review method for social science findings on just sustainability transition, relying on the collaborative model and tools of Wikimedia projects notably Wikidata, Wikiversity and Wikipedia.

=== The problem of scientific results dissemination ===
There is urgent need to disseminate knowledge on impactful topics like sustainability transition while proprietary publication models, disinformation and censorship (e.g. US) is threatening access to free and reliable knowledge. In parallel, social scientists struggle to make their work impactful&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, notably for social science in which women are more present. Our project proposes to improve expert contribution by making wikimedia projects (notably wikidata) useful tools that can facilitate research work, in addition to a key knowledge dissemination platform that is not country or institution-dependent. We propose to approach Wikimedia projects as a powerful (and free) knowledge management infrastructure that researchers could use. The Wikimedia ecosystem offers solutions that have strong potential to put open science principles into practices, including [[wikipedia:FAIR_data|FAIR]] principles and [[wikipedia:Linked_data#Linked_open_data|linked open data]].  

== Toward a living review on just sustainability transition ==

=== Just sustainability transition ===
Just sustainability transition transition is &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

Developping living reviews seem particularly relevant for the just transition literature: first, modeling knowledge and building graphs allows to take into account the complexity of sustainability transitions which involve multiple levels of analysis&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; would allow researchers to be less subject to information overload through a more systematic accumulation of knowledge. Finally, conducting this review with an open science philosophy aswers the challenge of knowledge dissemination, which is crucial in a context of socio-ecological emergency when decision-makers need to rapidely access reliable information on possible sustainability transition trajectories.  

=== Living reviews ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. Literature review methods are currently evolving with new technological possibilities. Generative artificial intelligence such as ChatGPT are expected to have a strong influence on literature review activities&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. While AI can appear as a solution for scaling literature reviews, we are in the present project exploring another possible scenario which is to use more crowdsourcing in the literature review process.

=== Wikimedia projects ===
Wikipedia is a successfull example of large-scaled crowdsourcing of reliable knowledge synthesis. That is why this project proposes to explore the potential of the Wikimedia ecosystem for conducting living reviews. Since Wikipedia does aim to host original research&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, we are working on two sister projects : Wikidata and Wikiversity. [[wikipedia:Wikidata|Wikidata]] is a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&lt;ref&gt;{{Cite news|last=Chalabi|first=Mona|date=April 26, 2013|title=Welcome to Wikidata! Now what?|url=https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|access-date=October 2, 2021|archive-date=2 October 2021|archive-url=https://web.archive.org/web/20211002152920/https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|url-status=live}}&lt;/ref&gt;&quot;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt;. &quot;A  [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;. In complement to using Wikidata to model knowledge, we decided to use Wikiversity to report and write our research results. [[wikipedia:Wikiversity|Wikiversity]] is another Wikimedia project hosting pedagogical content, original research, and even a publishing house ([[WikiJournal|WikiJournals]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like Wikipedia pages, Wikiversity pages are editable by everyone, have a discussion tab and a history log tab. 

Our research question is : '''How can Wikimedia projects contribute to building a collaborative living review on just sustainability transition ?'''

In this project, we aim to test 4 hypothesis : 

● '''Hypothesis 1:''' Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations.

● '''Hypothesis 2:''' Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (e.g. conceptual typologies, cause-effect chains…).

● '''Hypothesis 3:''' SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs.

● '''Hypothesis 4''': Wikimedia or Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links (following the ideal of linked open data).

We also have 2 assumptions :

● '''Assumption 1:''' Wikimedia projects have to be integrated into validated scientific protocols in order to be a valuable research tool.

● '''Assumption 2:''' Wikimedia project contribution has to be made interoperable with tools, methods and data types already used by researchers.
== Methodology ==
Our study rely on a meta-review, that is a review of existing literature reviews. Data presented in literature reviews are usually presented as tables or diagrams, and sometimes provided as supplementary materials in publications. However, these data are not made interoperable and are not used to update prior literature reviews. Our goal was to synthesize results of previous literature reviews by making their findings compatible with linked open data and open science standards using Wikidata, Wikiversity, and other open-science infrastructures. The first step was to build and enrich the bibliographic metadata of a corpus of articles we selected into Wikidata. The second step was to model the content  of these articles in Wikidata (e.g. typologies, causes-effects relationships...). The third step was to experiment relevant visualization of this content (e.g. causes-effects graphs). The las step was to write our report on a Wikiversity page, including links to our knowledge graph, following a linked open data philosophy.

== 1. Building an academic corpus and enriching bibliographic metadata ==
The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations). To do so we imported academic references into Wikidata, and explored the advantages of constituting a scholarly corpus on Wikidata in comparison (or in complementarity) to existing tools used by researchers such as reference management softwares and knowledge management softwares. Reference management software (Zenodo, Mendeley…) are used to collect scientific item metadata and integrate them into academic writing. They can also be used to analyze and annotate academic articles and can include export functions making the data interoperable with other analysis tools. Knowledge management software (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…) are used by some researchers to organize their ideas. To build and enrich our academic corpus on Wikidata, we searched existing databases, selected the sample of articles we wanted to study, imported these articles metadata into Wikidata, enriched these metadata and finally reflected on the advantages and limitations of Wikidata to build a rich academic corpus.

=== Database search ===
Doing a systematic review on all aspects of just transition would have resulted in too many articles to review. We thus decided to first explore one aspect of justice : procedural justice. Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). Procedural justice can include issues of community and citizen participation in decision making, their political representation, their consultation, or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For our search, we selected keywords related to procedural justice (procedural justice OR procedural fairness OR democracy OR participation OR participatory) and keywords related to sustainability transition (sustainability OR energy OR climate) AND (transition OR transitions).

We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article selection ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were : 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies/initiatives/cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors 
* Including justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper
The files with the lists of included and excluded articles are available on the archive plateform Zenodo : https://zenodo.org/records/20749974

=== Importing selected articles into Wikidata ===
Before importing the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in Wikidata. Next, we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through metadata enrichement ===
Metadatas are data describing other data. The metadata of academic items usually include title, author, publication outlet, publication date, pages, DOI, URL... and can be structured following specific standards (e.g. [[wikipedia:Dublin_Core|Dublin Core]]). In academic databases such as WOS or OpenAlex, the only metadata available regarding the content of an academic article are the abstract and sometimes keywords. However, researchers conducting literature reviews need more precise informations. An important part of literature review work can thus be about describing what the articles are about. For example, describing industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt) (e.g. &lt;ref name=&quot;:5&quot; /&gt;). 

By metadata enrichment, we mean completing metadata to include additional information about the content of an academic piece. In Wikidata, each type of information is added using a specific property. A property is the edge that links two entities in the Wikidata knowledge graph. We selected three Wikidata properties to describe the content of our selected articles : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe the geographical scope of the study. We also worked on adding {{Wikidata entity link|P50}}.

==== Adding {{Wikidata entity link|P921}} ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
!Description
|-
|[[d:Q42377797|Q42377797]]
|acceptability
|characteristic of a thing being subject to acceptance for some purpose
|-
|[[d:Q2798912|Q2798912]]
|accountability
|concept of responsibility in ethics, governance and decision-making
|-
|[[d:Q421953|Q421953]]
|actor–network theory
|theory within social science
|-
|[[d:Q84459973|Q84459973]]
|affordability
|
|-
|[[d:Q185836|Q185836]]
|age of a person
|time elapsed since a person was born
|-
|[[d:Q4764988|Q4764988]]
|animal studies
|field in which animals are studied in a variety of cross-disciplinary ways
|-
|[[d:Q4338318|Q4338318]]
|awareness
|state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns
|-
|[[d:Q4930066|Q4930066]]
|blue carbon
|carbon captured by the world's coastal ocean ecosystems
|-
|[[d:Q430460|Q430460]]
|capability approach
|economic theory
|-
|[[d:Q7569|Q7569]]
|child
|human between birth and puberty
|-
|[[d:Q4116870|Q4116870]]
|civic engagement
|individual or group activity addressing issues of public concern
|-
|[[d:Q125928|Q125928]]
|climate change
|human-caused changes to climate on Earth
|-
|[[d:Q260607|Q260607]]
|climate change 
adaptation
|process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities
|-
|[[d:Q1291678|Q1291678]]
|climate justice
|term linking the climate crisis with environmental and social justice
|-
|[[d:Q2270945|Q2270945]]
|co-creation
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q16972712|Q16972712]]
|co-design
|approach to design attempting to actively involve all stakeholders
|-
|[[d:Q16324410|Q16324410]]
|coproduction
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q11024|Q11024]]
|communication
|act of conveying intended meaning
|-
|[[d:Q177634|Q177634]]
|community
|social unit of human organisms who share common values
|-
|[[d:Q5154673|Q5154673]]
|community choice aggregation
|alternative energy supply system
|-
|[[d:Q113514984|Q113514984]]
|community energy
|delivery of community-led renewable energy, energy demand reduction and energy supply projects
|-
|[[d:Q65807646|Q65807646]]
|community participation
|The taking part by members of a community in decisionmaking processes related to the development of their community
|-
|[[d:Q188843|Q188843]]
|cosmopolitanism
|ideology that all human beings belong to a single community, based on a shared morality
|-
|[[d:Q11693783|Q11693783]]
|decarbonization
|change of economy, especially of energy industries, towards lower carbon dioxide emissions
|-
|[[d:Q284289|Q284289]]
|deliberative democracy
|form of democracy focusing on consensus
|-
|[[d:Q7174|Q7174]]
|democracy
|form of government
|-
|[[d:Q552284|Q552284]]
|distributive justice
|concept of the socially just allocation of goods
|-
|[[d:Q1230584|Q1230584]]
|diversity
|concept in sociology and political studies
|-
|[[d:Q1049066|Q1049066]]
|ecological economics
|research field on the interdependence of human economies and natural ecosystems
|-
|[[d:Q8134|Q8134]]
|economics
|social science that studies the production, distribution, and consumption of goods and services
|-
|[[d:Q868575|Q868575]]
|empowerment
|providing increased autonomy
|-
|[[d:Q295865|Q295865]]
|ecosystem service
|benefits created by nature, forests and environmental systems
|-
|[[d:Q138359220|Q138359220]]
|energy citizenship
|involvement of citizens in energy-related decisions
|-
|[[d:Q16869822|Q16869822]]
|energy consumption
|amount of energy or power used
|-
|[[d:Q1358789|Q1358789]]
|senior
|elderly person
|-
|[[d:Q14944319|Q14944319]]
|energy democracy
|concept in environmental justice movement
|-
|[[d:Q192704|Q192704]]
|energy efficiency
|ratio between the useful energy output and the input of a machine
|-
|[[d:Q24965464|Q24965464]]
|energy modeling
|process of building computer models of energy systems in order to analyze them
|-
|[[d:Q1805337|Q1805337]]
|energy policy
|policy addressing energy issues
|-
|[[d:Q1341244|Q1341244]]
|energy poverty
|lack of access to modern energy services
|-
|[[d:Q3406659|Q3406659]]
|energy production
|conversion of energy from a primary source into a form useful to humans
|-
|[[d:Q117091181|Q117091181]]
|energy justice
|subconcept of economic equality
|-
|[[d:Q3456219|Q3456219]]
|energy renovation
|building works aimed at reducing energy consumption and decarbonising the energy sources used
|-
|[[d:Q2700433|Q2700433]]
|energy security
|national security considerations of energy availability
|-
|[[d:Q837718|Q837718]]
|energy storage
|capture of energy produced at one time for use at a later time
|-
|[[d:Q795757|Q795757]]
|energy transition
|long-term structural change towards sustainable energy systems
|-
|[[d:Q1479527|Q1479527]]
|environmental justice
|system of fairness
|-
|[[d:Q771773|Q771773]]
|fairness
|concept in sociology and generally the interaction of society
|-
|[[d:Q56395513|Q56395513]]
|farming system
|method of agricultural production defined by its physical practices and economic characteristics
|-
|[[d:Q5465532|Q5465532]]
|food system
|all processes and infrastructure involved in feeding a population
|-
|[[d:Q4421|Q4421]]
|forest
|dense collection of trees covering a relatively large area
|-
|[[d:Q48277|Q48277]]
|gender
|social concept which distinguish the different gender categories
|-
|[[d:Q1553864|Q1553864]]
|governance
|all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society
|-
|[[d:Q8458|Q8458]]
|human rights
|inalienable fundamental rights to which a person is inherently entitled
|-
|[[d:Q11376059|Q11376059]]
|human rights violation
|act or omission which contravene the principles of human rights
|-
|[[d:Q103817|Q103817]]
|indigenous people
|first inhabitants of an area and their descendants
|-
|[[d:Q113561794|Q113561794]]
|indigenous science
|indigenous knowledge applied to the scientific method
|-
|[[d:Q770480|Q770480]]
|injustice
|quality relating to unfairness or undeserved outcomes
|-
|[[d:Q17142211|Q17142211]]
|interactional justice
|the perceived appropriateness of interpersonal treatment
|-
|[[d:Q1516555|Q1516555]]
|intersectionnality
|theoretical framework of multidimensional oppression
|-
|[[d:Q6316391|Q6316391]]
|just transition
|Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy.
|-
|[[d:Q366139|Q366139]]
|legitimation
|the process of making something acceptable and normative to a group
|-
|[[d:Q3027857|Q3027857]]
|living lab
|user-centered, open innovation ecosystem integrating research and innovation in real life communities
|-
|[[d:Q59679511|Q59679511]]
|low income
|home with little money
|-
|[[d:Q43619|Q43619]]
|natural environment
|all living and non-living things occurring naturally on Earth or some region thereof
|-
|[[d:Q127514833|Q127514833]]
|nature-positive
|global goal to halt and reverse nature loss by 2030
|-
|[[d:Q13023682|Q13023682]]
|non-human
|organism not in the genus Homo
|-
|[[d:Q728646|Q728646]]
|partnership
|arrangement in which parties agree to cooperate to advance their mutual interests
|-
|[[d:Q3907287|Q3907287]]
|policy making
|the act of developing policy
|-
|[[d:Q9357091|Q9357091]]
|political theory
|class of theory
|-
|[[d:Q265425|Q265425]]
|postcolonialism
|academic discipline
|-
|[[d:Q25107|Q25107]]
|power
|ability to influence the behavior of others
|-
|[[d:Q442100|Q442100]]
|procedural justice
|fairness in the processes that resolve disputes and allocate resources
|-
|[[d:Q7249406|Q7249406]]
|project governance
|management framework
|-
|[[d:Q7257735|Q7257735]]
|public engagement
|Policy-making practice
|-
|[[d:Q541936|Q541936]]
|public participation
|participation of citizens in various policy decisions and planning processes
|-
|[[d:Q6142016|Q6142016]]
|recognition justice
|social philosophy theory
|-
|[[d:Q10509953|Q10509953]]
|renewable electricity
|electricity from renweable sources
|-
|[[d:Q12705|Q12705]]
|renewable energy
|energy collected from renewable resources
|-
|[[d:Q56510941|Q56510941]]
|renewable energy policy
|
|-
|[[d:Q1165392|Q1165392]]
|restorative justice
|approach to justice where victims and perpetrators mediate a restitution agreement
|-
|[[d:Q4414036|Q4414036]]
|rural population
|inhabitants of rural areas or of small towns classified as rural
|-
|[[d:Q17152351|Q17152351]]
|smart system
|adaptive intelligent systems
|-
|[[d:Q187588|Q187588]]
|social class
|group of people categorized in a hierarchy based on socioeconomic factors
|-
|[[d:Q264892|Q264892]]
|social justice
|concept that discrimination recognized in society should be remedied
|-
|[[d:Q34749|Q34749]]
|social science
|academic disciplines concerned with society and the relationships between individuals in society
|-
|[[d:Q2930198|Q2930198]]
|stakeholder participation
|involvement of groups or individuals affected by the actions of an entity
|-
|[[d:Q125359881|Q125359881]]
|sustainability transition
|
|-
|[[d:Q219416|Q219416]]
|sustainability
|ability of human civilization to coexist with the biosphere in a steady state
|-
|[[d:Q131201|Q131201]]
|sustainable development 
|mode of human development that meets current demands without compromising the needs of future generations
|-
|[[d:Q7649586|Q7649586]]
|Sustainable Development Goals
|set of United Nations-defined global development goals and climate change
|-
|[[d:Q69883|Q69883]]
|urban planning
|technical and political process concerned with the use of land and design of the urban environment
|-
|[[d:Q920600|Q920600]]
|urban renewal
|program of land redevelopment in cities, often where there is urban decay
|-
|[[d:Q3376054|Q3376054]]
|vulnerable population
|group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent
|-
|[[d:Q107389921|Q107389921]]
|water-management
|
|-
|[[d:Q7981051|Q7981051]]
|well-being
|measure of how well life is to someone or a group with factors such as health, happiness and satisfaction
|-
|[[d:Q467|Q467]]
|woman
|female adult human
|-
|[[d:Q188867|Q188867]]
|future studies
|study of possible, probable, and preferable social, technological and political futures
|-
|[[d:Q1038171|Q1038171]]
|participatory design
|active involvement of all stakeholders in the design process
|}Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. 

==== Adding {{Wikidata entity link|P8363}} ====
The sample of article we selected included only litterature reviews, but we wanted to describ what kind of literature review it was. We first read abstracts to identify all the different types of litterature reviews present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved some of these method items using the methodological references cited in the reviewed papers. For example, we added a statement saying that {{Wikidata entity link|Q101116078}} can have {{Wikidata entity link|Q653137}} as {{Wikidata entity link|P13391}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Study type
!Description
|-
|[[d:Q603441|Q603441]]
|bibliometrics
|statistical analysis of written publications, such as books or articles
|-
|[[d:Q472342|Q472342]]
|scientometrics
|study of measuring and analysing science, technology and innovation
|-
|[[d:Q815382|Q815382]]
|meta-analysis
|statistical method that summarizes data from multiple sources
|-
|[[d:Q1504425|Q1504425]]
|systematic review
|publication type, study that gathers, analyzes, and communicates the results of research and information on a topic
|-
|[[d:Q2412849|Q2412849]]
|literature review
|process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic
|-
|[[d:Q6822263|Q6822263]]
|meta-regression
|statistical tool used in meta-analyses
|-
|[[d:Q7301211|Q7301211]]
|realist evaluation
|theory-driven evaluation used in evaluating social programmes
|-
|[[d:Q17007303|Q17007303]]
|combinatorial meta-analysis 
|study of the statistical properties of combinations of studies from a meta-analytic dataset
|-
|[[d:Q70470634|Q70470634]]
|network meta-analysis
|meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions
|-
|[[d:Q101116078|Q101116078]]
|scoping review
|search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry
|-
|[[d:Q110665014|Q110665014]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|-
|[[d:Q137174203|Q137174203]]
|conceptual review
|academic research aiming to review existing concepts and definitions in the litterature
|-
|[[d:Q137174450|Q137174450]]
|critical review
|type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research
|-
|[[d:Q137209848|Q137209848]]
|integrative literature review
|type of literature review
|-
|[[d:Q110665014|Q137211242]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|}For each article, we added the {{Wikidata entity link|P8363}} based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Adding {{Wikidata entity link|P6153}} ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

==== Adding {{Wikidata entity link|P50}} ====
When scholarly metadata are imported into Wikidata, the name of authors are stored as a chain of characters and linked to the property {{Wikidata entity link|P2093}}. The property {{Wikidata entity link|P50}} allows to make a link with a Wikidata item representing the author. This avoids the problem of homonym authors by linking a unique identifyer to authors in Wikidata and linking this identifier to existing ones such as ORCID. We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers by categorizing scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages.

==== Adding {{Wikidata entity link|P6977}} ====
We explored the possibility to model review networks, that is linking review paper to reviewed papers (and not broadly cited papers). We selected one of the paper which had the most detailed list of reviewed papers ({{Wikidata entity link|Q114306483}}), then we added this list in Wikidata using Open refine. Then we snowballed this process for the reviewed papers which were also litterature reviews.

=== Advantages and limitations of Wikidata to build a rich living academic corpus ===
To share the result of our work, we exported the dataset we build on Wikidata and shared it on the open archive Zenodo : https://doi.org/10.5281/zenodo.20749973. The data is also available directly in Wikidata.

The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations)'''.''' The sections below discuss the advantages and limitations of wikidata regarding this hypothesis.

==== Advantages of Wikidata ====
Key advantages of Wikidata are its flexible and collaborative nature as well as its interoperability. Wikidata ontology (that is how the data are structured) is collaboratively defined and properties can be added if relevant (after validation by the community). Compared to global databases like WOS or OpenAlex, Wikidata allows anyone to enter more metadata about each academic articles. Another notable advantage is that Wikidata items can be used as an interoperable [[wikipedia:Controlled_vocabulary|controlled vocabulary]]. For example, when we stated that the {{Wikidata entity link|P921}} of the article {{Wikidata entity link|Q114306483}} was {{Wikidata entity link|Q795757}}, &quot;energy transition&quot; was not just a word but a concept with its unique identifyer, linked to identifiers in other databases such as the Google Knowledge Graph ID or BNCF Thesaurus ID. Wikidata's collaborative nature is here adain an advantage. Contrary to institutional thesaurus, Wikidata allows anyone to add new concepts. This is particularly interesting as existing controlled vocabularies rarely reflect the degree of precision that researchers need in their work. The multilingual nature of Wikidata was also a strengh, some Wikidata contributors added labels for the concepts we used into different languages (For example, contributors added labels for {{Wikidata entity link|Q14944319}} in Armenian and Slovenian, languages we do not speak at all). 

==== Limitations of Wikidata ====
The limitations of Wikidata is that it is not yet well integrated with the tools researchers use to do literature reviews (kowledge management softwares and bibliographic databases).    

Compared to reference management softwares (Zenodo, Mendeley…) and knowledge management softwares (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…), Wikidata is too general and does not allow to work on full texts. References and knowledge management softwares allow researcher to build their own specialised knowledge base, by taking notes and highlighting the content of the full texts. Wikidata is not connected to this process and there is a missing tool to facilitate the construction of graphs from the qualitative analysis of texts. In addition, when one is working on a specific corpus of item in Wikidata, it is also difficult to keep track of this corpus. We linked each academic item we were working on to our research project by adding a statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}, but it was still relatively difficult to &quot;filter&quot; the part of the knowledge graph we were working on.   

Compared to bilbiographic catalogues (OpenAlex, Web Of Science, GoTriple...), Wikidata will never be as exhaustive and do not offer user-friendly search functions. Since 2014, an important amount of bibliographic data was imported in Wikidata with the project [[d:Wikidata:WikiCite|Wikicite]]. At the time of its creation, Wikicite was adressing the issue of closed bibliographic data and was trying to make these data open, many academic items were imported automatically in Wikidata through scraping. This practice was abandoned because the large amont of bibliographic data congested queries on Wikidata (this led to the decision to split the Wikidata graph between academic and non academic entities), and because new open science initiatives, notably OpenAlex (2022), are now taking on the task of creating a exhaustive catalogues of all scholarly production. In this context, the community has to rethink the purpose of Wikidata regarding bilbiographic data.   

==== Future possbilities ====
A solution to the limitations would be to developp the role of Wikidata as a link between other tools of the open science ecosystem. For example, developping and maintaining plugins or extensions for specialised softwares like Zotero, Wikibase, and Omeka could connect Wikidata with more specialised graphs. Such extensions could help building local graphs by allowing the reuse of wikidata item (eg. autocompletion), but also help contributing to Wikidata thanks to export features. (Example :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  Wikidata could then be an intermediary between locally curated corpus and more exhaustive bilbiographic catalogues such as OpenAlex. For example, Wikidata items could be used to tag articles in a more precise way instead of using keywords and sharing enriched corpus in Wikidata could help to train more precise taging algorythms.

== 2.Modelling the content of litterature reviews ==
The goal of this step was to test '''Hypothesis 2''' (Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference) by modelling the content of our selected articles into Wikidata. [[wikipedia:Knowledge_modeling|Knowledge modelling]] is the process of making a machine readable model of knowledge. As we have a background in social sciences, we felt the need to question the relationship between this process and other methodologies such as concept mapping, thematic networks and causal networks. We present these methodologies before describing our current knowledge modelling experimentations.

=== Concept mapping, thematic networks and causal networks ===
This section presents social science methodology that presents similarities with knowledge modelling.

==== Concept maps ====
[[File:Conceptual_Diagram_-_Example.svg|link=https://en.wikipedia.org/wiki/File:Conceptual_Diagram_-_Example.svg|thumb|Example conceptual diagram|251x251px]]Concept maps are ''concepts'' (boxes) and ''propositions'' (arrow indicating the relationship between two boxes)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; logic is similar to how knowledge is modelled on Wikidata : the equivalent of concepts is ''item'' and the equivalent of propositions are ''statements''. The difference between a softwares like Cmap and Wikidata is the underlying format of the data.  
  
==== Thematic networks ====
[[File:Thematic network example.jpg|thumb|447x447px|Structure of a thematic network (Source: based on Attride-Stirling 2001)]]
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified.   
[[File:Adoption_CLD.svg|link=https://en.wikipedia.org/wiki/File:Adoption_CLD.svg|thumb|421x421px|Causal loop diagram of ''Adoption'' model, used to demonstrate systems dynamics]]

==== Causal diagrams ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== Knowledge modelling in Wikidata ===
This section presents our knowledge modelling experimentation in Wikidata.

==== Conceptual modelling ====
Our first step was to reflect on what is a &quot;concept&quot; and what kind of wikidata properties could be used to model concepts in Wikidata. Scholars in management have called for more rigorous ways to define concepts&lt;ref name=&quot;:22&quot; /&gt; and modelling concepts in Wikidata could help to build less ambiguous concepts. Concept definition encompass various aspects such as the nature of the phenomenon, its characteristics, the links with prototypical cases or examples, the contrast with other concepts, the links with causes and consequences...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. We summarize these approaches below and examine which wikidata properties exist to represent them.
*Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
*Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. (The closer a phenomenon is to the prototype, the more likely it belong to the category). Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}} (see discussion here https://www.wikidata.org/wiki/Help:Basic_membership_properties).
*Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}, {{P|1478}}, {{P|P9353}} (see discussions here : https://www.wikidata.org/wiki/Help:Modeling_causes/en).
*Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can use it before really knowing what its properties will be.
*Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

==== Testing concept modelling on {{Wikidata entity link|Q14944319}} ====
To test concept modelling, we started by experimenting with the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic :

*{{Wikidata entity link|Q137901202}}
*{{Wikidata entity link|Q137901196}}
*{{Wikidata entity link|Q137901182}}
*{{Wikidata entity link|Q136447761}}
*{{Wikidata entity link|Q129652515}}
*{{Wikidata entity link|Q114306483}}

We read each paper and used them as source in manually entered statements in the item {{Wikidata entity link|Q14944319}}. For example, Droubi et. Al stated &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}} and {{Wikidata entity link|Q3249551}}, using the paper as reference (see screenshot below). The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&amp;oldid=2495982191.

[[File:Wikidata statement- energy democracy is an instance of ideal.png|915x915px|border]]

We listed the difficulties encountered as we worked and we also asked the Wikidata community to give us feedback on our modelling on the item discussion page (https://www.wikidata.org/wiki/Talk:Q14944319). These issues were related to contradictions, precision, concision and ontology. We discuss each issue and draft recommandations to refine our modelling process.  

===== Contradictions =====
Wikidata contributor's feedback highlighted some apparent contradictions (The values in &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata&quot;&lt;ref&gt;{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;.Besides, in the case of goals versus effects statements, it is not contradictory because one can have a goal and fail to achieve it. In the case of energy democracy, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

Recommandations : Contradictions are allowed in Wikidata. 

===== Precision =====
We noted that conceptual modelling requires an important degree of formalization and precision. This is a key advantage of Wikidata to be able to create links toward precise concepts which have their own identifiers. For example, we were able to create statements about specific laws and their unique identifiers in legal databases (e.g. {{Wikidata entity link|Q139764294}} and its identifier in the EUR-Lex database). However, the sources we are working with are not always precise enough and when concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, {{Wikidata entity link|Q12888920}}... We can see here that academic texts are using natural language and thus are using ambiguous terms. As a result, we received feedbacks regarding a lack of precision in our statements (too many and too vague statements). In addition to the ambiguity of sources, a reason why we ended up with very general statements is because we avoided the creation of new Wikidata items. While following this implicit rule allowed us to focus on the most notable concepts, creating new items could also help make the statements more precise. 

Recommandations : If a concept is ambiguous it should not be included in the modelled statements. Create more precise relevant concepts if they do not exist in Wikidata.  [[File:Wikidata visualisation screenshot of subclasses relationships including the item political concept.png|thumb|298x298px|Subclass relationships between &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. The rule we take from this feeback is a need of logical simplification. Two examples illustrate possible logical simplification :
*We stated that {{Wikidata entity link|Q14944319}} was an {{Wikidata entity link|P31}} {{Wikidata entity link|Q33104069}} and an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}. But in that case, it is not necessary to state that it is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}, because {{Wikidata entity link|Q33104069}} is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q131362181}}, which is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q151885}} (see diagram on the right). Here, we have to keep only the more precise item. This reasonning is based on the assumption that {{Wikidata entity link|P279}} is transitive. It seems this reasonning could be generalized (we opened a discussion about this here https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning/Use_cases#Parcimonious_statement_constraints_based_on_subclass_of_(P279)_and_part_of_(P361)_transitivity&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*We stated that {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P2670}} {{Wikidata entity link|Q15991216}} and {{Wikidata entity link|Q113514984}}. But if we consider that {{Wikidata entity link|Q15991216}} is a {{Wikidata entity link|P279}} of {{Wikidata entity link|Q113514984}}, then the inclusion of {{Wikidata entity link|Q15991216}} is implied. Here we could keep only the broader item {{Wikidata entity link|Q113514984}}. But unlike the reasonning above, we could not identify a way to generalize this reasonning. For example, {{Wikidata entity link|Q113514984}} can be a subclass of {{Wikidata entity link|Q43229}}, but using the former item is more informative than using the later. For the property {{Wikidata entity link|P2670}}, using a value that is too precise results in too many statements and using a value that is too general results in too trivial statements. Choosing the degree of precision is a problem similar to choosing whether or not to create a new item.
Recommandations : When possible, use logical simplification to make statements more parcimonious. The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize redundancies. 

===== Quantification =====
Modelling quantitative statements was challenging. We mostly skipped those but made an attempt for one case : the paper {{Wikidata entity link|Q137901196}} states that &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. There are such properties in Wikidata : for example, for renewable energy subsidies, there is {{Wikidata entity link|P6826}}. This type of property is notably useful to display information about an item  (e.g. the population of a town) on Wikipedia through dynamic infoboxes that are updated with Wikidata information. But the current informal rule on Wikidata property creation is to prefer the use of general properties and avoid the creation of specific properties. Therefore creating properties for every quantifiable characteristic may not be possible. An alternative way to model quantity is to create a specific item to represent what is quantified and use other properties to represent quantities (see example below). 

[[File:Wikidata screenshot of percentage modelling.png|border|783x783px]]

Despite this possibility, since quantitative data are usually stored elsewhere in more convenient formats, it may not be always relevant to include quantitative data into Wikidata. Instead, it might be more useful to link the item that is quantified to an external dataset about it. This can be done using the property {{Wikidata entity link|P1325}}. 

Recommandation : Use existing properties for the quantity if it exists. If it does not exist, create an item representing what is measured, link it toward an external datasets and add quantitative statements if relevant.

===== Ontology =====
Ultimately, several of our difficulties were linked to some ontology challenges reflecting the complexity of the concept we were working on. According to the litterature, {{Wikidata entity link|Q14944319}} refer to multiple types of entities. It represents a concept, an ideal, a process and an outcome. We reflected this with multiple {{Wikidata entity link|P31}} statements, but encountered struggles to model differences between ''process versus outcomes'', ''ideal versus reality'', ''phenomenon versus theory'' and ''discourses versus practices''. The wikidata community recommands having different items for ''process and outcome'' (criterion &quot;{{Wikidata entity link|Q127270577}}&quot;). For example, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork). However, this distinction is less straightforward for social processes which are are ongoing and evolving without a clear ending. Another problem was to separate ''ideal versus reality,'' we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes attempting to reach it. It was also difficult to separate ''phenomenon versus theory,'' for example, we did not manage to model the idea that the literature on energy democracy is fragmented. Finally, we also would have needed more distinctions between ''discourses and practices.'' Our sources suggested that energy democracy discourses and practices may have different causes, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movments promoting it, but this was not a very precise way to model this idea.

Our first attempt to multiple &quot;natures&quot; of energy democracy into a single item is not satisfactory, and implementing the modelling recommandations we identified above may not be sufficient to build more parcimonious statements. Creating new items to reflect the different aspects of energy democracy may here be necessary. To do so we have to decide how to split the different natures {{Wikidata entity link|Q14944319}} and what will be the relationship between the resulting items. Here the definition of an {{Wikidata entity link|Q324254}} (formal representation) in Wikidata can quickly escalate into questionning {{Wikidata entity link|Q44325}} (metaphysical reflexion on the nature of things), and especially {{Wikidata entity link|Q1713511}} (the nature of the social world). Indeed, the relations between the different types of entities constituting the social worls are not simple nor consensual among philosophers of social science.  

For example, {{Wikidata entity link|Q15080858}} posits that different things have different ways of being (modes of reality). They propose to classify entities in four categories : material entities (that can exist independently of humans), conceptual entities (concepts, discourses, ideas, meaning…), artefactual entities (human-made and combining conceptual and material elements) and social entities (that depends on human activity to exist)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. A complex concept like {{Wikidata entity link|Q14944319}} is concerned with all these types of entities. The energy system include many material entities such as oil fields, the sun, seas, trees... and artefacts such as energy production unit, power lines, home appliances, trucks... (Including conceptual entities such as the name of these artefacts or the knowledge to make them function.). There are the social entities in which they are encompassed (the enregy sectors, energy businesses, energy policies...). There are conceptual entities like normative/political discourses discussing how these artefact and social system should work and there are conceptual entities in the academic sphere building theories about how all this works or should work. 

In theory, such ontology can be implemented in Wikidata as it &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; Current Wikidata ontology is structured with dischotomies such as {{Wikidata entity link|Q7048977}}/{{Wikidata entity link|Q4406616}}, {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}, {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q23958946}}, {{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}, {{Wikidata entity link|Q30241068}}/{{Wikidata entity link|Q3799040}}... A quick search seem to show that entities related to the social reality can quickly fall into heterogeneous and sometime contradictory root classes (see table below). 
{| class=&quot;wikitable&quot;
|+
!Item
!Existing root class entity
|-
|{{Wikidata entity link|Q190539}}
|{{Wikidata entity link|Q7048977}}

{{Wikidata entity link|Q30241068}}

{{Wikidata entity link|Q67518978}}
|-
|{{Wikidata entity link|Q34770}}
|{{Wikidata entity link|Q7048977}}
|-
|{{Wikidata entity link|Q9081}}
|{{Wikidata entity link|Q7048977}}
|-
|{{Wikidata entity link|Q43229}}
|{{Wikidata entity link|Q99527517}}{{Wikidata entity link|Q30241068}}

{{Wikidata entity link|Q103940464}}
|-
|{{Wikidata entity link|Q49773}}
|{{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}

{{Wikidata entity link|Q99527517}}

{{Wikidata entity link|Q30241068}}

{{Wikidata entity link|Q7048977}}
|-
|{{Wikidata entity link|Q12705}}
|{{Wikidata entity link|Q4406616}}/{{Wikidata entity link|Q7048977}}

{{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}
{{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q132907471}}
|}

=== Future steps ===
Current {{Wikidata entity link|Q3882785}} in Wikidata do not seem very informative to classify the aspects of our concept in a top-down manner. Our next steps will thus be to :  

* Export the existing model of the concept to keep track of each modelling steps (files and graphs screenshots)

*Explore if existing standards offer more relevant classes to categorize social entities (we opened the discussion about social ontology with the wikidata community here https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Ontology#Social_ontology).
**CRMSoc : https://cidoc-crm.org/crmsoc/fm_releases
**Criteria for choosing an ontology = choose something that is used and maintained by a community (colleague feedback)
*Create new items to reflect the different facets of {{Wikidata entity link|Q14944319}} : the social movement advocating for it, the political concept theorizing it, the concrete initiatives implementing it, the litterature theorizing it, etc... based on a new reading of our sample of article and aligned with existing ontologies.
*Compare grounded theory/CAQDAS with knowledge modelling methods, test annotation softwares : https://inception-project.github.io/ ; https://pacte.crim.ca/index_en.html
*Read articles again and model more precise statements, including quotation of the original text in the reference field.

== 3. Data visualisation ==
The goal of this step is to test '''Hypothesis 3''' (SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs). At this stage we started to identify visualization use cases we would like to test, they are listed below.

=== Filter statements ===

*Visualize only statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*Visualize only items which are part of the present project (all the academic items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; of a property used in an item : Wikidata graph builder seem to be the most user friendly, robust and versatile tool to visualise a graph of a single property (https://angryloki.github.io)

=== Concepts visualisation ===

*Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request &quot;topic in context&quot;)
*Map the statements that two items have in common to make comparisons.

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 4. Writing ==
The goal of this step is to test '''Hypothesis 4''' (Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links). 

=== Advantages of Wikiversity ===
Writing on a Wikiversity page offers some advantages to implement the principles of open linked data in text format. We could cite academic items using their Wikidata QID to generate the citations below (but if we use an URL to cite a paper, it does not automatically check if there is an existing QID for it), and also link toward Wikidata entities using a template ([[Template:Wikidata entity link|Wikidata entity link]]). Although we did not translate the present page for now, Wikiversity offers the possibility to translate a page in multiple languages (an interesting possibility in the context of the The FAIRisation of scholarly communication&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). The possibility to view the page history provide an exhaustive versionning of a paper, which constitute interesting data documenting the academic writing process (at first we did not comment the nature of our modifications but after thinking about this reuse possibility, we described it with more details). The contribution statistics based on the history also offers a new way to track author's contributions to a paper in a transparent way.

=== The issue of text interoperability ===
A key issue we are encountering is the question of the interoperability of texts. While the interoperability of data is starting to be well discussed in the open science community, the interoperability of texts do not seem to benefit from the same level of discussion. We encountered several interoperability issues regarding our writing. First, copying texts written on a word processor software (e.g. microsoft word) into a wiki page (or the other way around) is relatively seamless in terms of formatting, but the management of references is lacking. Reformatting references is very time consuming and a real barrier for text interoperability in academic context : it is difficult to copy text from an academic publication into a wiki text, and difficult to turn a wiki text into a publication (the only possibility is to export the page as a PDF and you may be reading this article in this format).   

There are also uncertaineties regarding how to reuse texts published under creative common licences. Academic texts published under CC-BY-SA licences can in theory be remixed and reused. But academia does not have established practices regarding how this can be done. If we want to reuse a whole page, should we put it in quotation marks and simply cite the paper ? Should the original authors be listed as co-authors ? To what extent academic publisher will accept to publish &quot;remixed&quot; texts while they usually require that publications contain mainly unpublished content ? The norms of what is appropriate remix and reuse practices in academia has yet to be decided... and this could be an interesting conversation to have in the open science community.  

== Discussion ==
Our goal was to assess the potential of Wikidata and Wikiversity to build a living literature review method and tackle issues of information overload, knowledge synthesis and knowledge dissemination, following open science principles. By conducting a meta-review on just sustainability transition (that is a review of existing literature reviews), and experimenting with existing technical solutions, we were able to identify the strenghs and limitations of the Wikidata ecosystem. First Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations (Hypothesis 1 is verified). However, the technical barriers to do so are still high and the Wikidata database is too general to allow a community to work on a specific curated corpus of scientific items. Secondly, Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (Hypothesis 2 is verified). However, this requires developping new methodological standards regarding what is a rigorous modelling in social science, as well as building specific {{Wikidata entity link|Q324254}}. An unexpected finding is that the formalization necessary for knowledge modelling in wikidata invites to formalize the reasonning behind modelling choices, a process that is rarely done in qualitative analysis even though such reasonning are likely present. The modelling process also responds to calls in management sciences to engage in more systematic categorizing to avoid semantic confusion&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; it (Hypothesis 3 is partially verified). SPARQL-based queries are powerful but they require technical knowledge, especially now that the split between academic and non-academic items in Wikidata requires to write federated queries. To be able to &quot;navigate&quot; scientific corpora and scientific knowledge graphs on Wikidata, more user-friendly tools would be needed. Finally, we did see advantaged in using Wikiversity pages to write collaboratively in text format augmented by hypertext links, but there are still important technical and instittional barriers (Hypothesis 3 is partially verified) : the interoperability of text is a key issue to reuse publications. The main issues we encountered were the management of references and uncertainty regarding the compatibilities between writing publicly on a wiki page and engaging in classical publication processes.

A potential solution to the issues encountered could be to develop a specialised literature review software that would allow researchers to build living literature reviews including knowledge graphs and wiki pages. This kind of tool should be user-friendly and include the missing tools we identified. In addition, it could include more advanced analysis functions such as logical reasonning based on the knowledge graph&lt;ref name=&quot;:9&quot; /&gt;. 

=== Future research ===
- Can we &quot;reason&quot; with Wikidata  (See: https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning) : neurosymbolic AI ?

== Funding ==
This project is funded by the [[m:Grants:Programs/Wikimedia_Research_&amp;_Technology_Fund/Wikimedia_Research_Fund|Wikimedia Research Fund]], Grant ID: G-RS-2504-18935. The text of the initial research proposal is available here : https://doi.org/10.5281/zenodo.20760603.

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
| [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape
|-
| [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models
|-
| [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review
|-
| [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions
|-
| [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions
|-
| [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation
|-
| [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings
|-
| [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda
|-
| [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review
|-
| [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework
|-
| [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende
|-
| [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa
|-
| [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities
|-
| [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion
|-
| [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review
|-
| [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights
|-
| [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review
|-
| [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations
|-
| [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance
|-
| [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions
|-
| [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review
|-
| [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice
|-
| [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice
|-
| [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review
|-
| [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review
|-
| [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions
|-
| [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition
|-
| [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy
|-
| [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends
|-
| [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience
|}


== References ==
{{References}}</text>
      <sha1>kf5obo64kokh7ckpkkihj6vwuylrgk9</sha1>
    </revision>
    <revision>
      <id>2818219</id>
      <parentid>2818218</parentid>
      <timestamp>2026-07-13T09:37:27Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Future steps */ adding a future step : refine research question</comment>
      <origin>2818219</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="108615" sha1="4tfgb0ts7y61sjm41wnb4rf2xr6w28d" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

== Acknowledgements ==
The present text was originally written on a Wikiversity page, if you are reading it in another format, you can find this page here : [[Just sustainability transitions: a living review|https://en.wikiversity.org/wiki/Just_sustainability_transitions:_a_living_review]]. You are free to add your comments on the page discussion section. 

=== Contributors ===
{| class=&quot;wikitable&quot;
|+
!Name
!Affiliation
!ORCID
!Contribution
|-
|Adélie Ranville
|IAE de Grenoble, CERAG lab (https://ror.org/0509qp208)
|https://orcid.org/0000-0002-3993-6135
|Research design, database search, article screening, knowledge modelling, article writing
|-
|Amélie E. Pereira
|Laboratoire DICEN IDF
|https://orcid.org/0009-0005-5928-5586
|Meta-data enrichement, article writing
|-
|Finn Nielsen
|Technical University of Denmark
|https://orcid.org/0000-0001-6128-3356
|Data visualisation
|}
Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review

== Introduction ==
Just sustainability transition refers to the process of shifting towards sustainable practices in a way that is equitable and inclusive. It includes dimensions of procedural, recognition, distributive and reparative justice and the concept is related to climate justice, environmental justice and energy justice&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-3-030-89460-3_2|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021|publisher=Springer International Publishing|isbn=978-3-030-89460-3|editor-last=Heffron|editor-first=Raphael J.|location=Cham|pages=9–19|language=en|doi=10.1007/978-3-030-89460-3_2}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.sciencedirect.com/science/article/pii/S0301421518302301|journal=Energy Policy|volume=119|pages=1–7|doi=10.1016/j.enpol.2018.04.014|issn=0301-4215}}&lt;/ref&gt;. The study of sustainability transitions in social sciences requires dynamic and adaptive research synthesis methods. Sustainability transitions involve complex, multi-level processes influenced by technological, economic, social, and policy factors&lt;ref name=&quot;:15&quot;&gt;{{Cite journal|date=2020-03-01|title=Micro-foundations of the multi-level perspective on socio-technical transitions: Developing a multi-dimensional model of agency through crossovers between social constructivism, evolutionary economics and neo-institutional theory|url=https://www.sciencedirect.com/science/article/abs/pii/S0040162518316111|journal=Technological Forecasting and Social Change|language=en-US|volume=152|pages=119894|doi=10.1016/j.techfore.2019.119894|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2023-08-01|title=A socio-technical transition perspective on positive tipping points in climate change mitigation: Analysing seven interacting feedback loops in offshore wind and electric vehicles acceleration|url=https://www.sciencedirect.com/science/article/pii/S0040162523003244|journal=Technological Forecasting and Social Change|language=en-US|volume=193|pages=122639|doi=10.1016/j.techfore.2023.122639|issn=0040-1625}}&lt;/ref&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Sovacool|first=Benjamin K.|last2=Geels|first2=Frank W.|last3=Andersen|first3=Allan Dahl|last4=Grubb|first4=Michael|last5=Jordan|first5=Andrew J.|last6=Kern|first6=Florian|last7=Kivimaa|first7=Paula|last8=Lockwood|first8=Matthew|last9=Markard|first9=Jochen|date=2025-03-01|title=The acceleration of low-carbon transitions: Insights, concepts, challenges, and new directions for research|url=https://www.sciencedirect.com/science/article/pii/S2214629625000295|journal=Energy Research &amp; Social Science|volume=121|pages=103948|doi=10.1016/j.erss.2025.103948|issn=2214-6296}}&lt;/ref&gt;. Given the rapidly evolving nature of sustainability-related research, static literature reviews often become outdated, limiting their usefulness for policymakers, scholars, and practitioners. A living literature review – continuously updated with new findings – ensures that emerging insights, case studies, and theoretical developments are integrated cumulatively into the knowledge base. Developing such review will answer the call for more evidence-based practices in management sciences&lt;ref&gt;{{Cite journal|last=Kepes|first=Sven|last2=Bennett|first2=Andrew A.|last3=McDaniel|first3=Michael A.|date=2014-09|title=Evidence-Based Management and the Trustworthiness of Our Cumulative Scientific Knowledge: Implications for Teaching, Research, and Practice|url=https://journals.aom.org/doi/10.5465/amle.2013.0193|journal=Academy of Management Learning &amp; Education|volume=13|issue=3|pages=446–466|doi=10.5465/amle.2013.0193|issn=1537-260X}}&lt;/ref&gt;&lt;ref&gt;Pfeffer, J., &amp; Sutton, R. I. (2006). Evidence-Based Management. Harvard Business Review, 13. &lt;/ref&gt;. Our project assesses the potential of Wikidata to build living review workflow on sustainability transition. We address three issues encountered by scientists: information overload, knowledge synthesis and results dissemination.

=== The problem of academic information overload ===
Global scientific output doubles every nine years&lt;ref&gt;{{Cite web|url=http://blogs.nature.com/news/2014/05/global-scientific-output-doubles-every-nine-years.html|title=Global scientific output doubles every nine years : News blog|website=blogs.nature.com|language=en-US|access-date=2026-06-23}}&lt;/ref&gt;, pushed by the “publish or perish” model incentivizing researchers to increase the quantity of research outputs. Researchers are subject to information overload as the number of publications to read is beyond what a human brain can handle, they are expected to produce high-quality research under an increasing time pressure. This intensification of academic work is being denounced as detrimental to the deep cognitive process needed to actually produce interesting knowledge&lt;ref&gt;{{Cite journal|last=Hartman|first=Yvonne|last2=Darab|first2=Sandy|date=2012-01-01|title=A Call for Slow Scholarship: A Case Study on the Intensification of Academic Life and Its Implications for Pedagogy|url=https://doi.org/10.1080/10714413.2012.643740|journal=Review of Education, Pedagogy, and Cultural Studies|volume=34|issue=1-2|pages=49–60|doi=10.1080/10714413.2012.643740|issn=1071-4413}}&lt;/ref&gt;. “Wikifying science” may in this context contribute to facilitating researcher’s work while preserving scientific quality. That is why in this project, we aim to build a searchable academic publication database with enriched meta-data that will allow scholars to navigate the existing publications corpus related to just sustainability transition more easily.

=== The problem of knowledge synthesis ===
The volume of academic production is rendering knowledge synthesis difficult. Scholars have thus called for making literature reviews cumulative and updatable&lt;ref&gt;{{Citation|title=Day 2 {{!}} Arnaud Vaganay: Reproducible Literature Reviews|url=https://www.youtube.com/watch?v=Nspd_1cx9kc|date=2017-10-19|accessdate=2026-06-23|last=Berkeley Initiative for Transparency in the Social Sciences (BITSS)}}&lt;/ref&gt; and for shifting from static text format publications to dynamic knowledge mapping&lt;ref name=&quot;:11&quot;&gt;{{Cite web|url=https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/|title=The death of the literature review and the rise of the dynamic knowledge map - LSE Impact|last=Taster|date=2019-05-14|website=LSE Impact - Understanding impact and practice in academic research|access-date=2026-06-23}}&lt;/ref&gt;. This call is being answered through the development of living literature reviews that can be updated dynamically with new knowledge (examples : &lt;ref&gt;{{Cite journal|last=Elliott|first=Julian H.|last2=Synnot|first2=Anneliese|last3=Turner|first3=Tari|last4=Simmonds|first4=Mark|last5=Akl|first5=Elie A.|last6=McDonald|first6=Steve|last7=Salanti|first7=Georgia|last8=Meerpohl|first8=Joerg|last9=MacLehose|first9=Harriet|date=2017-11|title=Living systematic review: 1. Introduction—the why, what, when, and how|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435617306364|journal=Journal of Clinical Epidemiology|volume=91|pages=23–30|doi=10.1016/j.jclinepi.2017.08.010|issn=0895-4356}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Uttley|first=Lesley|last2=Quintana|first2=Daniel S.|last3=Montgomery|first3=Paul|last4=Carroll|first4=Christopher|last5=Page|first5=Matthew J.|last6=Falzon|first6=Louise|last7=Sutton|first7=Anthea|last8=Moher|first8=David|date=2023-04|title=The problems with systematic reviews: a living systematic review|url=https://linkinghub.elsevier.com/retrieve/pii/S0895435623000112|journal=Journal of Clinical Epidemiology|volume=156|pages=30–41|doi=10.1016/j.jclinepi.2023.01.011|issn=0895-4356}}&lt;/ref&gt;,&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;). While such reviews method exist for quantitative research producing standardized results, they are not adapted to synthetize social science studies on sustainability transitions that involve diverse methodologies and various disciplinary perspectives. The goal of the project is to propose a demonstration of a living review method for social science findings on just sustainability transition, relying on the collaborative model and tools of Wikimedia projects notably Wikidata, Wikiversity and Wikipedia.

=== The problem of scientific results dissemination ===
There is urgent need to disseminate knowledge on impactful topics like sustainability transition while proprietary publication models, disinformation and censorship (e.g. US) is threatening access to free and reliable knowledge. In parallel, social scientists struggle to make their work impactful&lt;ref&gt;{{Cite journal|last=Haley|first=Usha C. V.|date=2023-09-01|title=Triviality and the Search for Scholarly Impact|url=https://doi.org/10.1177/01708406231175292|journal=Organization Studies|language=EN|volume=44|issue=9|pages=1547–1550|doi=10.1177/01708406231175292|issn=0170-8406}}&lt;/ref&gt;. Wikipedia is a key knowledge dissemination platform widely used by students&lt;ref&gt;{{Cite journal|last=Sunvy|first=Ahmed Shafkat|last2=Reza|first2=Raiyan Bin|date=2023-04-12|title=Students’ Perception of Wikipedia as an Academic Information Source|url=https://ejournal.undiksha.ac.id/index.php/IJERR/article/view/57572|journal=Indonesian Journal Of Educational Research and Review|volume=6|issue=1|pages=134–147|doi=10.23887/ijerr.v6i1.57572|issn=2621-8984}}&lt;/ref&gt; and scientists themselves, as shown by the fact that articles used as sources on Wikipedia are more cited in the literature&lt;ref&gt;{{Cite journal|last=Thompson|first=Neil|last2=Hanley|first2=Douglas|date=2017|title=Science Is Shaped by Wikipedia: Evidence from a Randomized Control Trial|url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3039505|journal=SSRN Electronic Journal|doi=10.2139/ssrn.3039505|issn=1556-5068}}&lt;/ref&gt; and that some scholars cite directly Wikipedia&lt;ref&gt;{{Cite journal|last=Dooley|first=Patricia L.|date=2010-07-07|title=Wikipedia and the two-faced professoriate|url=https://doi.org/10.1145/1832772.1832803|journal=Proceedings of the 6th International Symposium on Wikis and Open Collaboration|series=WikiSym '10|location=New York, NY, USA|publisher=Association for Computing Machinery|pages=1–2|doi=10.1145/1832772.1832803|isbn=978-1-4503-0056-8}}&lt;/ref&gt;. However, scientists do not naturally contribute to wikimedia projects as part of their work because of lack of incentives&lt;ref&gt;{{Cite journal|last=Chen|first=Yan|last2=Farzan|first2=Rosta|last3=Kraut|first3=Robert|last4=YeckehZaare|first4=Iman|last5=Zhang|first5=Ark Fangzhou|date=2024-05|title=Motivating Experts to Contribute to Digital Public Goods: A Personalized Field Experiment on Wikipedia|url=https://pubsonline.informs.org/doi/10.1287/mnsc.2023.4852|journal=Management Science|volume=70|issue=5|pages=3264–3280|doi=10.1287/mnsc.2023.4852|issn=0025-1909}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Kincaid|first=Dustin W.|last2=Beck|first2=Whitney S.|last3=Brandt|first3=Jessica E.|last4=Mars Brisbin|first4=Margaret|last5=Farrell|first5=Kaitlin J.|last6=Hondula|first6=Kelly L.|last7=Larson|first7=Erin I.|last8=Shogren|first8=Arial J.|date=2021|title=Wikipedia can help resolve information inequality in the aquatic sciences|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/lol2.10168|journal=Limnology and Oceanography Letters|language=en|volume=6|issue=1|pages=18–23|doi=10.1002/lol2.10168|issn=2378-2242}}&lt;/ref&gt;, but also other factors such as lack of time, lack of recognition and fit with scholarly workflow&lt;ref name=&quot;:10&quot;&gt;Taraborelli, D., Mietchen, D., Alevizou, P., &amp; Gill, A. (2011, August). Expert participation on Wikipedia: Barriers and opportunities. Wikimania 2011, Haifa, Israel. &lt;nowiki&gt;http://upload.wikimedia.org/wikipedia/commons/4/4f/Expert_Participation_Survey_-_Wikimania_2011.pdf&lt;/nowiki&gt; &lt;/ref&gt;. In addition, expert participation is not immune to the gender gap&lt;ref name=&quot;:10&quot; /&gt;. Because of gender segregation in disciplines&lt;ref&gt;{{Cite journal|last=Ceci|first=Stephen J.|last2=Ginther|first2=Donna K.|last3=Kahn|first3=Shulamit|last4=Williams|first4=Wendy M.|date=2014-12-01|title=Women in Academic Science: A Changing Landscape|url=https://doi.org/10.1177/1529100614541236|journal=Psychological Science in the Public Interest|language=EN|volume=15|issue=3|pages=75–141|doi=10.1177/1529100614541236|issn=1529-1006}}&lt;/ref&gt;, this may be detrimental to the content coverage on “female” topics&lt;ref&gt;{{Cite journal|last=Lam|first=Shyong (Tony) K.|last2=Uduwage|first2=Anuradha|last3=Dong|first3=Zhenhua|last4=Sen|first4=Shilad|last5=Musicant|first5=David R.|last6=Terveen|first6=Loren|last7=Riedl|first7=John|date=2011-10-03|title=WP:clubhouse?: an exploration of Wikipedia's gender imbalance|url=https://dl.acm.org/doi/10.1145/2038558.2038560|language=en|publisher=ACM|pages=1–10|doi=10.1145/2038558.2038560|isbn=978-1-4503-0909-7}}&lt;/ref&gt;, notably for social science in which women are more present. Our project proposes to improve expert contribution by making wikimedia projects (notably wikidata) useful tools that can facilitate research work, in addition to a key knowledge dissemination platform that is not country or institution-dependent. We propose to approach Wikimedia projects as a powerful (and free) knowledge management infrastructure that researchers could use. The Wikimedia ecosystem offers solutions that have strong potential to put open science principles into practices, including [[wikipedia:FAIR_data|FAIR]] principles and [[wikipedia:Linked_data#Linked_open_data|linked open data]].  

== Toward a living review on just sustainability transition ==

=== Just sustainability transition ===
Just sustainability transition transition is &quot;a fair and equitable process of moving towards a post-carbon society&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 

Developping living reviews seem particularly relevant for the just transition literature: first, modeling knowledge and building graphs allows to take into account the complexity of sustainability transitions which involve multiple levels of analysis&lt;ref name=&quot;:15&quot; /&gt;&lt;ref name=&quot;:16&quot; /&gt;&lt;ref name=&quot;:17&quot; /&gt; and fragmented results coming from various disciplines&lt;ref name=&quot;:20&quot;&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;. Then, making literature reviews &quot;living&quot; would allow researchers to be less subject to information overload through a more systematic accumulation of knowledge. Finally, conducting this review with an open science philosophy aswers the challenge of knowledge dissemination, which is crucial in a context of socio-ecological emergency when decision-makers need to rapidely access reliable information on possible sustainability transition trajectories.  

=== Living reviews ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. Literature review methods are currently evolving with new technological possibilities. Generative artificial intelligence such as ChatGPT are expected to have a strong influence on literature review activities&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref name=&quot;:12&quot; /&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but not yet integrated into tested and validated methodologies&lt;ref name=&quot;:13&quot;&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;. Human validation stays notably necessary&lt;ref&gt;{{Cite journal|last=Alshami|first=Ahmad|last2=Elsayed|first2=Moustafa|last3=Ali|first3=Eslam|last4=Eltoukhy|first4=Abdelrahman E. E.|last5=Zayed|first5=Tarek|date=2023-07-09|title=Harnessing the Power of ChatGPT for Automating Systematic Review Process: Methodology, Case Study, Limitations, and Future Directions|url=https://www.mdpi.com/2079-8954/11/7/351|journal=Systems|language=en|volume=11|issue=7|pages=351|doi=10.3390/systems11070351|issn=2079-8954}}&lt;/ref&gt;,&lt;ref name=&quot;:13&quot; /&gt;. While AI can appear as a solution for scaling literature reviews, we are in the present project exploring another possible scenario which is to use more crowdsourcing in the literature review process.

=== Wikimedia projects ===
Wikipedia is a successfull example of large-scaled crowdsourcing of reliable knowledge synthesis. That is why this project proposes to explore the potential of the Wikimedia ecosystem for conducting living reviews. Since Wikipedia does aim to host original research&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikipedia:No original research|url=https://en.wikipedia.org/w/index.php?title=Wikipedia:No_original_research&amp;oldid=1360514388|journal=Wikipedia|language=en}}&lt;/ref&gt;, we are working on two sister projects : Wikidata and Wikiversity. [[wikipedia:Wikidata|Wikidata]] is a &quot;collaboratively edited multilingual knowledge graph hosted by the Wikimedia Foundation&lt;ref&gt;{{Cite news|last=Chalabi|first=Mona|date=April 26, 2013|title=Welcome to Wikidata! Now what?|url=https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|access-date=October 2, 2021|archive-date=2 October 2021|archive-url=https://web.archive.org/web/20211002152920/https://www.theguardian.com/news/datablog/2013/apr/26/wikidata-launch|url-status=live}}&lt;/ref&gt;&quot;&lt;ref&gt;{{Cite journal|date=2026-06-21|title=Wikidata|url=https://en.wikipedia.org/w/index.php?title=Wikidata&amp;oldid=1360462340|journal=Wikipedia|language=en}}&lt;/ref&gt;. &quot;A  [[wikidata:Q33002955|knowledge graph]] is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt; Such graphs have a strong potential to conduct knowledge synthesis&lt;ref name=&quot;:11&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;&lt;ref name=&quot;:18&quot; /&gt;. They are especially usefull to build the ontologies (formal representations of concepts) that are necessary to organize and represent existing knowledge&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Spadaro|first=Giuliana|last2=Tiddi|first2=Ilaria|last3=Columbus|first3=Simon|last4=Jin|first4=Shuxian|last5=ten Teije|first5=Annette|last6=Balliet|first6=Daniel|date=2022-09-01|title=The Cooperation Databank: Machine-Readable Science Accelerates Research Synthesis|url=https://doi.org/10.1177/17456916211053319|journal=Perspectives on Psychological Science|language=EN|volume=17|issue=5|pages=1472–1489|doi=10.1177/17456916211053319|issn=1745-6916|pmc=9442633|pmid=35580271}}&lt;/ref&gt;. In complement to using Wikidata to model knowledge, we decided to use Wikiversity to report and write our research results. [[wikipedia:Wikiversity|Wikiversity]] is another Wikimedia project hosting pedagogical content, original research, and even a publishing house ([[WikiJournal|WikiJournals]])&lt;ref&gt;{{Cite journal|date=2026-06-09|title=Wikiversity|url=https://en.wikipedia.org/w/index.php?title=Wikiversity&amp;oldid=1358552930|journal=Wikipedia|language=en}}&lt;/ref&gt;. Like Wikipedia pages, Wikiversity pages are editable by everyone, have a discussion tab and a history log tab. 

Our research question is : '''How can Wikimedia projects contribute to building a collaborative living review on just sustainability transition ?'''

In this project, we aim to test 4 hypothesis : 

● '''Hypothesis 1:''' Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations.

● '''Hypothesis 2:''' Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (e.g. conceptual typologies, cause-effect chains…).

● '''Hypothesis 3:''' SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs.

● '''Hypothesis 4''': Wikimedia or Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links (following the ideal of linked open data).

We also have 2 assumptions :

● '''Assumption 1:''' Wikimedia projects have to be integrated into validated scientific protocols in order to be a valuable research tool.

● '''Assumption 2:''' Wikimedia project contribution has to be made interoperable with tools, methods and data types already used by researchers.
== Methodology ==
Our study rely on a meta-review, that is a review of existing literature reviews. Data presented in literature reviews are usually presented as tables or diagrams, and sometimes provided as supplementary materials in publications. However, these data are not made interoperable and are not used to update prior literature reviews. Our goal was to synthesize results of previous literature reviews by making their findings compatible with linked open data and open science standards using Wikidata, Wikiversity, and other open-science infrastructures. The first step was to build and enrich the bibliographic metadata of a corpus of articles we selected into Wikidata. The second step was to model the content  of these articles in Wikidata (e.g. typologies, causes-effects relationships...). The third step was to experiment relevant visualization of this content (e.g. causes-effects graphs). The las step was to write our report on a Wikiversity page, including links to our knowledge graph, following a linked open data philosophy.

== 1. Building an academic corpus and enriching bibliographic metadata ==
The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations). To do so we imported academic references into Wikidata, and explored the advantages of constituting a scholarly corpus on Wikidata in comparison (or in complementarity) to existing tools used by researchers such as reference management softwares and knowledge management softwares. Reference management software (Zenodo, Mendeley…) are used to collect scientific item metadata and integrate them into academic writing. They can also be used to analyze and annotate academic articles and can include export functions making the data interoperable with other analysis tools. Knowledge management software (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…) are used by some researchers to organize their ideas. To build and enrich our academic corpus on Wikidata, we searched existing databases, selected the sample of articles we wanted to study, imported these articles metadata into Wikidata, enriched these metadata and finally reflected on the advantages and limitations of Wikidata to build a rich academic corpus.

=== Database search ===
Doing a systematic review on all aspects of just transition would have resulted in too many articles to review. We thus decided to first explore one aspect of justice : procedural justice. Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt; (e.g. the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;). Procedural justice can include issues of community and citizen participation in decision making, their political representation, their consultation, or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities)&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For our search, we selected keywords related to procedural justice (procedural justice OR procedural fairness OR democracy OR participation OR participatory) and keywords related to sustainability transition (sustainability OR energy OR climate) AND (transition OR transitions).

We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article selection ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were : 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies/initiatives/cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors 
* Including justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper
The files with the lists of included and excluded articles are available on the archive plateform Zenodo : https://zenodo.org/records/20749974

=== Importing selected articles into Wikidata ===
Before importing the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in Wikidata. Next, we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through metadata enrichement ===
Metadatas are data describing other data. The metadata of academic items usually include title, author, publication outlet, publication date, pages, DOI, URL... and can be structured following specific standards (e.g. [[wikipedia:Dublin_Core|Dublin Core]]). In academic databases such as WOS or OpenAlex, the only metadata available regarding the content of an academic article are the abstract and sometimes keywords. However, researchers conducting literature reviews need more precise informations. An important part of literature review work can thus be about describing what the articles are about. For example, describing industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt) (e.g. &lt;ref name=&quot;:5&quot; /&gt;). 

By metadata enrichment, we mean completing metadata to include additional information about the content of an academic piece. In Wikidata, each type of information is added using a specific property. A property is the edge that links two entities in the Wikidata knowledge graph. We selected three Wikidata properties to describe the content of our selected articles : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe the geographical scope of the study. We also worked on adding {{Wikidata entity link|P50}}.

==== Adding {{Wikidata entity link|P921}} ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
!Description
|-
|[[d:Q42377797|Q42377797]]
|acceptability
|characteristic of a thing being subject to acceptance for some purpose
|-
|[[d:Q2798912|Q2798912]]
|accountability
|concept of responsibility in ethics, governance and decision-making
|-
|[[d:Q421953|Q421953]]
|actor–network theory
|theory within social science
|-
|[[d:Q84459973|Q84459973]]
|affordability
|
|-
|[[d:Q185836|Q185836]]
|age of a person
|time elapsed since a person was born
|-
|[[d:Q4764988|Q4764988]]
|animal studies
|field in which animals are studied in a variety of cross-disciplinary ways
|-
|[[d:Q4338318|Q4338318]]
|awareness
|state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns
|-
|[[d:Q4930066|Q4930066]]
|blue carbon
|carbon captured by the world's coastal ocean ecosystems
|-
|[[d:Q430460|Q430460]]
|capability approach
|economic theory
|-
|[[d:Q7569|Q7569]]
|child
|human between birth and puberty
|-
|[[d:Q4116870|Q4116870]]
|civic engagement
|individual or group activity addressing issues of public concern
|-
|[[d:Q125928|Q125928]]
|climate change
|human-caused changes to climate on Earth
|-
|[[d:Q260607|Q260607]]
|climate change 
adaptation
|process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities
|-
|[[d:Q1291678|Q1291678]]
|climate justice
|term linking the climate crisis with environmental and social justice
|-
|[[d:Q2270945|Q2270945]]
|co-creation
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q16972712|Q16972712]]
|co-design
|approach to design attempting to actively involve all stakeholders
|-
|[[d:Q16324410|Q16324410]]
|coproduction
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q11024|Q11024]]
|communication
|act of conveying intended meaning
|-
|[[d:Q177634|Q177634]]
|community
|social unit of human organisms who share common values
|-
|[[d:Q5154673|Q5154673]]
|community choice aggregation
|alternative energy supply system
|-
|[[d:Q113514984|Q113514984]]
|community energy
|delivery of community-led renewable energy, energy demand reduction and energy supply projects
|-
|[[d:Q65807646|Q65807646]]
|community participation
|The taking part by members of a community in decisionmaking processes related to the development of their community
|-
|[[d:Q188843|Q188843]]
|cosmopolitanism
|ideology that all human beings belong to a single community, based on a shared morality
|-
|[[d:Q11693783|Q11693783]]
|decarbonization
|change of economy, especially of energy industries, towards lower carbon dioxide emissions
|-
|[[d:Q284289|Q284289]]
|deliberative democracy
|form of democracy focusing on consensus
|-
|[[d:Q7174|Q7174]]
|democracy
|form of government
|-
|[[d:Q552284|Q552284]]
|distributive justice
|concept of the socially just allocation of goods
|-
|[[d:Q1230584|Q1230584]]
|diversity
|concept in sociology and political studies
|-
|[[d:Q1049066|Q1049066]]
|ecological economics
|research field on the interdependence of human economies and natural ecosystems
|-
|[[d:Q8134|Q8134]]
|economics
|social science that studies the production, distribution, and consumption of goods and services
|-
|[[d:Q868575|Q868575]]
|empowerment
|providing increased autonomy
|-
|[[d:Q295865|Q295865]]
|ecosystem service
|benefits created by nature, forests and environmental systems
|-
|[[d:Q138359220|Q138359220]]
|energy citizenship
|involvement of citizens in energy-related decisions
|-
|[[d:Q16869822|Q16869822]]
|energy consumption
|amount of energy or power used
|-
|[[d:Q1358789|Q1358789]]
|senior
|elderly person
|-
|[[d:Q14944319|Q14944319]]
|energy democracy
|concept in environmental justice movement
|-
|[[d:Q192704|Q192704]]
|energy efficiency
|ratio between the useful energy output and the input of a machine
|-
|[[d:Q24965464|Q24965464]]
|energy modeling
|process of building computer models of energy systems in order to analyze them
|-
|[[d:Q1805337|Q1805337]]
|energy policy
|policy addressing energy issues
|-
|[[d:Q1341244|Q1341244]]
|energy poverty
|lack of access to modern energy services
|-
|[[d:Q3406659|Q3406659]]
|energy production
|conversion of energy from a primary source into a form useful to humans
|-
|[[d:Q117091181|Q117091181]]
|energy justice
|subconcept of economic equality
|-
|[[d:Q3456219|Q3456219]]
|energy renovation
|building works aimed at reducing energy consumption and decarbonising the energy sources used
|-
|[[d:Q2700433|Q2700433]]
|energy security
|national security considerations of energy availability
|-
|[[d:Q837718|Q837718]]
|energy storage
|capture of energy produced at one time for use at a later time
|-
|[[d:Q795757|Q795757]]
|energy transition
|long-term structural change towards sustainable energy systems
|-
|[[d:Q1479527|Q1479527]]
|environmental justice
|system of fairness
|-
|[[d:Q771773|Q771773]]
|fairness
|concept in sociology and generally the interaction of society
|-
|[[d:Q56395513|Q56395513]]
|farming system
|method of agricultural production defined by its physical practices and economic characteristics
|-
|[[d:Q5465532|Q5465532]]
|food system
|all processes and infrastructure involved in feeding a population
|-
|[[d:Q4421|Q4421]]
|forest
|dense collection of trees covering a relatively large area
|-
|[[d:Q48277|Q48277]]
|gender
|social concept which distinguish the different gender categories
|-
|[[d:Q1553864|Q1553864]]
|governance
|all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society
|-
|[[d:Q8458|Q8458]]
|human rights
|inalienable fundamental rights to which a person is inherently entitled
|-
|[[d:Q11376059|Q11376059]]
|human rights violation
|act or omission which contravene the principles of human rights
|-
|[[d:Q103817|Q103817]]
|indigenous people
|first inhabitants of an area and their descendants
|-
|[[d:Q113561794|Q113561794]]
|indigenous science
|indigenous knowledge applied to the scientific method
|-
|[[d:Q770480|Q770480]]
|injustice
|quality relating to unfairness or undeserved outcomes
|-
|[[d:Q17142211|Q17142211]]
|interactional justice
|the perceived appropriateness of interpersonal treatment
|-
|[[d:Q1516555|Q1516555]]
|intersectionnality
|theoretical framework of multidimensional oppression
|-
|[[d:Q6316391|Q6316391]]
|just transition
|Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy.
|-
|[[d:Q366139|Q366139]]
|legitimation
|the process of making something acceptable and normative to a group
|-
|[[d:Q3027857|Q3027857]]
|living lab
|user-centered, open innovation ecosystem integrating research and innovation in real life communities
|-
|[[d:Q59679511|Q59679511]]
|low income
|home with little money
|-
|[[d:Q43619|Q43619]]
|natural environment
|all living and non-living things occurring naturally on Earth or some region thereof
|-
|[[d:Q127514833|Q127514833]]
|nature-positive
|global goal to halt and reverse nature loss by 2030
|-
|[[d:Q13023682|Q13023682]]
|non-human
|organism not in the genus Homo
|-
|[[d:Q728646|Q728646]]
|partnership
|arrangement in which parties agree to cooperate to advance their mutual interests
|-
|[[d:Q3907287|Q3907287]]
|policy making
|the act of developing policy
|-
|[[d:Q9357091|Q9357091]]
|political theory
|class of theory
|-
|[[d:Q265425|Q265425]]
|postcolonialism
|academic discipline
|-
|[[d:Q25107|Q25107]]
|power
|ability to influence the behavior of others
|-
|[[d:Q442100|Q442100]]
|procedural justice
|fairness in the processes that resolve disputes and allocate resources
|-
|[[d:Q7249406|Q7249406]]
|project governance
|management framework
|-
|[[d:Q7257735|Q7257735]]
|public engagement
|Policy-making practice
|-
|[[d:Q541936|Q541936]]
|public participation
|participation of citizens in various policy decisions and planning processes
|-
|[[d:Q6142016|Q6142016]]
|recognition justice
|social philosophy theory
|-
|[[d:Q10509953|Q10509953]]
|renewable electricity
|electricity from renweable sources
|-
|[[d:Q12705|Q12705]]
|renewable energy
|energy collected from renewable resources
|-
|[[d:Q56510941|Q56510941]]
|renewable energy policy
|
|-
|[[d:Q1165392|Q1165392]]
|restorative justice
|approach to justice where victims and perpetrators mediate a restitution agreement
|-
|[[d:Q4414036|Q4414036]]
|rural population
|inhabitants of rural areas or of small towns classified as rural
|-
|[[d:Q17152351|Q17152351]]
|smart system
|adaptive intelligent systems
|-
|[[d:Q187588|Q187588]]
|social class
|group of people categorized in a hierarchy based on socioeconomic factors
|-
|[[d:Q264892|Q264892]]
|social justice
|concept that discrimination recognized in society should be remedied
|-
|[[d:Q34749|Q34749]]
|social science
|academic disciplines concerned with society and the relationships between individuals in society
|-
|[[d:Q2930198|Q2930198]]
|stakeholder participation
|involvement of groups or individuals affected by the actions of an entity
|-
|[[d:Q125359881|Q125359881]]
|sustainability transition
|
|-
|[[d:Q219416|Q219416]]
|sustainability
|ability of human civilization to coexist with the biosphere in a steady state
|-
|[[d:Q131201|Q131201]]
|sustainable development 
|mode of human development that meets current demands without compromising the needs of future generations
|-
|[[d:Q7649586|Q7649586]]
|Sustainable Development Goals
|set of United Nations-defined global development goals and climate change
|-
|[[d:Q69883|Q69883]]
|urban planning
|technical and political process concerned with the use of land and design of the urban environment
|-
|[[d:Q920600|Q920600]]
|urban renewal
|program of land redevelopment in cities, often where there is urban decay
|-
|[[d:Q3376054|Q3376054]]
|vulnerable population
|group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent
|-
|[[d:Q107389921|Q107389921]]
|water-management
|
|-
|[[d:Q7981051|Q7981051]]
|well-being
|measure of how well life is to someone or a group with factors such as health, happiness and satisfaction
|-
|[[d:Q467|Q467]]
|woman
|female adult human
|-
|[[d:Q188867|Q188867]]
|future studies
|study of possible, probable, and preferable social, technological and political futures
|-
|[[d:Q1038171|Q1038171]]
|participatory design
|active involvement of all stakeholders in the design process
|}Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. 

==== Adding {{Wikidata entity link|P8363}} ====
The sample of article we selected included only litterature reviews, but we wanted to describ what kind of literature review it was. We first read abstracts to identify all the different types of litterature reviews present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved some of these method items using the methodological references cited in the reviewed papers. For example, we added a statement saying that {{Wikidata entity link|Q101116078}} can have {{Wikidata entity link|Q653137}} as {{Wikidata entity link|P13391}}&lt;ref&gt;{{Cite journal|last=Paré|first=Guy|last2=Trudel|first2=Marie-Claude|last3=Jaana|first3=Mirou|last4=Kitsiou|first4=Spyros|date=2015-03|title=Synthesizing information systems knowledge: A typology of literature reviews|url=https://linkinghub.elsevier.com/retrieve/pii/S0378720614001116|journal=Information &amp; Management|language=en|volume=52|issue=2|pages=183–199|doi=10.1016/j.im.2014.08.008}}&lt;/ref&gt;. After this step, the {{Wikidata entity link|P279}} of {{Wikidata entity link|Q2412849}} in Wikidata were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Study type
!Description
|-
|[[d:Q603441|Q603441]]
|bibliometrics
|statistical analysis of written publications, such as books or articles
|-
|[[d:Q472342|Q472342]]
|scientometrics
|study of measuring and analysing science, technology and innovation
|-
|[[d:Q815382|Q815382]]
|meta-analysis
|statistical method that summarizes data from multiple sources
|-
|[[d:Q1504425|Q1504425]]
|systematic review
|publication type, study that gathers, analyzes, and communicates the results of research and information on a topic
|-
|[[d:Q2412849|Q2412849]]
|literature review
|process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic
|-
|[[d:Q6822263|Q6822263]]
|meta-regression
|statistical tool used in meta-analyses
|-
|[[d:Q7301211|Q7301211]]
|realist evaluation
|theory-driven evaluation used in evaluating social programmes
|-
|[[d:Q17007303|Q17007303]]
|combinatorial meta-analysis 
|study of the statistical properties of combinations of studies from a meta-analytic dataset
|-
|[[d:Q70470634|Q70470634]]
|network meta-analysis
|meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions
|-
|[[d:Q101116078|Q101116078]]
|scoping review
|search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry
|-
|[[d:Q110665014|Q110665014]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|-
|[[d:Q137174203|Q137174203]]
|conceptual review
|academic research aiming to review existing concepts and definitions in the litterature
|-
|[[d:Q137174450|Q137174450]]
|critical review
|type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research
|-
|[[d:Q137209848|Q137209848]]
|integrative literature review
|type of literature review
|-
|[[d:Q110665014|Q137211242]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|}For each article, we added the {{Wikidata entity link|P8363}} based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Adding {{Wikidata entity link|P6153}} ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

==== Adding {{Wikidata entity link|P50}} ====
When scholarly metadata are imported into Wikidata, the name of authors are stored as a chain of characters and linked to the property {{Wikidata entity link|P2093}}. The property {{Wikidata entity link|P50}} allows to make a link with a Wikidata item representing the author. This avoids the problem of homonym authors by linking a unique identifyer to authors in Wikidata and linking this identifier to existing ones such as ORCID. We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers by categorizing scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages.

==== Adding {{Wikidata entity link|P6977}} ====
We explored the possibility to model review networks, that is linking review paper to reviewed papers (and not broadly cited papers). We selected one of the paper which had the most detailed list of reviewed papers ({{Wikidata entity link|Q114306483}}), then we added this list in Wikidata using Open refine. Then we snowballed this process for the reviewed papers which were also litterature reviews.

=== Advantages and limitations of Wikidata to build a rich living academic corpus ===
To share the result of our work, we exported the dataset we build on Wikidata and shared it on the open archive Zenodo : https://doi.org/10.5281/zenodo.20749973. The data is also available directly in Wikidata.

The goal of this step was to test '''Hypothesis 1''' (Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations)'''.''' The sections below discuss the advantages and limitations of wikidata regarding this hypothesis.

==== Advantages of Wikidata ====
Key advantages of Wikidata are its flexible and collaborative nature as well as its interoperability. Wikidata ontology (that is how the data are structured) is collaboratively defined and properties can be added if relevant (after validation by the community). Compared to global databases like WOS or OpenAlex, Wikidata allows anyone to enter more metadata about each academic articles. Another notable advantage is that Wikidata items can be used as an interoperable [[wikipedia:Controlled_vocabulary|controlled vocabulary]]. For example, when we stated that the {{Wikidata entity link|P921}} of the article {{Wikidata entity link|Q114306483}} was {{Wikidata entity link|Q795757}}, &quot;energy transition&quot; was not just a word but a concept with its unique identifyer, linked to identifiers in other databases such as the Google Knowledge Graph ID or BNCF Thesaurus ID. Wikidata's collaborative nature is here adain an advantage. Contrary to institutional thesaurus, Wikidata allows anyone to add new concepts. This is particularly interesting as existing controlled vocabularies rarely reflect the degree of precision that researchers need in their work. The multilingual nature of Wikidata was also a strengh, some Wikidata contributors added labels for the concepts we used into different languages (For example, contributors added labels for {{Wikidata entity link|Q14944319}} in Armenian and Slovenian, languages we do not speak at all). 

==== Limitations of Wikidata ====
The limitations of Wikidata is that it is not yet well integrated with the tools researchers use to do literature reviews (kowledge management softwares and bibliographic databases).    

Compared to reference management softwares (Zenodo, Mendeley…) and knowledge management softwares (Obsidian, Zettlr, Room Research, Notion, Logseq, Reflect…), Wikidata is too general and does not allow to work on full texts. References and knowledge management softwares allow researcher to build their own specialised knowledge base, by taking notes and highlighting the content of the full texts. Wikidata is not connected to this process and there is a missing tool to facilitate the construction of graphs from the qualitative analysis of texts. In addition, when one is working on a specific corpus of item in Wikidata, it is also difficult to keep track of this corpus. We linked each academic item we were working on to our research project by adding a statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}, but it was still relatively difficult to &quot;filter&quot; the part of the knowledge graph we were working on.   

Compared to bilbiographic catalogues (OpenAlex, Web Of Science, GoTriple...), Wikidata will never be as exhaustive and do not offer user-friendly search functions. Since 2014, an important amount of bibliographic data was imported in Wikidata with the project [[d:Wikidata:WikiCite|Wikicite]]. At the time of its creation, Wikicite was adressing the issue of closed bibliographic data and was trying to make these data open, many academic items were imported automatically in Wikidata through scraping. This practice was abandoned because the large amont of bibliographic data congested queries on Wikidata (this led to the decision to split the Wikidata graph between academic and non academic entities), and because new open science initiatives, notably OpenAlex (2022), are now taking on the task of creating a exhaustive catalogues of all scholarly production. In this context, the community has to rethink the purpose of Wikidata regarding bilbiographic data.   

==== Future possbilities ====
A solution to the limitations would be to developp the role of Wikidata as a link between other tools of the open science ecosystem. For example, developping and maintaining plugins or extensions for specialised softwares like Zotero, Wikibase, and Omeka could connect Wikidata with more specialised graphs. Such extensions could help building local graphs by allowing the reuse of wikidata item (eg. autocompletion), but also help contributing to Wikidata thanks to export features. (Example :&lt;ref&gt;{{Cite journal|last=Nielsen|first=Finn Årup|last2=Lyhne|first2=Ivar|last3=Garigliotti|first3=Dario|last4=Butzbach|first4=Annika|last5=Ravn Boess|first5=Emilia|last6=Hose|first6=Katja|last7=Kørnøv|first7=Lone|date=2023|title=Environmental impact assessment reports in Wikidata and a Wikibase|url=https://repositum.tuwien.at/handle/20.500.12708/193492|language=en|publisher=CEUR-WS.org|volume=3443|pages=1–8|doi=10.34726/5421}}&lt;/ref&gt;)  Wikidata could then be an intermediary between locally curated corpus and more exhaustive bilbiographic catalogues such as OpenAlex. For example, Wikidata items could be used to tag articles in a more precise way instead of using keywords and sharing enriched corpus in Wikidata could help to train more precise taging algorythms.

== 2.Modelling the content of litterature reviews ==
The goal of this step was to test '''Hypothesis 2''' (Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference) by modelling the content of our selected articles into Wikidata. [[wikipedia:Knowledge_modeling|Knowledge modelling]] is the process of making a machine readable model of knowledge. As we have a background in social sciences, we felt the need to question the relationship between this process and other methodologies such as concept mapping, thematic networks and causal networks. We present these methodologies before describing our current knowledge modelling experimentations.

=== Concept mapping, thematic networks and causal networks ===
This section presents social science methodology that presents similarities with knowledge modelling.

==== Concept maps ====
[[File:Conceptual_Diagram_-_Example.svg|link=https://en.wikipedia.org/wiki/File:Conceptual_Diagram_-_Example.svg|thumb|Example conceptual diagram|251x251px]]Concept maps are ''concepts'' (boxes) and ''propositions'' (arrow indicating the relationship between two boxes)&lt;ref name=&quot;:19&quot;&gt;Cañas, Alberto J., et al. &quot;CmapTools: A knowledge modeling and sharing environment.&quot; (2004): 125-135. https://thomaseskridge.com/assets/pdf/Canas-2004.pdf&lt;/ref&gt;. Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. They can be built using specialised softwares (e.g. [https://cmap.ihmc.us/ Cmap]&lt;ref name=&quot;:19&quot; /&gt;, Altas.Ti &quot;network&quot; feature...). The &quot;box and arrow&quot; logic is similar to how knowledge is modelled on Wikidata : the equivalent of concepts is ''item'' and the equivalent of propositions are ''statements''. The difference between a softwares like Cmap and Wikidata is the underlying format of the data.  
  
==== Thematic networks ====
[[File:Thematic network example.jpg|thumb|447x447px|Structure of a thematic network (Source: based on Attride-Stirling 2001)]]
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified.   
[[File:Adoption_CLD.svg|link=https://en.wikipedia.org/wiki/File:Adoption_CLD.svg|thumb|421x421px|Causal loop diagram of ''Adoption'' model, used to demonstrate systems dynamics]]

==== Causal diagrams ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows.&quot;&lt;ref name=&quot;:21&quot;&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; After identifying themes, qualitative researchers are expected to theorize the &quot;arrows&quot; between themes&lt;ref name=&quot;:21&quot; /&gt;R.esearchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;. 

=== Knowledge modelling in Wikidata ===
This section presents our knowledge modelling experimentation in Wikidata.

==== Conceptual modelling ====
Our first step was to reflect on what is a &quot;concept&quot; and what kind of wikidata properties could be used to model concepts in Wikidata. Scholars in management have called for more rigorous ways to define concepts&lt;ref name=&quot;:22&quot; /&gt; and modelling concepts in Wikidata could help to build less ambiguous concepts. Concept definition encompass various aspects such as the nature of the phenomenon, its characteristics, the links with prototypical cases or examples, the contrast with other concepts, the links with causes and consequences...&lt;ref name=&quot;:22&quot;&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;, and scholars have advised to take insight from philosophy to work on concepts&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;. We thus read work in cognitive science which was summarizing psychology and philsosophy approaches on the determination of the content of concepts&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt;. We summarize these approaches below and examine which wikidata properties exist to represent them.
*Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
*Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. (The closer a phenomenon is to the prototype, the more likely it belong to the category). Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}} (see discussion here https://www.wikidata.org/wiki/Help:Basic_membership_properties).
*Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}, {{P|1478}}, {{P|P9353}} (see discussions here : https://www.wikidata.org/wiki/Help:Modeling_causes/en).
*Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can use it before really knowing what its properties will be.
*Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

==== Testing concept modelling on {{Wikidata entity link|Q14944319}} ====
To test concept modelling, we started by experimenting with the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic :

*{{Wikidata entity link|Q137901202}}
*{{Wikidata entity link|Q137901196}}
*{{Wikidata entity link|Q137901182}}
*{{Wikidata entity link|Q136447761}}
*{{Wikidata entity link|Q129652515}}
*{{Wikidata entity link|Q114306483}}

We read each paper and used them as source in manually entered statements in the item {{Wikidata entity link|Q14944319}}. For example, Droubi et. Al stated &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}} and {{Wikidata entity link|Q3249551}}, using the paper as reference (see screenshot below). The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&amp;oldid=2495982191.

[[File:Wikidata statement- energy democracy is an instance of ideal.png|915x915px|border]]

We listed the difficulties encountered as we worked and we also asked the Wikidata community to give us feedback on our modelling on the item discussion page (https://www.wikidata.org/wiki/Talk:Q14944319). These issues were related to contradictions, precision, concision and ontology. We discuss each issue and draft recommandations to refine our modelling process.  

===== Contradictions =====
Wikidata contributor's feedback highlighted some apparent contradictions (The values in &quot;does not have effect&quot; seems contrary to what is listed in &quot;has goal&quot;.) We would however argue this is not a problem because &quot;statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata&quot;&lt;ref&gt;{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}&lt;/ref&gt;. Wikidata essentially supports epistemic pluralism : different worldviews can be represented in wikidata&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;.Besides, in the case of goals versus effects statements, it is not contradictory because one can have a goal and fail to achieve it. In the case of energy democracy, the discrepancy between the stated goals of this movement and what it actually achieves is precisely what some authors are critiquing&lt;ref name=&quot;:20&quot; /&gt;.  

Recommandations : Contradictions are allowed in Wikidata. 

===== Precision =====
We noted that conceptual modelling requires an important degree of formalization and precision. This is a key advantage of Wikidata to be able to create links toward precise concepts which have their own identifiers. For example, we were able to create statements about specific laws and their unique identifiers in legal databases (e.g. {{Wikidata entity link|Q139764294}} and its identifier in the EUR-Lex database). However, the sources we are working with are not always precise enough and when concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, {{Wikidata entity link|Q12888920}}... We can see here that academic texts are using natural language and thus are using ambiguous terms. As a result, we received feedbacks regarding a lack of precision in our statements (too many and too vague statements). In addition to the ambiguity of sources, a reason why we ended up with very general statements is because we avoided the creation of new Wikidata items. While following this implicit rule allowed us to focus on the most notable concepts, creating new items could also help make the statements more precise. 

Recommandations : If a concept is ambiguous it should not be included in the modelled statements. Create more precise relevant concepts if they do not exist in Wikidata.  [[File:Wikidata visualisation screenshot of subclasses relationships including the item political concept.png|thumb|298x298px|Subclass relationships between &quot;concept&quot; and &quot;political concept&quot;.]]
===== Concision =====
Wikidata contributor's feedback indicated a lack of concision. Some of it coming from the fact that some values were &quot;in the tree of another value&quot;. The rule we take from this feeback is a need of logical simplification. Two examples illustrate possible logical simplification :
*We stated that {{Wikidata entity link|Q14944319}} was an {{Wikidata entity link|P31}} {{Wikidata entity link|Q33104069}} and an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}. But in that case, it is not necessary to state that it is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q151885}}, because {{Wikidata entity link|Q33104069}} is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q131362181}}, which is a {{Wikidata entity link|P279}} {{Wikidata entity link|Q151885}} (see diagram on the right). Here, we have to keep only the more precise item. This reasonning is based on the assumption that {{Wikidata entity link|P279}} is transitive. It seems this reasonning could be generalized (we opened a discussion about this here https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning/Use_cases#Parcimonious_statement_constraints_based_on_subclass_of_(P279)_and_part_of_(P361)_transitivity&lt;nowiki/&gt;) The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize this kind of relationship (using &quot;instance of&quot; as transversal property and checking &quot;instance of or subclass of&quot;).

*We stated that {{Wikidata entity link|Q14944319}} {{Wikidata entity link|P2670}} {{Wikidata entity link|Q15991216}} and {{Wikidata entity link|Q113514984}}. But if we consider that {{Wikidata entity link|Q15991216}} is a {{Wikidata entity link|P279}} of {{Wikidata entity link|Q113514984}}, then the inclusion of {{Wikidata entity link|Q15991216}} is implied. Here we could keep only the broader item {{Wikidata entity link|Q113514984}}. But unlike the reasonning above, we could not identify a way to generalize this reasonning. For example, {{Wikidata entity link|Q113514984}} can be a subclass of {{Wikidata entity link|Q43229}}, but using the former item is more informative than using the later. For the property {{Wikidata entity link|P2670}}, using a value that is too precise results in too many statements and using a value that is too general results in too trivial statements. Choosing the degree of precision is a problem similar to choosing whether or not to create a new item.
Recommandations : When possible, use logical simplification to make statements more parcimonious. The [https://angryloki.github.io/wikidata-graph-builder/ Wikidata graph builder] is usefull to visualize redundancies. 

===== Quantification =====
Modelling quantitative statements was challenging. We mostly skipped those but made an attempt for one case : the paper {{Wikidata entity link|Q137901196}} states that &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;&lt;ref&gt;{{Cite journal|last=Vanegas-Cantarero|first=María M.|date=2020-12-01|title=Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries|url=https://www.wikidata.org/wiki/Q137901196|journal=Energy Research &amp; Social Science|language=English|volume=70|doi=10.1016/J.ERSS.2020.101716}}&lt;/ref&gt;. Modelling this sentence by adding a statement in the item {{Wikidata entity link|Q177323}} would require creating a specific property for &quot;final energy consumption&quot;. There are such properties in Wikidata : for example, for renewable energy subsidies, there is {{Wikidata entity link|P6826}}. This type of property is notably useful to display information about an item  (e.g. the population of a town) on Wikipedia through dynamic infoboxes that are updated with Wikidata information. But the current informal rule on Wikidata property creation is to prefer the use of general properties and avoid the creation of specific properties. Therefore creating properties for every quantifiable characteristic may not be possible. An alternative way to model quantity is to create a specific item to represent what is quantified and use other properties to represent quantities (see example below). 

[[File:Wikidata screenshot of percentage modelling.png|border|783x783px]]

Despite this possibility, since quantitative data are usually stored elsewhere in more convenient formats, it may not be always relevant to include quantitative data into Wikidata. Instead, it might be more useful to link the item that is quantified to an external dataset about it. This can be done using the property {{Wikidata entity link|P1325}}. 

Recommandation : Use existing properties for the quantity if it exists. If it does not exist, create an item representing what is measured, link it toward an external datasets and add quantitative statements if relevant.

===== Ontology =====
Ultimately, several of our difficulties were linked to some ontology challenges reflecting the complexity of the concept we were working on. According to the litterature, {{Wikidata entity link|Q14944319}} refer to multiple types of entities. It represents a concept, an ideal, a process and an outcome. We reflected this with multiple {{Wikidata entity link|P31}} statements, but encountered struggles to model differences between ''process versus outcomes'', ''ideal versus reality'', ''phenomenon versus theory'' and ''discourses versus practices''. The wikidata community recommands having different items for ''process and outcome'' (criterion &quot;{{Wikidata entity link|Q127270577}}&quot;). For example, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork). However, this distinction is less straightforward for social processes which are are ongoing and evolving without a clear ending. Another problem was to separate ''ideal versus reality,'' we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes attempting to reach it. It was also difficult to separate ''phenomenon versus theory,'' for example, we did not manage to model the idea that the literature on energy democracy is fragmented. Finally, we also would have needed more distinctions between ''discourses and practices.'' Our sources suggested that energy democracy discourses and practices may have different causes, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movments promoting it, but this was not a very precise way to model this idea.

Our first attempt to multiple &quot;natures&quot; of energy democracy into a single item is not satisfactory, and implementing the modelling recommandations we identified above may not be sufficient to build more parcimonious statements. Creating new items to reflect the different aspects of energy democracy may here be necessary. To do so we have to decide how to split the different natures {{Wikidata entity link|Q14944319}} and what will be the relationship between the resulting items. Here the definition of an {{Wikidata entity link|Q324254}} (formal representation) in Wikidata can quickly escalate into questionning {{Wikidata entity link|Q44325}} (metaphysical reflexion on the nature of things), and especially {{Wikidata entity link|Q1713511}} (the nature of the social world). Indeed, the relations between the different types of entities constituting the social worls are not simple nor consensual among philosophers of social science.  

For example, {{Wikidata entity link|Q15080858}} posits that different things have different ways of being (modes of reality). They propose to classify entities in four categories : material entities (that can exist independently of humans), conceptual entities (concepts, discourses, ideas, meaning…), artefactual entities (human-made and combining conceptual and material elements) and social entities (that depends on human activity to exist)&lt;ref&gt;Fleetwood, S. (2004). An ontology for organisation and management studies. ''Critical Realist Applications in Organisation and Management Studies'', 27–53.&lt;/ref&gt;. A complex concept like {{Wikidata entity link|Q14944319}} is concerned with all these types of entities. The energy system include many material entities such as oil fields, the sun, seas, trees... and artefacts such as energy production unit, power lines, home appliances, trucks... (Including conceptual entities such as the name of these artefacts or the knowledge to make them function.). There are the social entities in which they are encompassed (the enregy sectors, energy businesses, energy policies...). There are conceptual entities like normative/political discourses discussing how these artefact and social system should work and there are conceptual entities in the academic sphere building theories about how all this works or should work. 

In theory, such ontology can be implemented in Wikidata as it &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot; /&gt; Current Wikidata ontology is structured with dischotomies such as {{Wikidata entity link|Q7048977}}/{{Wikidata entity link|Q4406616}}, {{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}, {{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q23958946}}, {{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}, {{Wikidata entity link|Q30241068}}/{{Wikidata entity link|Q3799040}}... A quick search seem to show that entities related to the social reality can quickly fall into heterogeneous and sometime contradictory root classes (see table below). 
{| class=&quot;wikitable&quot;
|+
!Item
!Existing root class entity
|-
|{{Wikidata entity link|Q190539}}
|{{Wikidata entity link|Q7048977}}

{{Wikidata entity link|Q30241068}}

{{Wikidata entity link|Q67518978}}
|-
|{{Wikidata entity link|Q34770}}
|{{Wikidata entity link|Q7048977}}
|-
|{{Wikidata entity link|Q9081}}
|{{Wikidata entity link|Q7048977}}
|-
|{{Wikidata entity link|Q43229}}
|{{Wikidata entity link|Q99527517}}{{Wikidata entity link|Q30241068}}

{{Wikidata entity link|Q103940464}}
|-
|{{Wikidata entity link|Q49773}}
|{{Wikidata entity link|Q67518978}}/{{Wikidata entity link|Q103940464}}

{{Wikidata entity link|Q99527517}}

{{Wikidata entity link|Q30241068}}

{{Wikidata entity link|Q7048977}}
|-
|{{Wikidata entity link|Q12705}}
|{{Wikidata entity link|Q4406616}}/{{Wikidata entity link|Q7048977}}

{{Wikidata entity link|Q1970309}}/{{Wikidata entity link|Q16686448}}
{{Wikidata entity link|Q99527517}}/{{Wikidata entity link|Q132907471}}
|}

=== Future steps ===
Current {{Wikidata entity link|Q3882785}} in Wikidata do not seem very informative to classify the aspects of our concept in a top-down manner. Our next steps will thus be to :  

* Export the existing model of the concept to keep track of each modelling steps (files and graphs screenshots)

*Explore if existing standards offer more relevant classes to categorize social entities (we opened the discussion about social ontology with the wikidata community here https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Ontology#Social_ontology).
**CRMSoc : https://cidoc-crm.org/crmsoc/fm_releases
**Criteria for choosing an ontology = choose something that is used and maintained by a community (colleague feedback)
*Create new items to reflect the different facets of {{Wikidata entity link|Q14944319}} : the social movement advocating for it, the political concept theorizing it, the concrete initiatives implementing it, the litterature theorizing it, etc... based on a new reading of our sample of article and aligned with existing ontologies.
*Compare grounded theory/CAQDAS with knowledge modelling methods, test annotation softwares : https://inception-project.github.io/ ; https://pacte.crim.ca/index_en.html
*Read articles again and model more precise statements, including quotation of the original text in the reference field.
*Write a more precise research question to orient modelling : What are the main concepts of just transition, their definition and their relationships ?

== 3. Data visualisation ==
The goal of this step is to test '''Hypothesis 3''' (SPARQL-based queries and visualizations can be used to navigate  scientific corpora and scientific knowledge graphs). At this stage we started to identify visualization use cases we would like to test, they are listed below.

=== Filter statements ===

*Visualize only statements using a specitic source in order to map the content of a single academic article. First test here: https://w.wiki/PFqH
*Visualize only items which are part of the present project (all the academic items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}).

=== Properties visualisation ===

*Visualise the &quot;tree&quot; of a property used in an item : Wikidata graph builder seem to be the most user friendly, robust and versatile tool to visualise a graph of a single property (https://angryloki.github.io)

=== Concepts visualisation ===

*Map all statements related to a single item (e.g. [[d:Wikidata:Scholia|Scholia]] request &quot;topic in context&quot;)
*Map the statements that two items have in common to make comparisons.

=== Mapping sources consensus ===

*Visualise graphs and use the number of references to determine edge thickness/weight in order to make consensual statements more visible.

== 4. Writing ==
The goal of this step is to test '''Hypothesis 4''' (Wikiversity pages can be used to write literature reviews collaboratively in text format augmented by interwiki links). 

=== Advantages of Wikiversity ===
Writing on a Wikiversity page offers some advantages to implement the principles of open linked data in text format. We could cite academic items using their Wikidata QID to generate the citations below (but if we use an URL to cite a paper, it does not automatically check if there is an existing QID for it), and also link toward Wikidata entities using a template ([[Template:Wikidata entity link|Wikidata entity link]]). Although we did not translate the present page for now, Wikiversity offers the possibility to translate a page in multiple languages (an interesting possibility in the context of the The FAIRisation of scholarly communication&lt;ref&gt;{{Cite journal|last=Maryl|first=Maciej|last2=Blaszczyńska|first2=Marta|last3=Zalotyńska|first3=Agnieszka|last4=Taylor|first4=Laurence|last5=Avanço|first5=Karla|last6=Balula|first6=Ana|last7=Buchner|first7=Anna|last8=Caliman|first8=Lorena|last9=Clivaz|first9=Claire|date=2021-01|title=Future of Scholarly Communication|url=https://hal.science/hal-03277615}}&lt;/ref&gt;). The possibility to view the page history provide an exhaustive versionning of a paper, which constitute interesting data documenting the academic writing process (at first we did not comment the nature of our modifications but after thinking about this reuse possibility, we described it with more details). The contribution statistics based on the history also offers a new way to track author's contributions to a paper in a transparent way.

=== The issue of text interoperability ===
A key issue we are encountering is the question of the interoperability of texts. While the interoperability of data is starting to be well discussed in the open science community, the interoperability of texts do not seem to benefit from the same level of discussion. We encountered several interoperability issues regarding our writing. First, copying texts written on a word processor software (e.g. microsoft word) into a wiki page (or the other way around) is relatively seamless in terms of formatting, but the management of references is lacking. Reformatting references is very time consuming and a real barrier for text interoperability in academic context : it is difficult to copy text from an academic publication into a wiki text, and difficult to turn a wiki text into a publication (the only possibility is to export the page as a PDF and you may be reading this article in this format).   

There are also uncertaineties regarding how to reuse texts published under creative common licences. Academic texts published under CC-BY-SA licences can in theory be remixed and reused. But academia does not have established practices regarding how this can be done. If we want to reuse a whole page, should we put it in quotation marks and simply cite the paper ? Should the original authors be listed as co-authors ? To what extent academic publisher will accept to publish &quot;remixed&quot; texts while they usually require that publications contain mainly unpublished content ? The norms of what is appropriate remix and reuse practices in academia has yet to be decided... and this could be an interesting conversation to have in the open science community.  

== Discussion ==
Our goal was to assess the potential of Wikidata and Wikiversity to build a living literature review method and tackle issues of information overload, knowledge synthesis and knowledge dissemination, following open science principles. By conducting a meta-review on just sustainability transition (that is a review of existing literature reviews), and experimenting with existing technical solutions, we were able to identify the strenghs and limitations of the Wikidata ecosystem. First Wikidata can be used to enrich scientific item metadata and build living scientific corpora with rich annotations (Hypothesis 1 is verified). However, the technical barriers to do so are still high and the Wikidata database is too general to allow a community to work on a specific curated corpus of scientific items. Secondly, Wikidata can be used for scientific knowledge modeling through statements using scientific items as reference (Hypothesis 2 is verified). However, this requires developping new methodological standards regarding what is a rigorous modelling in social science, as well as building specific {{Wikidata entity link|Q324254}}. An unexpected finding is that the formalization necessary for knowledge modelling in wikidata invites to formalize the reasonning behind modelling choices, a process that is rarely done in qualitative analysis even though such reasonning are likely present. The modelling process also responds to calls in management sciences to engage in more systematic categorizing to avoid semantic confusion&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;. Thirdly, SPARQL-based queries can be used to vizualised scientific corpora and scientific knowledge graphs, but we cannot completely say they allow users to &quot;navigate&quot; it (Hypothesis 3 is partially verified). SPARQL-based queries are powerful but they require technical knowledge, especially now that the split between academic and non-academic items in Wikidata requires to write federated queries. To be able to &quot;navigate&quot; scientific corpora and scientific knowledge graphs on Wikidata, more user-friendly tools would be needed. Finally, we did see advantaged in using Wikiversity pages to write collaboratively in text format augmented by hypertext links, but there are still important technical and instittional barriers (Hypothesis 3 is partially verified) : the interoperability of text is a key issue to reuse publications. The main issues we encountered were the management of references and uncertainty regarding the compatibilities between writing publicly on a wiki page and engaging in classical publication processes.

A potential solution to the issues encountered could be to develop a specialised literature review software that would allow researchers to build living literature reviews including knowledge graphs and wiki pages. This kind of tool should be user-friendly and include the missing tools we identified. In addition, it could include more advanced analysis functions such as logical reasonning based on the knowledge graph&lt;ref name=&quot;:9&quot; /&gt;. 

=== Future research ===
- Can we &quot;reason&quot; with Wikidata  (See: https://www.wikidata.org/wiki/Wikidata:WikiProject_Reasoning) : neurosymbolic AI ?

== Funding ==
This project is funded by the [[m:Grants:Programs/Wikimedia_Research_&amp;_Technology_Fund/Wikimedia_Research_Fund|Wikimedia Research Fund]], Grant ID: G-RS-2504-18935. The text of the initial research proposal is available here : https://doi.org/10.5281/zenodo.20760603.

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
| [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape
|-
| [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models
|-
| [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review
|-
| [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions
|-
| [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions
|-
| [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation
|-
| [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings
|-
| [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda
|-
| [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review
|-
| [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework
|-
| [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende
|-
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== References ==
{{References}}</text>
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="86552" sha1="6bjhg6a6ux2dcvnso9s6bn0x20a8iys" 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

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

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

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

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

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

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

== The 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}+r_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;-r_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;-r_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;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 great circle 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 the &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 six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 left 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 left 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 helix is an 8-rung ladder twisted 3 times, 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, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in 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 left 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° chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

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 central 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.

[[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 a single &lt;math&gt;r_{1}&lt;/math&gt; edge 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 skew&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 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 skew 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 every 180° of isoclinic rotation each vertex circles a skew 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 skew 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 two twisted parallel strands that intersects each 24-cell vertex once. In every 360° of isoclinic rotation each vertex circles a skew great square 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 triangles 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.  

...  

{{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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      <id>2818174</id>
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        <username>Dc.samizdat</username>
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      <text bytes="86543" sha1="ptg98w7mc0l1kaxnvh8ilz5v9d1zc2t" 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

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

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

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

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

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

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

== The 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}+r_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;-r_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;-r_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;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 the &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 six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 left 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 left 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 helix is an 8-rung ladder twisted 3 times, 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, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in 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 left 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° chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

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 central 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.

[[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 a single &lt;math&gt;r_{1}&lt;/math&gt; edge 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 skew&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 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 skew 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 every 180° of isoclinic rotation each vertex circles a skew 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 skew 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 two twisted parallel strands that intersects each 24-cell vertex once. In every 360° of isoclinic rotation each vertex circles a skew great square 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 triangles 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.  

...  

{{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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      <id>2818176</id>
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      <text bytes="86577" sha1="1ufv8h5tof67i8zvcqeo8ekvsao9yhb" 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). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

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

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

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

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

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

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

== The 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}+r_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;-r_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;-r_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;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 the &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 six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &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 left 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 left 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 helix is an 8-rung ladder twisted 3 times, 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, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in 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 left 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° chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once.

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 central 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.

[[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 skew&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 skew 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 skew 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 skew 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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  </page>
  <page>
    <title>User:Juandev/R/Wikicode</title>
    <ns>2</ns>
    <id>330443</id>
    <revision>
      <id>2818172</id>
      <parentid>2818126</parentid>
      <timestamp>2026-07-12T17:09:01Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>+1</comment>
      <origin>2818172</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1271" sha1="0h3jubupspq4wqu5dxysq4yycuomd3t" xml:space="preserve">== What is the name of &lt;code&gt;&lt;nowiki&gt;{{{obec|}}}&lt;/nowiki&gt;&lt;/code&gt;? ==
This is a template parameter reference. The string &quot;obec&quot; is the parameter name and the space after the vertical bar is the empty default value.
*Note. The default parameter reference is only printed if the user does not enter it in the template. This means that &lt;code&gt;&lt;nowiki&gt;{{{obec|unfilled}}}&lt;/nowiki&gt;&lt;/code&gt; will insert &quot;unfilled&quot; into the page after saving, unless this parameter is specified.

== Can I use template parameter reference more times in the template? ==
Yes.

== Is there a way to add notes to wikicode template? ==
Yes, the same way as in wikitext, surrounding strings with &lt;code&gt;&lt;nowiki&gt;&lt;!-- string --&gt;&lt;/nowiki&gt;&lt;/code&gt;.

== Who we call &lt;code&gt;&lt;nowiki&gt;state =&lt;/nowiki&gt;&lt;/code&gt; of a template ?==
Its called parameter.

== What is &lt;code&gt;&lt;nowiki&gt;#if:&lt;/nowiki&gt;&lt;/code&gt; and how it works? ==
&lt;nowiki&gt;&lt;code&gt;#&lt;/code&gt;&lt;/nowiki&gt; means parser function and the generic syntax is &lt;code&gt;&lt;nowiki&gt;{{#if: condition | add this if the condition is true | add this if the condition is false }}&lt;/nowiki&gt;&lt;/code&gt;

== What is the syntax of &lt;code&gt;&lt;nowiki&gt;#ifeq&lt;/nowiki&gt;&lt;/code&gt;? ==
&lt;code&gt;&lt;nowiki&gt;{{#ifeq: value 1| value 2 | result if both values are equal| result if both values are not equal }}&lt;/nowiki&gt;&lt;/code&gt;</text>
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  <page>
    <title>Category:GNU Octave</title>
    <ns>14</ns>
    <id>330561</id>
    <revision>
      <id>2818162</id>
      <timestamp>2026-07-12T16:16:05Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <comment>create</comment>
      <origin>2818162</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21" sha1="5s0t1ov3cjpb2r56onhqpqo2c1elnv5" xml:space="preserve">[[Category:Software]]</text>
      <sha1>5s0t1ov3cjpb2r56onhqpqo2c1elnv5</sha1>
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  </page>
  <page>
    <title>Artificial Intelligence/Ethical Concerns</title>
    <ns>0</ns>
    <id>330563</id>
    <revision>
      <id>2818179</id>
      <timestamp>2026-07-12T22:07:09Z</timestamp>
      <contributor>
        <username>MaylenGomez And SilvanaRoldan</username>
        <id>3100552</id>
      </contributor>
      <comment>Created page with &quot;&lt;nowiki&gt;'''Artificial Intelligence in Modern Society'''  '''Overview'''   Artificial intelligence (AI) is a branch of computer science concerned with developing systems capable of performing tasks that normally require human intelligence. Modern AI is primarily based on narrow AI rather than hypothetical general AI and is widely used in areas such as automation, communication, education, and data analysis. Although AI offers numerous practical benefits, it also presents...&quot;</comment>
      <origin>2818179</origin>
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      <format>text/x-wiki</format>
      <text bytes="3445" sha1="79nmotfzgrf6whx2s9i2m8bqyesdzzv" xml:space="preserve">&lt;nowiki&gt;'''Artificial Intelligence in Modern Society'''

'''Overview''' 

Artificial intelligence (AI) is a branch of computer science concerned with developing systems capable of performing tasks that normally require human intelligence. Modern AI is primarily based on narrow AI rather than hypothetical general AI and is widely used in areas such as automation, communication, education, and data analysis. Although AI offers numerous practical benefits, it also presents important challenges, including algorithmic bias, ethical concerns, and responsible use. 

'''Fiction vs Reality'''

Artificial intelligence is frequently represented in films and popular culture as machines capable of human-like reasoning, 
emotions, and self-awareness. This representation corresponds to Artificial General Intelligence (AGI), a hypothetical form of AI that has not yet been developed.
Current AI systems are examples of Narrow Artificial Intelligence, which is designed to perform specific tasks by identifying statistical patterns in large amounts of data. Rather than understanding information in the same way humans do, these systems generate predictions based on previously learned patterns. Consequently, current AI performs specialized functions without possessing consciousness, reasoning abilities, or self-awareness.
Applications: Practical Benefits

Artificial intelligence is widely used to automate repetitive tasks and improve efficiency across different fields. AI applications assist users in organizing workflows, generating text, creating outlines, summarizing information, translating languages, and supporting decision-making processes. According to the IFTTT Team (2024), AI applications can save time by automating routine activities, allowing individuals to dedicate more attention to complex and creative tasks. 

'''Limitations: Algorithmic Bias'''

One of the most significant challenges associated with artificial intelligence is algorithmic bias. Because AI systems are trained on human-generated data, they reproduce patterns of discrimination or inequality already present in those data. Biases related to race, gender, socioeconomic status, or other social characteristics can therefore be reflected in algorithmic decisions. As Broussard (2022) explains, algorithms are not inherently neutral because they are created by humans and inevitably reflect the assumptions embedded in their design and training data. 

'''Responsible Use'''

The increasing integration of artificial intelligence into everyday life has highlighted the importance of responsible use. International organizations such as UNESCO emphasize that AI should be used ethically and critically rather than accepted uncritically. Responsible use involves recognizing the capabilities and limitations of AI systems, verifying the information they generate, protecting privacy, promoting fairness, and developing AI literacy. These practices contribute to the ethical and informed use of artificial intelligence in society. 


'''References'''

Broussard, M. (2022). Popular Doesn´t Mean Good. In Artificial Intelligence: How Computers Misunderstand the World (pp. 146–157). The MIT Press. 

The IFTTT Team. (2024, August 2). How does ChatGPT work? (link)

UNESCO. (2025, October 20). AI and society: How to build a more responsible future.(link)

Broussard, M. (2023, February 9). Remote Daily #263 – with Meredith Broussard [Video]. YouTube.(link)</text>
      <sha1>79nmotfzgrf6whx2s9i2m8bqyesdzzv</sha1>
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  </page>
  <page>
    <title>User talk:Ibanezcalixtristanb</title>
    <ns>3</ns>
    <id>330564</id>
    <revision>
      <id>2818180</id>
      <timestamp>2026-07-12T23:41:33Z</timestamp>
      <contributor>
        <username>~2026-39466-24</username>
        <id>3100628</id>
      </contributor>
      <comment>/* Give 500000 */ new section</comment>
      <origin>2818180</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="184" sha1="i50a17es7aii1zkxq2pvopuj7bkmjal" xml:space="preserve">== Give 500000 ==

Pay you@[[User:Yunan12|Yunan12]] [[Special:Contributions/&amp;#126;2026-39466-24|&amp;#126;2026-39466-24]] ([[User talk:&amp;#126;2026-39466-24|talk]]) 23:41, 12 July 2026 (UTC)</text>
      <sha1>i50a17es7aii1zkxq2pvopuj7bkmjal</sha1>
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  </page>
  <page>
    <title>User:DaraganSergio/Social Synaptic Plasticity</title>
    <ns>2</ns>
    <id>330565</id>
    <revision>
      <id>2818181</id>
      <timestamp>2026-07-13T00:07:27Z</timestamp>
      <contributor>
        <username>DaraganSergio</username>
        <id>3100629</id>
      </contributor>
      <comment>Created page with &quot;  = Social Synaptic Plasticity =  == Overview == '''Social Synaptic Plasticity (SSP)''' is a conceptual framework introduced by Sergey Daragan for exploring how social environments, group interactions, and interpersonal relationships may influence psychological adaptation, resilience, and behavioral change.  The concept is inspired by the principle of neural plasticity. Just as neural networks can reorganize in response to experience, social environments may contribute t...&quot;</comment>
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      <text bytes="3478" sha1="8d3s8trfbbrekr3y7ugmqn4gp3ysp45" xml:space="preserve">

= Social Synaptic Plasticity =

== Overview ==
'''Social Synaptic Plasticity (SSP)''' is a conceptual framework introduced by Sergey Daragan for exploring how social environments, group interactions, and interpersonal relationships may influence psychological adaptation, resilience, and behavioral change.

The concept is inspired by the principle of neural plasticity. Just as neural networks can reorganize in response to experience, social environments may contribute to the development of new behavioral patterns, emotional responses, coping strategies, and social competencies.

Social Synaptic Plasticity proposes that repeated social experiences, supportive relationships, and group dynamics can play a significant role in shaping individual adaptation to changing life circumstances.

== Background ==
The concept emerged from practical observations of group-based support programs, community initiatives, and social rehabilitation activities. Particular attention was given to situations involving social stress, crisis, displacement, and collective recovery processes.

The framework seeks to examine the relationship between social interaction and psychological adaptation, with an emphasis on how communities and support networks may facilitate resilience and personal development.

== Core Principles ==
Human behavior is influenced by social interaction.

Social environments can contribute to psychological adaptation.

Groups may accelerate learning, recovery, and behavioral change.

Social support can strengthen resilience during periods of crisis.

Repeated social experiences may reinforce new patterns of behavior and emotional response.

Community participation may increase a sense of belonging, meaning, and social competence.

== Theoretical Foundations ==
The framework draws inspiration from several fields of study, including:

Neural plasticity

Social psychology

Community psychology

Group dynamics

Resilience studies

Social learning theory

Behavioral adaptation research

Social Synaptic Plasticity is intended as an interdisciplinary conceptual model rather than a replacement for existing psychological or neuroscientific theories.

== Areas of Application ==
Potential areas of application include:

Group support programs

Social rehabilitation

Community development

Crisis intervention

Psychological resilience training

Educational environments

Volunteer and humanitarian initiatives

Peer-support networks

== Research Questions ==
The framework raises several research questions:

How do social networks influence psychological adaptation?

Can group dynamics facilitate long-term behavioral change?

What mechanisms connect social support and resilience?

How do communities contribute to recovery after crisis or trauma?

Can social environments be intentionally designed to improve adaptation outcomes?

What indicators can be used to measure social adaptation processes?

== Future Development ==
Future work may include:

Development of practical methodologies.

Collection of case studies.

Comparative analysis with existing theories of adaptation and resilience.

Evaluation of group-based intervention programs.

Exploration of measurable indicators of social adaptation.

== References ==
Daragan, S. ''Social Synaptic Plasticity''. 2024.

Literature on neural plasticity.

Literature on social psychology.

Literature on resilience and community support.

[[Category]]

[[Category psychology]]

[[Category studies]]</text>
      <sha1>8d3s8trfbbrekr3y7ugmqn4gp3ysp45</sha1>
    </revision>
    <revision>
      <id>2818185</id>
      <parentid>2818181</parentid>
      <timestamp>2026-07-13T01:38:16Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>Atcovi moved page [[Social Synaptic Plasticity]] to [[User:DaraganSergio/Social Synaptic Plasticity]] without leaving a redirect: not suitable for mainspace, lacks scholarly research</comment>
      <origin>2818181</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3478" sha1="8d3s8trfbbrekr3y7ugmqn4gp3ysp45" xml:space="preserve">

= Social Synaptic Plasticity =

== Overview ==
'''Social Synaptic Plasticity (SSP)''' is a conceptual framework introduced by Sergey Daragan for exploring how social environments, group interactions, and interpersonal relationships may influence psychological adaptation, resilience, and behavioral change.

The concept is inspired by the principle of neural plasticity. Just as neural networks can reorganize in response to experience, social environments may contribute to the development of new behavioral patterns, emotional responses, coping strategies, and social competencies.

Social Synaptic Plasticity proposes that repeated social experiences, supportive relationships, and group dynamics can play a significant role in shaping individual adaptation to changing life circumstances.

== Background ==
The concept emerged from practical observations of group-based support programs, community initiatives, and social rehabilitation activities. Particular attention was given to situations involving social stress, crisis, displacement, and collective recovery processes.

The framework seeks to examine the relationship between social interaction and psychological adaptation, with an emphasis on how communities and support networks may facilitate resilience and personal development.

== Core Principles ==
Human behavior is influenced by social interaction.

Social environments can contribute to psychological adaptation.

Groups may accelerate learning, recovery, and behavioral change.

Social support can strengthen resilience during periods of crisis.

Repeated social experiences may reinforce new patterns of behavior and emotional response.

Community participation may increase a sense of belonging, meaning, and social competence.

== Theoretical Foundations ==
The framework draws inspiration from several fields of study, including:

Neural plasticity

Social psychology

Community psychology

Group dynamics

Resilience studies

Social learning theory

Behavioral adaptation research

Social Synaptic Plasticity is intended as an interdisciplinary conceptual model rather than a replacement for existing psychological or neuroscientific theories.

== Areas of Application ==
Potential areas of application include:

Group support programs

Social rehabilitation

Community development

Crisis intervention

Psychological resilience training

Educational environments

Volunteer and humanitarian initiatives

Peer-support networks

== Research Questions ==
The framework raises several research questions:

How do social networks influence psychological adaptation?

Can group dynamics facilitate long-term behavioral change?

What mechanisms connect social support and resilience?

How do communities contribute to recovery after crisis or trauma?

Can social environments be intentionally designed to improve adaptation outcomes?

What indicators can be used to measure social adaptation processes?

== Future Development ==
Future work may include:

Development of practical methodologies.

Collection of case studies.

Comparative analysis with existing theories of adaptation and resilience.

Evaluation of group-based intervention programs.

Exploration of measurable indicators of social adaptation.

== References ==
Daragan, S. ''Social Synaptic Plasticity''. 2024.

Literature on neural plasticity.

Literature on social psychology.

Literature on resilience and community support.

[[Category]]

[[Category psychology]]

[[Category studies]]</text>
      <sha1>8d3s8trfbbrekr3y7ugmqn4gp3ysp45</sha1>
    </revision>
  </page>
  <page>
    <title>User:Eunice Ameh/sandbox</title>
    <ns>2</ns>
    <id>330566</id>
    <revision>
      <id>2818183</id>
      <timestamp>2026-07-13T01:14:25Z</timestamp>
      <contributor>
        <username>Eunice Ameh</username>
        <id>3026207</id>
      </contributor>
      <comment>CATACTS EXPLAINED</comment>
      <origin>2818183</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1940" sha1="crfxkv92mcu0oycqu6yk1zord89g8dg" xml:space="preserve">CATARACT

Cataract is opacity of the crystalline lens of the eye. Usually the lens is clear. Cataracts develop when protein cells in the break down and form cloudy patches. Over time, they can get big enough to make vision blurry.

Cataract develop naturally as the eyes age, injuries and some health conditions can bring about cataract. Cataract  is prevalent in people over 65  years of age. but they is treatable as an ophthalmologist can perform surgery to remove cataracts and restore a clear vision.

TYPES OF CATARACTS

This classification is based on the location of the cataract in the lens.

* '''Nuclear cataracts:''' The nucleus is the center of the lens. Nuclear cataracts are the most common type and primarily affect  the nucleus. These cataracts usually develop with age. They form deep in the central part of the lens and increase gradually, causing the centre of the lens to become yellow or brown. As the cataract progresses, it can greatly impact both near and distance vision. Nuclear cataracts can change the power of the natural lens in the eye.

The most common cause of nuclear cataracts is the natural ageing process. with ag, the proteins within the eye’s lens can break down and clump together. This causes opacity of the lens. Other risk factors include diabetes, genetics, and smoking.

Common symptoms include blurred or reduced vision, increased sensitivity to glare, distorted vision, and a gradual shift toward greater short-sightedness (myopia) or reduced long-sightedness (hyperopia).

'''Treatment:''' The main and most effective treatment for a nuclear cataract is '''cataract surgery'''. This procedure is highly successful in restoring clear vision and is among the most frequently performed surgical operations worldwide.


== REFERENCES ==
https://my.clevelandclinic.org/health/diseases/8589-cataracts-age-related

https://practiceplusgroup.com/knowledge-hub/different-types-cataracts-explained/</text>
      <sha1>crfxkv92mcu0oycqu6yk1zord89g8dg</sha1>
    </revision>
    <revision>
      <id>2818184</id>
      <parentid>2818183</parentid>
      <timestamp>2026-07-13T01:16:40Z</timestamp>
      <contributor>
        <username>Eunice Ameh</username>
        <id>3026207</id>
      </contributor>
      <minor />
      <comment>EDITED</comment>
      <origin>2818184</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1945" sha1="6vzz99nsuepycst5fzhr9v1ho9inpzy" xml:space="preserve">CATARACT

Cataract is opacity of the crystalline lens of the eye. Usually the lens is clear. Cataracts develop when protein cells in the break down and form cloudy patches. Over time, they can get big enough to make vision blurry.

Cataract develop naturally as the eyes age, injuries and some health conditions can bring about cataract. Cataract  is prevalent in people over 65  years of age. but they is treatable as an ophthalmologist can perform surgery to remove cataracts and restore a clear vision.

== TYPES OF CATARACTS ==
This classification is based on the location of the cataract in the lens.

* '''Nuclear cataracts:''' The nucleus is the center of the lens. Nuclear cataracts are the most common type and primarily affect  the nucleus. These cataracts usually develop with age. They form deep in the central part of the lens and increase gradually, causing the centre of the lens to become yellow or brown. As the cataract progresses, it can greatly impact both near and distance vision. Nuclear cataracts can change the power of the natural lens in the eye.

The most common cause of nuclear cataracts is the natural ageing process. with ag, the proteins within the eye’s lens can break down and clump together. This causes opacity of the lens. Other risk factors include diabetes, genetics, and smoking.

Common symptoms include blurred or reduced vision, increased sensitivity to glare, distorted vision, and a gradual shift toward greater short-sightedness (myopia) or reduced long-sightedness (hyperopia).

'''Treatment:''' The main and most effective treatment for a nuclear cataract is '''cataract surgery'''. This procedure is highly successful in restoring clear vision and is among the most frequently performed surgical operations worldwide.


== REFERENCES ==
https://my.clevelandclinic.org/health/diseases/8589-cataracts-age-related

https://practiceplusgroup.com/knowledge-hub/different-types-cataracts-explained/</text>
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|Source={{own|Young1lim}}
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{{Information
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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    <title>Draft:Experimental Verification of the NKTg Law Using NASA Mercury Data in 2025</title>
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      <comment>Created page with &quot;Experimental Verification of the NKTg Law Using NASA Mercury Data in 2025 Independent researcher:  Nguyễn Khánh Tùng     ORCID: 0009-0002-9877-4137	 				 Table 1 – Actual orbital and mass data of Mercury in 2024 from the NASA JPL Horizons System  Thời điểm	x (m)	v (m/s)	m (kg)	p = m•v 1/1/2024	5.16E+10	5.33E+04	3.30E+23	1.76E+28 4/1/2024	6.97E+10	3.90E+04	3.30E+23	1.29E+28 7/1/2024	5.36E+10	5.20E+04	3.30E+23	1.72E+28 10/1/2024	6.95E+10	3.92E+04	3.30E+23	1.30...&quot;</comment>
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      <text bytes="3727" sha1="1r8706jmg5hawpe3c4aiq9xvms6y3yv" xml:space="preserve">Experimental Verification of the NKTg Law Using NASA Mercury Data in 2025
Independent researcher:  Nguyễn Khánh Tùng   

ORCID: 0009-0002-9877-4137	
				
Table 1 – Actual orbital and mass data of Mercury in 2024 from the NASA JPL Horizons System

Thời điểm	x (m)	v (m/s)	m (kg)	p = m•v
1/1/2024	5.16E+10	5.33E+04	3.30E+23	1.76E+28
4/1/2024	6.97E+10	3.90E+04	3.30E+23	1.29E+28
7/1/2024	5.36E+10	5.20E+04	3.30E+23	1.72E+28
10/1/2024	6.95E+10	3.92E+04	3.30E+23	1.30E+28
12/31/2024	4.64E+10	5.81E+04	3.30E+23	1.92E+28
				
Table 2 – Simulated orbital and mass data of Mercury in 2025 according to the NKTg law

Thời điểm	x (m)	v (m/s)	m (kg)	p = m•v (kg•m/s)
1/1/2025	5.16E+10	5.22E+04	3.301E+23	1.72E+28
4/1/2025	6.97E+10	3.87E+04	3.301E+23	1.28E+28
7/1/2025	5.49E+10	4.91E+04	3.301E+23	1.62E+28
10/1/2025	6.83E+10	3.95E+04	3.301E+23	1.30E+28
12/31/2025	4.61E+10	5.85E+04	3.301E+23	1.93E+28
				
Table 3 – Actual orbital and mass data of Mercury in 2025 from the NASA JPL Horizons System

Thời điểm	x (m)	v (m/s)	m (kg)	
1/1/2025	5.16E+10	5.34E+04	3.30E+23	
4/1/2025	6.97E+10	3.89E+04	3.30E+23	
7/1/2025	5.49E+10	5.04E+04	3.30E+23	
1/10/2025	6.83E+10	3.98E+04	3.30E+23	
12/31/2025	4.61E+10	5.89E+04	3.30E+23	
				
Table 4 – Comparison between NKTg Simulation and NASA Data in 2025

Thời điểm	v – NKTg	v – NASA	Sai lệch v tương đối (%)	
1/1/2025	5.22E+04	53400	-2.15461	
4/1/2025	3.87E+04	38900	-0.562786	
7/1/2025	4.91E+04	50400	-2.561981	
1/10/2025	3.95E+04	39800	-0.819213	
12/31/2025	5.85E+04	58900	-0.707806	
SCIENTIFIC CONCLUSION
On the experimental verification of the NKTg Law on planetary orbits
Abstract: This report presents the results of the experimental verification of the NKTg law through the simulation of Mercury’s orbit in 2025, using reference data from NASA JPL Horizons and the MESSENGER spacecraft.
1. Conservation of motion tendency through the quantity NKTg₁
The study confirms that the quantity NKTg₁ = x × p (the product of position and momentum) is the fundamental parameter determining the motion tendency of an object in space. The experimental results show that:
• Establishing NKTg₁ = x × p as a target constant (8.90E+38 NKTm) allows the prediction of orbital velocity with high accuracy, with an average relative error of only 1.3% compared to NASA data.
• The interaction between position (x) and momentum (p) self-adjusts to maintain this constant, enabling the object to preserve its motion structure and maintain a stable state without the need for external force parameters.
2. Experimental verification of mass variation (dm/dt) from MESSENGER data
The model integrates the actual mass variation coefficient dm/dt = -0.5 kg/s, established from measured data of the atmospheric escape phenomenon by the MESSENGER spacecraft. According to the NKTg law, this variation leads to:
• The formation of the quantity NKTg₂: The product of the rate of mass change and momentum (NKTg₂ = (dm/dt) × p) carries a negative value throughout the orbit.
• Resisting effect: The negative value of NKTg₂ confirms that the mass loss process plays a resisting role against changes in the state of motion, accurately reflecting the nature of varying inertia.
3. Consistency and applicability
The experimental results on Mercury confirm that the NKTg law has consistency and high applicability. The system demonstrates the capability to:
• Fully describe planetary dynamics solely through the interaction between fundamental quantities (x, v, m).
• Eliminate dependence on approximation methods, without using force expressions in the calculation process, ensuring the reproducibility of results across different computational platforms.</text>
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      <text bytes="4140" sha1="161li9582z4z021cs0le9xdoyd0e2wo" xml:space="preserve">= Experimental Verification of the NKTg Law Using NASA Mercury Data in 2025 =

'''Independent researcher:''' Nguyễn Khánh Tùng

'''ORCID:''' 0009-0002-9877-4137

== Table 1 – Actual orbital and mass data of Mercury in 2024 from the NASA JPL Horizons System ==

{| class=&quot;wikitable&quot;
! Thời điểm
! x (m)
! v (m/s)
! m (kg)
! p = m•v
|-
| 1/1/2024
| 5.16E+10
| 5.33E+04
| 3.30E+23
| 1.76E+28
|-
| 4/1/2024
| 6.97E+10
| 3.90E+04
| 3.30E+23
| 1.29E+28
|-
| 7/1/2024
| 5.36E+10
| 5.20E+04
| 3.30E+23
| 1.72E+28
|-
| 10/1/2024
| 6.95E+10
| 3.92E+04
| 3.30E+23
| 1.30E+28
|-
| 12/31/2024
| 4.64E+10
| 5.81E+04
| 3.30E+23
| 1.92E+28
|}

== Table 2 – Simulated orbital and mass data of Mercury in 2025 according to the NKTg law ==

{| class=&quot;wikitable&quot;
! Thời điểm
! x (m)
! v (m/s)
! m (kg)
! p = m•v (kg•m/s)
|-
| 1/1/2025
| 5.16E+10
| 5.22E+04
| 3.301E+23
| 1.72E+28
|-
| 4/1/2025
| 6.97E+10
| 3.87E+04
| 3.301E+23
| 1.28E+28
|-
| 7/1/2025
| 5.49E+10
| 4.91E+04
| 3.301E+23
| 1.62E+28
|-
| 10/1/2025
| 6.83E+10
| 3.301E+23
| 3.95E+04
| 1.30E+28
|-
| 12/31/2025
| 4.61E+10
| 5.85E+04
| 3.301E+23
| 1.93E+28
|}

== Table 3 – Actual orbital and mass data of Mercury in 2025 from the NASA JPL Horizons System ==

{| class=&quot;wikitable&quot;
! Thời điểm
! x (m)
! v (m/s)
! m (kg)
|-
| 1/1/2025
| 5.16E+10
| 5.34E+04
| 3.30E+23
|-
| 4/1/2025
| 6.97E+10
| 3.89E+04
| 3.30E+23
|-
| 7/1/2025
| 5.49E+10
| 5.04E+04
| 3.30E+23
|-
| 1/10/2025
| 6.83E+10
| 3.98E+04
| 3.30E+23
|-
| 12/31/2025
| 4.61E+10
| 5.89E+04
| 3.30E+23
|}

== Table 4 – Comparison between NKTg Simulation and NASA Data in 2025 ==

{| class=&quot;wikitable&quot;
! Thời điểm
! v – NKTg
! v – NASA
! Sai lệch v tương đối (%)
|-
| 1/1/2025
| 5.22E+04
| 53400
| -2.15461
|-
| 4/1/2025
| 3.87E+04
| 38900
| -0.562786
|-
| 7/1/2025
| 4.91E+04
| 50400
| -2.561981
|-
| 1/10/2025
| 3.95E+04
| 39800
| -0.819213
|-
| 12/31/2025
| 5.85E+04
| 58900
| -0.707806
|}

= SCIENTIFIC CONCLUSION =
== On the experimental verification of the NKTg Law on planetary orbits ==

'''Abstract:'''
This report presents the results of the experimental verification of the NKTg law through the simulation of Mercury’s orbit in 2025, using reference data from NASA JPL Horizons and the MESSENGER spacecraft.

== 1. Conservation of motion tendency through the quantity NKTg₁ ==

The study confirms that the quantity NKTg₁ = x × p (the product of position and momentum) is the fundamental parameter determining the motion tendency of an object in space. The experimental results show that:

* Establishing NKTg₁ = x × p as a target constant (8.90E+38 NKTm) allows the prediction of orbital velocity with high accuracy, with an average relative error of only 1.3% compared to NASA data.
* The interaction between position (x) and momentum (p) self-adjusts to maintain this constant, enabling the object to preserve its motion structure and maintain a stable state without the need for external force parameters.

== 2. Experimental verification of mass variation (dm/dt) from MESSENGER data ==

The model integrates the actual mass variation coefficient dm/dt = -0.5 kg/s, established from measured data of the atmospheric escape phenomenon by the MESSENGER spacecraft. According to the NKTg law, this variation leads to:

* The formation of the quantity NKTg₂: The product of the rate of mass change and momentum (NKTg₂ = (dm/dt) × p) carries a negative value throughout the orbit.
* Resisting effect: The negative value of NKTg₂ confirms that the mass loss process plays a resisting role against changes in the state of motion, accurately reflecting the nature of varying inertia.

== 3. Consistency and applicability ==

The experimental results on Mercury confirm that the NKTg law has consistency and high applicability. The system demonstrates the capability to:

* Fully describe planetary dynamics solely through the interaction between fundamental quantities (x, v, m).
* Eliminate dependence on approximation methods, without using force expressions in the calculation process, ensuring the reproducibility of results across different computational platforms.</text>
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