Wikiversity:Colloquium

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&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

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

Dear all,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hi all,

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

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

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

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

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

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

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

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

Hello all,

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

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

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Best regards,

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

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

Good morning,

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

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

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

Many thanks,

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

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

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

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[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; 16:53, 18 September 2024 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

''Other effects'':

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

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

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

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

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

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

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

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

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

== Protected template bug for Pp ==

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

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

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

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

Dear all,


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

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

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

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

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


Warm regards,

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

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

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

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

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

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

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

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

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

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

== Active editors ==

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

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

== Intentionally incorrect resource ==

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

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

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

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

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

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

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

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

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

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

The following four candidates were the most voted:

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

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

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

Best regards,

The Elections Committee and Board Selection Working Group

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

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

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

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

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

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

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

:What sort of interactive elements are you thinking about?

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

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

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)
:I guess the mention in mass media might be a cause. Someone metions it and then thousands go and look. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:02, 17 December 2024 (UTC)

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

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

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

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

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

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

Dear all,

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

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

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

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

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


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

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

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)
:I dont think there is an app.  [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:01, 17 December 2024 (UTC)
:This would be useful, however we do not have an app for Wikiversity yet. I am thinking of helping out with no-code or low code tools, but I am working on some courses here. I might be able to do some contributions though. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 14:14, 17 December 2024 (UTC)

== Import Resource From Wikibooks? ==

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

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

== Language translation requests? ==

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

:I am not too sure about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:44, 17 December 2024 (UTC)

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

Hello everyone,

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

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

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

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

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)
:I am affraid, that creation of educational resources on certain topic is way harder then wikipedia. Secondly while wikipedia stub does not matter, education resource stub is uselless completly. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 17 December 2024 (UTC)
::How is it useless, you can contribute to other learning resources and maybe improve it as such, if you have some knowledge on a particular topic or something else. This is to increase diversity. Just a kind notice. It's also pretty hard to do it on Wikivoyage, but that's the same for every platform. Stubs may be improved on, and this is the concept. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:19, 17 December 2024 (UTC)
::There are lots of stubs here, on Wikiversity. So the whole purpose of this event is to increase engagement and willingness to edit these pages. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:24, 17 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

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

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

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)
::::::I'm thinking of the community entering their projects, and discussing those in the newsletter. It depends on what they want, though. There would be a dedicated page for giving the information about their projects [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:24, 16 December 2024 (UTC)
:::::::I might start working on this soon, depending on the projects being created on Wikiversity. @[[User:Ottawahitech|Ottawahitech]] @[[User:Atcovi|Atcovi]] [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:25, 17 December 2024 (UTC)
::::::::I'd recommend you start off with putting this under a userspace page (something like [[User:RockTransport/Wikiversity Newsletter]]), and drafting what you desire. Let us know once it's done, and the community can provide their input. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:30, 17 December 2024 (UTC)
:::::::::I will try and make one for this month. This is supposed to be a monthly newsletter, showcasing the different projects mentioned there. Users can put their projects, and we will document them on the newsletter. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:33, 17 December 2024 (UTC)
:::::::::I am hoping for it to be released by January 2025. There's no rush to get it done; it's still in it's planning stage. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:43, 17 December 2024 (UTC)
::::I '''might''' be able to icnrease the frequency there, but it doesn't go into detail about these topics. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:30, 18 December 2024 (UTC)
:Where you are going to get the audience for your website and Wikiversity newsletter? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 18 December 2024 (UTC)
::It's on Wikiversity, not on an outside platform. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 13:51, 18 December 2024 (UTC)
::The audience will be Wikiversity contributors. There will be a dedicated page for it on Wikiversity. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 13:55, 18 December 2024 (UTC)

== Describing Wikiversity content on Wikidata ==

Anyone knows how to properly describe Wikiversity pages on Wikidata? Any examples for some content pages like courses, supplement materials etc.? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 18 December 2024 (UTC)

:For general topics that will have other Wikimedia Foundation project links (e.g. [[astronomy]]), there will probably be a sufficient short description already, but for subpages or more obscure topics, you could plausibly use &quot;Wikimedia content page&quot; or some such. —[[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; 15:52, 18 December 2024 (UTC)
::Yeah, general topics are easy to map. While specific projects which does not have Wikipedia counterparts and which are quite specific it would be nice to have few examples - i.e. what are typical properties of a course or research project. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:00, 19 December 2024 (UTC)

== Degrees ==

Why does Wikiversity not provide degrees? I know it was a promise to the Wikimedia Foundation in the Wikiversity project proposal. But anyway, why is that? Wikiversity is about opening doors, i.e., removing obstacles. So, what kind of an obstacle was a paper? Was a certain body of knowledge that you learned well?! Because Wikiversity is not accredited for that? Yes, and do we need official US accreditation? We cannot create our system so that the learners who learn here and would like to continue their science career have a recognizable degree they can continue? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:19, 18 December 2024 (UTC)

:&quot;I know it was a promise to the Wikimedia Foundation in the Wikiversity project proposal.&quot; Was it? Becoming a degree-granting institution is an extremely high bar in the United States, but what is even the point in becoming a degree-granting institution in... Malawi? Tonga? Somewhere else where the servers aren't located or the WMF aren't incorporated? —[[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; 15:53, 18 December 2024 (UTC)</text>
      <sha1>655jab4dre1f6bh22w4ow1r3cohbevc</sha1>
    </revision>
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      <id>2692630</id>
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      <text bytes="85025" sha1="e139l01rs0te7gi453743cob0y0c4rt" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

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

Dear all,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Hi all,

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

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

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

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

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

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

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

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

Hello all,

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

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

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Best regards,

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

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

Good morning,

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

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

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

Many thanks,

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

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

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

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[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; 16:53, 18 September 2024 (UTC)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

''Other effects'':

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

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

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

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

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

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

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

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

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

== Protected template bug for Pp ==

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

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

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

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

Dear all,


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

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

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

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

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


Warm regards,

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

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

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

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

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

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

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

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

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

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

== Active editors ==

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

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

== Intentionally incorrect resource ==

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

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

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

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

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

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

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

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

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

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

The following four candidates were the most voted:

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

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

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

Best regards,

The Elections Committee and Board Selection Working Group

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

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

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

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

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

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

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

:What sort of interactive elements are you thinking about?

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

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

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)
:I guess the mention in mass media might be a cause. Someone metions it and then thousands go and look. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:02, 17 December 2024 (UTC)

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

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

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

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

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

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

Dear all,

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

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

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

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

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


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

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

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)
:I dont think there is an app.  [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:01, 17 December 2024 (UTC)
:This would be useful, however we do not have an app for Wikiversity yet. I am thinking of helping out with no-code or low code tools, but I am working on some courses here. I might be able to do some contributions though. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 14:14, 17 December 2024 (UTC)

== Import Resource From Wikibooks? ==

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

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

== Language translation requests? ==

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

:I am not too sure about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:44, 17 December 2024 (UTC)

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

Hello everyone,

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

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

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

== Events on Wikiversity ==

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

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)
:I am affraid, that creation of educational resources on certain topic is way harder then wikipedia. Secondly while wikipedia stub does not matter, education resource stub is uselless completly. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 17 December 2024 (UTC)
::How is it useless, you can contribute to other learning resources and maybe improve it as such, if you have some knowledge on a particular topic or something else. This is to increase diversity. Just a kind notice. It's also pretty hard to do it on Wikivoyage, but that's the same for every platform. Stubs may be improved on, and this is the concept. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:19, 17 December 2024 (UTC)
::There are lots of stubs here, on Wikiversity. So the whole purpose of this event is to increase engagement and willingness to edit these pages. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:24, 17 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

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

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

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)
::::::I'm thinking of the community entering their projects, and discussing those in the newsletter. It depends on what they want, though. There would be a dedicated page for giving the information about their projects [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:24, 16 December 2024 (UTC)
:::::::I might start working on this soon, depending on the projects being created on Wikiversity. @[[User:Ottawahitech|Ottawahitech]] @[[User:Atcovi|Atcovi]] [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:25, 17 December 2024 (UTC)
::::::::I'd recommend you start off with putting this under a userspace page (something like [[User:RockTransport/Wikiversity Newsletter]]), and drafting what you desire. Let us know once it's done, and the community can provide their input. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:30, 17 December 2024 (UTC)
:::::::::I will try and make one for this month. This is supposed to be a monthly newsletter, showcasing the different projects mentioned there. Users can put their projects, and we will document them on the newsletter. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:33, 17 December 2024 (UTC)
:::::::::I am hoping for it to be released by January 2025. There's no rush to get it done; it's still in it's planning stage. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 18:43, 17 December 2024 (UTC)
::::I '''might''' be able to icnrease the frequency there, but it doesn't go into detail about these topics. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:30, 18 December 2024 (UTC)
:Where you are going to get the audience for your website and Wikiversity newsletter? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:38, 18 December 2024 (UTC)
::It's on Wikiversity, not on an outside platform. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 13:51, 18 December 2024 (UTC)
::The audience will be Wikiversity contributors. There will be a dedicated page for it on Wikiversity. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 13:55, 18 December 2024 (UTC)

== Describing Wikiversity content on Wikidata ==

Anyone knows how to properly describe Wikiversity pages on Wikidata? Any examples for some content pages like courses, supplement materials etc.? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:36, 18 December 2024 (UTC)

:For general topics that will have other Wikimedia Foundation project links (e.g. [[astronomy]]), there will probably be a sufficient short description already, but for subpages or more obscure topics, you could plausibly use &quot;Wikimedia content page&quot; or some such. —[[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; 15:52, 18 December 2024 (UTC)
::Yeah, general topics are easy to map. While specific projects which does not have Wikipedia counterparts and which are quite specific it would be nice to have few examples - i.e. what are typical properties of a course or research project. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:00, 19 December 2024 (UTC)

== Degrees ==

Why does Wikiversity not provide degrees? I know it was a promise to the Wikimedia Foundation in the Wikiversity project proposal. But anyway, why is that? Wikiversity is about opening doors, i.e., removing obstacles. So, what kind of an obstacle was a paper? Was a certain body of knowledge that you learned well?! Because Wikiversity is not accredited for that? Yes, and do we need official US accreditation? We cannot create our system so that the learners who learn here and would like to continue their science career have a recognizable degree they can continue? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:19, 18 December 2024 (UTC)

:&quot;I know it was a promise to the Wikimedia Foundation in the Wikiversity project proposal.&quot; Was it? Becoming a degree-granting institution is an extremely high bar in the United States, but what is even the point in becoming a degree-granting institution in... Malawi? Tonga? Somewhere else where the servers aren't located or the WMF aren't incorporated? —[[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; 15:53, 18 December 2024 (UTC)
::I ment certificates. The question is the recognazibility of a certificate. I am not talking here about equal certification, which is provided by governmental institucians to universities, rather on Wikiversity own certification, which might may advocate itself over the time. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:05, 19 December 2024 (UTC)</text>
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  <page>
    <title>Wikiversity:Sandbox</title>
    <ns>4</ns>
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      <timestamp>2024-12-19T14:29:52Z</timestamp>
      <contributor>
        <ip>152.59.91.60</ip>
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      <comment>nothing</comment>
      <origin>2692624</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="51" sha1="gaw73wv9idvxaaleod4kgpewrss2yz4" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}.</text>
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    <revision>
      <id>2692738</id>
      <parentid>2692624</parentid>
      <timestamp>2024-12-19T22:46:58Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
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      <comment>reset sandbox</comment>
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      <format>text/x-wiki</format>
      <text bytes="50" sha1="phlij3i0lq7l17sacctmpzowd8epftu" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}</text>
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    <title>Wikiversity:Changing username</title>
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      <comment>should link to archive page for anyone who wants to see the original requests on this oage</comment>
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      <text bytes="112" sha1="1ehe88s7qtlj5l1a9vi2auy0k8j6iuu" xml:space="preserve">{{soft redirect|m:Special:GlobalRenameRequest}}

See [[Wikiversity:Changing username/Archive]] for the archives.</text>
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      <comment>edit</comment>
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See [[Wikiversity:Changing username/Archive]] for the archives of this page.</text>
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  <page>
    <title>Template:Index of core filmmaking courses</title>
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        <username>Regis ABAYISENGA</username>
        <id>2995249</id>
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      <text bytes="6250" sha1="4ivb8l2gxf5jy7vjnzvsefraunbih15" xml:space="preserve">[[Image:Mplayer.svg|right|96px]]
{{center top}}&lt;big&gt;Courses in Narrative Film Production&lt;/big&gt;{{center bottom}}
===Filmmaking===

;[[Filmmaking Basics|Course #01]] - A beginning course in pre-production.
* How to make a short motion picture.  Begin learning filmmaking by '''[[Filmmaking Basics/Formatting the Script|formatting the script]]''' using the '''{{font|color=purple|free}}''' demo version of Final Draft or CeltX. Your instructor is [[Special:Emailuser/Regis_ABAYISENGA]]. 


{{center top}}
{| cellpadding=&quot;3&quot; cellspacing=&quot;0&quot;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:red; color:white&quot; | &lt;big&gt;Start Here: &lt;/big&gt;
| style=&quot;background:red; color:red&quot; | .
| style=&quot;background:#ffffe0; color:red&quot; | →  
| style=&quot;background:#ffffff; border: 3px solid yellow; vertical-align: top; -moz-border-radius-topleft: 4px; -moz-border-radius-bottomleft: 4px; -moz-border-radius-topright: 4px; -moz-border-radius-bottomright: 4px;&quot; | &lt;big&gt;[[Filmmaking Basics|Course: Basic filmmaking]]&lt;/big&gt;
|}
{{center bottom}}
&lt;!--
{{center top}}
{| cellpadding=&quot;3&quot; cellspacing=&quot;0&quot;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:red; color:white&quot; | &lt;big&gt;Start Here: &lt;/big&gt;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:#ffffe0; color:red&quot; | →  
| style=&quot;background:#ffffff; border: 3px solid yellow; vertical-align: top; -moz-border-radius-topleft: 4px; -moz-border-radius-bottomleft: 4px; -moz-border-radius-topright: 4px; -moz-border-radius-bottomright: 4px;&quot; | &lt;big&gt;[[Filmmaking Basics/Formatting the Script|First lesson: Formatting the movie script]]&lt;/big&gt;
|}
{{center bottom}}
--&gt;
.

----
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[[Image:Nuvola apps knotify.png|right|96px]]

===Film Scoring===
;[[Film Scoring|Course #02]] - Film scoring for filmmakers (non-musicians)
*[[Film Scoring|Film scoring introduction for filmmakers]] - Lessons for filmmakers who are not musicians yet want to learn to make music for motion pictures.  This class shows you that music is used to create the proper mood in a motion pictures.  Therefore, simple musical sound effects is often all you need.  And with programs like GarageBand with Jam Pack:Symphony Orchestra, this is easy.  Your instructor is [[Special:Emailuser/Robert_Elliott|Robert Elliott]].


{{center top}}
{| cellpadding=&quot;3&quot; cellspacing=&quot;0&quot;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:red; color:white&quot; | &lt;big&gt;Start Here: &lt;/big&gt;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:#ffffe0; color:red&quot; | →  
| style=&quot;background:#ffffff; border: 3px solid yellow; vertical-align: top; -moz-border-radius-topleft: 4px; -moz-border-radius-bottomleft: 4px; -moz-border-radius-topright: 4px; -moz-border-radius-bottomright: 4px;&quot; | &lt;big&gt;[[Film Scoring|Course: Film scoring for filmmakers]]&lt;/big&gt;
|}
{{center bottom}}

.

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[[Image:Nuvola apps ksig.png|right|96px]]
===Script Writing===

;[[Filmmaking for High School Drama Departments|Course #03]] - Filmmaking for high school drama departments 
* [[Filmmaking for High School Drama Departments|Creating a short motion picture by a high school drama department]] - When finished, this course will take you through all the steps of filming a short motion picture by a high school drama department for worldwide distribution. For now, we just discuss script writing.  Your instructor is [[Special:Emailuser/Robert_Elliott|Robert Elliott]].


{{center top}}
{| cellpadding=&quot;3&quot; cellspacing=&quot;0&quot;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:red; color:white&quot; | &lt;big&gt;Start Here: &lt;/big&gt;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:#ffffe0; color:red&quot; | →  
| style=&quot;background:#ffffff; border: 3px solid yellow; vertical-align: top; -moz-border-radius-topleft: 4px; -moz-border-radius-bottomleft: 4px; -moz-border-radius-topright: 4px; -moz-border-radius-bottomright: 4px;&quot; | &lt;big&gt;[[Filmmaking for High School Drama Departments|Course: Script writing for drama departments]]&lt;/big&gt;
|}
{{center bottom}}
 
.
----
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[[Image:Crystal Clear action editcut.png|right|96px]]
===Film Editing - Motion picture style===
;[[Film editing|Course #04]] - Hands-on editing of feature films and television dramas 
* [[Film editing|The fundamentals of narrative film editing]] - Of all the post production skills, film editing is the most important.  The best place to start is learning how to edit dramatic scenes from motion pictures and television dramas.  Your instructor is  [[Special:Emailuser/Robert_Elliott|Robert Elliott]].


{{center top}}
{| cellpadding=&quot;3&quot; cellspacing=&quot;0&quot;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:red; color:white&quot; | &lt;big&gt;Start Here: &lt;/big&gt;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:#ffffe0; color:red&quot; | →  
| style=&quot;background:#ffffff; border: 3px solid yellow; vertical-align: top; -moz-border-radius-topleft: 4px; -moz-border-radius-bottomleft: 4px; -moz-border-radius-topright: 4px; -moz-border-radius-bottomright: 4px;&quot; | &lt;big&gt;[[Film editing|Course: Film editing for dramatic scenes]]&lt;/big&gt;
|}
{{center bottom}}

 
.
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[[Image:Toon Penguin Percy 045 135mm.png|right|96px]]
===The Storyboard Artwork Project===
;[[Storyboard Artwork Project | The Storyboard Artwork Project]] creates artwork for storyboarding 
* [[Storyboard Artwork Project | The Storyboard Artwork Project]] creates artwork for Tux Paint so that kids can quickly and easily create storyboards.  If you use Poser or DAZ Studio, you can help. We also need artists who like to draw human figures. The co-ordinator for this project is  [[Special:Emailuser/Robert_Elliott|Robert Elliott]].


{{center top}}
{| cellpadding=&quot;3&quot; cellspacing=&quot;0&quot;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:red; color:white&quot; | &lt;big&gt;Start Here: &lt;/big&gt;
| style=&quot;background:red; color:red&quot; | . 
| style=&quot;background:#ffffe0; color:red&quot; | →  
| style=&quot;background:#ffffff; border: 3px solid yellow; vertical-align: top; -moz-border-radius-topleft: 4px; -moz-border-radius-bottomleft: 4px; -moz-border-radius-topright: 4px; -moz-border-radius-bottomright: 4px;&quot; | &lt;big&gt;[[Storyboard Artwork Project | Project: Storyboard artwork]]&lt;/big&gt;
|}
{{center bottom}}

.
&lt;includeonly&gt;[[Category:Film School]]&lt;/includeonly&gt;
&lt;noinclude&gt;[[Category:Film School Templates]]&lt;/noinclude&gt;</text>
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    <title>Introduction to Computers/Personal</title>
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      <text bytes="43706" sha1="b6un7zzloj75a1r8hu6l8nj1ouspbq8" xml:space="preserve">&lt;big&gt;Course Navigation&lt;/big&gt;
{| class=&quot;wikitable&quot;
! [[Introduction_to_Computers/Application_software|'''&lt;&lt;  Previous - Application software''']]
!                                                        
!
!
! [[Introduction_to_Computers/Networks|'''Next - Networks  &gt;&gt; ''']]
|}

{{Introduction to computers/header}}

==Receive==
===Time shifting===
Time shifting refers to the saving of a program (audio/visual) to a storage medium for future use. Specific types of softwares are listed below. [[w:Time_shifting]]

====TiVo====
TiVo is a common brand of DVR (Digital Video Recorder) in the United States.  The DVR is a device which allows viewers to record television programs onto a hard disk, enabling the program to be watched at a different time than when it airs. The recording can also be done at the same time as viewing another program; this is called  &quot;time shifting&quot;. [[w:TiVo]] Tivo can be programed to automatically record specific shows, or even entire seasons of shows. Because it is set up with an account, it can be accessed anywhere in the world.

====VOD====

VOD, or Video on Demand is a system that allows viewers to watch videos or clips of videos over a network. It is part of an interactive television system. (Wikipedia 2007).

VOD has a hard drive around 200 hours and is able to save and play back video when you want at your convenience

Video on Demand has its advantage over pay per view because it allows you to start the movie you want as soon as you order. It also allows you to watch the movie as many times as you want in a 24 hr period.

The first commercial VOD service to take place was in Hong Kong in 1990. [[w:Video_on_demand]]

'''Learn More About IPTV:'''

To understand how IPTV is transforming entertainment, check out this comprehensive article on [https://iptvsubscriptionhub.com/the-role-of-iptv-in-modern-entertainment-revolutionizing-how-we-watch-tv/ The Role of IPTV in Modern Entertainment.]

===space shifting===
Is similar to time shifting only in the way of storage.  Space shifted materials are usually converted into a different file type, which leads to legal arguments about copyright infringement.[[w:Space_shifting]]

====MobiTV====
Is a global television and digital radio provider for mobile phones, palm devices, PCs and Windows mobiles.  This service makes it possible for individuals to have access several popular TV and radio stations right from their personal phone!  (wikipedia, 2007) [http://www.mobitv.com/products/products.php]

====TU Media====
TU Media stands for &quot;TV for you&quot;. It was the frist mobile service company and is based in Seoul, South Korea. Currently there are about a half million subscribers[[w:TU_Media]]

===digital radio===
[[Image:digital radio.jpg|thumb|200px|Digital Radio]]
Digital radio is the next generation of radio. It has the capacity to provide listeners with more services, clearer reception and sound quality and a range of other features, including tuning by station name, record and rewind capability, text information, graphics, pictures and web pages. [http://www.abc.net.au/radio/digital/]

'''Digital radio offers:'''

''Better Sound and Reception''

The digital radio receiver locks on to the strongest signal it can find and ignores everything else.

''More Choice''

As digital radio transmission is far more efficient than analogue AM/FM, more stations can be broadcast within the same amount spectrum. This means it has the potential to offer new services to expand listeners' choice - specialist program formats, for example, such as sport or specialist music.

''Easier Tuning''

With digital radio there's no need to remember the frequencies of stations, as you can tune by station name.

''Rewind and Record''

Digital radio receivers are essentially computers that receive and decode a digital program stream into a format that you can hear (and see in the small screen). Like computers, they also have the capacity to store information. Digital radio receivers can therefore pause, re-wind for a few minutes or store audio (via a memory card) for later listening. [http://www.abc.net.au/radio/digital/]

====satellite====
Satellite radio is also called digital radio. This radio service is sent from satelites that are orbiting the earth. Only people that subscribe to this service have access to decoding their signals. Exx. Sirius radio and XM radio
Sirius Canada has approx. 300,000 paid subscribers while XM radio has around 270,000.  (wikipedia, 2007)
Approx. 4 million subscribers in the states. Subscription is approx  $9.95/month - $12.95/month. There are are 2 SDARS (  satellite digital audio radio service ) providers in the states.
Satellite radio allows the subscriber to access more content from different parts of the globe, providing greater variety in their music/talk show selection.

====CD quality====

The quality is as good as a CD!

====hundreds of channels====
Compared to a regular radio, digital radio has many more channels available.

====No Commercials====
No commercials as the radio is sponsored by the satellite provider and of course, the subscribers also share the burden of the cost of operation.

===HD radio===
Is a higher quality of media output compared to the traditional radio.  Using better compression and better delivery methods to a propriety equipment, The client will be able to enjoy a wider range of programs at close to CD quality.
How it operates (four steps):
1. Radio Stations
2. Digital Signal Layer
3. Combined Analog
4. Receivers Reduce Interface

====analog + 2 digital====
Just because you want the latest and greatest, it doesn't mean you have to abandon the old analog format completely.  The new HD radio receivers are often HYBRID which will allow the client to listen to more choices of stations.

====better quality====
The compression of music is increase thus allowing close to CD quality medias to be streamed to the radio.
====hundreds of channels====
1200 stations to choose from AM and FM frequency
====free====
YES! Like traditional radio stations, all the programming in HD Radio both in FM and AM are FREE as they are supported by commercial advertising.

[[w:Hd_radio]]

===internet radio===
[[Image:O0A1.jpg|frame|200px|Internet Radio]]

Internet radio is a free service provided so that you can access a wider range of music just as easily as digital radio. Internet radio can be streamed and played by a software media player in the computer, such as iTunes.
Internet radio offers listeners with a continuous stream of audio that the listener cannot alter. It also provides listeners with the ability to listen in on international stations.[ http://en.wikipedia.org/wiki/Internet_radio]
[http://www.pcmag.com/encyclopedia_term/0,2542,t=Internet+radio&amp;i=45248,00.asp]

====most popular====
The most popular internet radio providers are Yahoo, AOL &amp; MSN.

====hundreds of channels====
Currently there are more than 4,000 broadcasts available on the Internet.
[http://www.pcmag.com/encyclopedia_term/0,2542,t=Internet+radio&amp;i=45248,00.asp]

====free====
Most internet radio broadcasts are free, such as SHOUT, Yahoo and Pandora.
[http://www.google.com/search?sourceid=navclient&amp;ie=UTF-8&amp;rlz=1T4GGLJ_enCA241CA241&amp;q=free+internet+radio]

===MP3 players===

[[File:Sony-NWZ-S616F.jpg|thumb|200px|Sony NWZ-S616F MP3 player]]

An MP3 is a device used to listen to music (or recordings) portably. The most famous of which is an Apple iPod. MP3 is a format that allows audio recordings to be compressed so they are small enough to be sent over the internet or stored as a digital file.

====hard drive vs flash====
A hard drive is much more common than a flash drive as it allows for more downloads, although a flash drive is susceptible to a drive crash.

====sampling rate====
This is the amount of times, in KB/second that a song is recorded and converted into a digital value. The better quality the song (or sampling rate) the more space it will take on your MP3.

====video====
Some MP3 players, such as the Apple iPod offer music video downloads along with the song. This obviously takes a lot more space.

====listening habits====
An MP3 makes listening to music in any environment possible. Some examples are: while working, during a boring lecture, jogging or while being forced to listen to someone's horrible taste in music in the car.
(Selena &amp; Mardi)

It is important because users have access to hundreds of songs while they are on the go!!!!

MP3 Webiste link: [http://www.mp3.com/hardware.php]

===DTV===

====clearer====
Digital television (DTV) is a television system that uses digital signals, rather than analog signals used by analog (traditional) TV.DTV (digital television) is much clearer than analog TV. Picture and sound quality are noticeably better because digital transmissions are free of snow, ghosts, or static noises.

http://dtvfacts.com/

Less prone to interference than an analog TV

====not pure digital====

DTV takes analog broadcast signals, coverts them into digital signals and then coverts them back into analog signals for viewing.

====digital broadcast by 2006====

The Federal Communications Commission (FCC) has mandated that all TV stations must be able to broadcast DTV by 2006. Viewers will be able to get movie-quality pictures, CD-quality sound as well as receive different kinds of information services.

The expected problems are that the old analog tv's will not work after the switch to digital. Some governments have allocated money for people to purchase the converter boxes that will soon be necessary. They also plan to use some of that money for public education on the digital transition.[http://www.pcworld.com/article/id,144139/article.html?tk=nl_dnxnws]

===HDTV===
[[File:Projection-screen-home2.jpg|thumb|200px|Example of HDTV]]

HDTV is short for &quot;high definition television&quot;.  It has a wider screen and a higher resolution than standard resolution.  The term &quot;high definition&quot; was first used in the 1930's and 1940's to describe the television systems in their infancy.  The High Definition works with digital broadcasting signals and has a wider screen and higher resolution than standard resolution.[[Image:144px-HDTV_example_-_Fish_40x46_squares.svg.pngjpg]|thumb|200|] HDTV technology was introduced in the United States in the 1990s by the Digital HDTV Grand Alliance, a group of television companies and MIT.  Terrestrial Analog Broadcasting will be terminated in all full power stations and replaced by Digital broadcasting, as of February 17, 2009.

====Way More Clear====
HDTV makes its picture clearer by using a wider screen and a greater resolution

HDTV allows for smoother motion, more natural colors, surround sound, and allows several input devices to work together. [[w:HDTV#Standard_resolutions]]

====pure digital====
Pure Digital refers to a constant stream of digital view as opposed to streams of digital-analog-digital

====16x9 like movies (not 4x3 like TV)====
Standard televisions have a 4:3 width:height ratio, whereas in contrast a HDTV has a 16:9 width:height ratio.

====expensive ====
Brand new HDTV's can cost anywhere between $500- $2000, but it all depends on the style of the TV, a tube system will cost less than a projection system
[http://tv.about.com/od/highdefinitionhdtv/f/cost_of_HDTV.htm]

====New York Leads in HD Uptake====
New Yorkers are leading in numbers when it comes to the HD movement. A recent survey shows that 17.5 percent of New Yorkers get HD broadcasts, above the 11.3 percent national average. [http://www.pcworld.com/article/id,139124-page,1-c,hdtv/article.html]

It is important because it has a wider screen and resolution than a regular tv.

====Aigo USB Dongle====

This new Dongle toy is one of many HD (High Definition) transmitters being installed in taxis, buses and public places all over China.  It uses a chip which picks up China's new HDTV mobile broadcast network. The network was set up specifically for Beijing's 2008 Olympics.  Just plug it into a laptop's USB port and receive the HD wireless feed.

[http://images.pcworld.com/news/graphics/144127-aigoDongle.jpg click here to see picture]

===SDTV===
Standard-definition television or SDTV refers to television systems that have a resolution that meets standards but not considered either enhanced definition or high definition. The term is usually used in reference to digital television, in particular when broadcasting at the same (or similar) resolution as analog systems.
http://en.wikipedia.org/wiki/Standard-definition_television

The difference between SDTV and HDTV is that the signal on SDTV is more compressed than that of HDTV. As the digital signal is compressed, broadcasters can transmit five SDTV programs, whereas HDTV can only broadcast one. Multiple program broadcasting, called multicasting, was not previously available with analog transmissions. The picture definition of SDTV is also slightly lower than on HDTV.

====DVD clearer (less than HDTV)====
Standard definition is equal to that of a DVD.

Quality of media that does not match high-definition or enhanced definition, so it is refered to as standard. [[w:SDTV]]

====more stations====
Because an HDTV is so expensive, broadcasters will use SDTV instead.

It is important because it enables broadcasters to transmit more info within the bandwidth allowing them to multicast their products.

===Smell-Phones===

Do you need to freshen up your room? Usually, to acquire a pleasant smell, one must either light up a scented candle or obtain the object that produces the desired smell. But wouldn't it be easier to simply press a few buttons on your cell phone? NTT Communication has decided to attempt to create such a phone.

This may seem a bit far fetched, sending smells from cell phone to cell phone. But sounds and images and even videos have traveled with cell phones, why not smells? NTT Communication has found a reasonable, reachable, believable method of allowing smells to travel, too. In their new cell-phones, sixteen base smells are stored. To send a fragrance, one must simply select the scent they would like to send. Then, the sending cell phone gives a &quot;recipe&quot; to the receiving cell phone. This recipe instructs the phone to release certain amounts of certain base smells. These base smells will combine to give off a familiar fragrance, such as roses or vanilla. This is much like how a printer combines the primary colours to make thousands of colours.

Furthermore, a holder of these new cell phones can create new smells with the base smells. If they wish they can post these recipes on the internet for fellow scented cell phone holders to use.

==Create==
===digital camera===
[[Image:800px-Small_sipix_ubt.jpeg|thumb|200px|[[w:Image:Small_sipix_ubt.jpeg|Example of a digital camera]]]A digital camera is an electronic device used to capture and store pictures digitally, instead of using film like regular cameras.
[[w:Digital_camera]] It then can be downloaded onto a computer to be edited and/or printed.[http://www.webopedia.com/TERM/d/digital_camera.html]

Older types of cameras tend to use 35mm photographic tape to store images.  Most digital cameras are multi-functional, being able to record video, sound as well as pictures in digital format. [[w:Digital_camera]]

====point n’ shoot====
Point n' Shoot cameras are (compact cameras) designed primarily for simple operation (eg taking pictures on a trip). Most of them use autofocus or focus free lenses and automatic exposure as well. Point-and-shoots are by far the best selling type of camera. They are really simple to use.
http://en.wikipedia.org/wiki/Point-and-shoot_camera

====SLR====
SLR stands for single lens reflex in digital cameras. This particular design
allows one camera to accommodate a very wide range of lens focal lengths.
SLR has only one lens, and a mirror diverts the image from the lens into the viewfinder; that mirror then retracts when the picture is taken so that the image can be recorded on the light sensor.
http://reviews.cnet.com/4520-7603_7-6241014-1.html
http://en.wikipedia.org/wiki/Point-and-shoot_camera

====megapixels====
a megapixel refers to one million pixels.  Each pixel is a single point or &quot;sample&quot; in a graphic image.  With context to the digital camera, megapixels refers to the number of sensor elements or number of display elements. With a greater number of megapixels, the higher resolution photos a camera will be able to take, because the image will contain more data that translates to greater detail in the photo. Most digitals cameras today have approximately 5-10 MP. (http://www.kenrockwell.com/tech/mpmyth.htm)

====storage====

Flash memory card stores pictures instead of being stored on film.
Flash card come in different formats, such as: Secure Digital, Multimedia, Compact Flash, Memory Sick and Smart Media.
Most digital cameras come with a starter card that holds a handful of photos. You can get reusable cards with at least 1 Gigabyte of storage space holding hundreds images.

====battery life====
====LCD screens====
LCD stands for Liquid Crystal Display. Liquid crystals are lined up between two sheets of glass. They are chemically altered to transmit or block light. LCD is used in digital cameras, cellular phones, and digital clocks.

====3D LCD TVs Could Debut Within Two Years====
A Taiwanese research group has developed 3D (three-dimensional) television picture technology that can be used on LCD (liquid crystal display) TVs as big as 42 inches, and believes such TVs will be on sale within two years. Users would be required to wear a specific polarized set of glasses for a clear 3D picture on the 3D LCD TV.
[http://www.pcworld.com]

====video clips====
The digital camera also has the ability to record images in video format. Whereas before, a digital camera could only shoot about 30 seconds of video, today some digital cameras can shoot as much as 44minutes of video.

====transfer images====
The principal methods for transferring images from the digital camera:
1) By using a direct connection between your camera and computer usually with a USB cable.
2) Insert the memory card into your computer or a card reader.
3) Putting the camera in a cradle attached to the PC
4) Using a photo printer with a built in card reader.
5) Use a portable hard drive-a Tripper.
6) Use a portable CD burner
7) Use a photo lab or a photo-printing kiosk.
[[Image:Example.jpg]]

===PDA===
Stands for 'Personal Digital Assistants' which are basically a pocket sized computer that's used for calculations, calendars, reminders, e-mails etc. [[w:Personal_digital_assistant]]
====flash memory====
A memory storage device in which the data is retained when power has been turned off.( non-volatile memory )  Lead to the introduction of small, light-weight items including PDA's. [http://pdaden.com/guide02.htm]

====synchronizing====
The ability to coordinate data with a PC.  Ensures that information stored on a PDA is up to date and consistent with the host computer. [[w:Personal_digital_assistant#Synchronization]]

===future PDA===
Ex. Smart phones
====TV====
Personal Digital Assistant might be able to play TV shows
Both broadcasting and receiving moving pictures and sound over a distance.
[[w:Television]]

====photographs====
Rather than using a PDA as an organizer, it may also become a handheld device that takes and displays photos, or powerpoint slides
[[w:Photograph]]

====weather meter====
Handheld weather meter can be useful to predict future weather conditions when going swimming, planning to play golf, or even going on a picnic.  Also, a handheld weather meter may even detect wind chill perfect for snowboarders

====GPS====
Wrist watch models of GPS (geo postioning system), a satellite based navigation system allows clients to monitor heart rates when jogging.  PDAs and even regular cellphones also have GPS capability to allow clients to know where they are. GPS locators have already become popular with serious hiking buffs.

===Tablets===

A Tablet PC is a notebook or slate-shaped mobile computer. Its touchscreen or graphics tablet/screen hybrid technology allows the user to operate the computer with a stylus or digital pen, or a fingertip, instead of a keyboard or mouse.A user can enter text using handwriting recognition, an on-screen (virtual) keyboard, speech recognition, or standard keyboard. Shorthand-like entry methods, which enable pen-driven input at speeds comparable to touch-typing.

There are different types of Tablets:

=====Slates=====
Slates, which resemble writing slates, are tablet PCs without a dedicated keyboard. Keyboards can usually be attached via a wireless or USB connection.

=====Thin-client slates=====
Thin-client slates consist of a touchscreen and an integrated wireless connection device. These units by design have limited processing power which is chiefly involved with Input/Output data processing such as video display, network communications, audio encoding/decoding, and input capture (touchscreen input, bar code reading, magnetic stripe reading (credit card swipe). The unit transmits data via a secured wireless connection to a remote server for processing

=====Convertibles=====
Convertible notebooks have a base body with an attached keyboard. They more closely resemble modern notebooks/laptops, and are usually heavier and larger than slates.

=====Hybrids=====
Hybrids share the features of the slate and convertible by using a detachable keyboard which operates in a similar fashion to a convertible when attached.

This is not to be confused with slate models that have a detachable keyboard—detachable keyboards for pure slate models do not rotate around to allow the tablet to rest on it like a convertible.

[[w:Tablet_PC]]

===mini tablet===
The mini tablet is a more advanced version of the tablet PC. The technology has improved to allow it to do things such as store electronic books. It is also significantly smaller, it measures 6&quot; x 8&quot;. [http://courseware.mymrc.ca/webapps/portal/frameset.jsp?tab=courses&amp;url=/bin/common/course.pl?course_id=_49548_1]

===smartphone===

Sim Chips

A Sim Chip is a chip that enables the use of GSM Cell phones in other countries.
This can be beneficial for students, and those in any profession that involves travel.

http://wso.williams.edu/wiki/index.php/Williams_SIM_Chip_Library

A smartphone is a mobile phone offering advanced capabilities mobile phone.
[[w:Smartphone]]

===iphone===

I-phone - The I-Phone is a multi-media device which combines a cellular phone, camera, personal media player,and an internet-enabled computer with a virtual keyboard. It is designed and marketed by Apple inc.

However, this device does not fit the category of UMPC as the end user only have read function and not edit function.  The document suites in the device will only allow you to open without making edit.

iPhone includes an SMS application with a full QWERTY soft keyboard to easily send and receive SMS messages in multiple sessions

===new phone===
Cellphones are now a form of PDA, they double as music players, TVs and personal organizers. Cellphones  are no longer limited to communications. Future cell phones will have the ability to provide live video conversation services. [http://www.pcworld.com/article/144127-1/10_cool_gadgets_you_cant_get_hereyet.html]
[http://mobilitytoday.com/vbmcms/images/a900duo.jpg]

&lt;big&gt;'''Ever dropped your cell phone in the lake or pool or &lt;u&gt;&lt;big&gt;toilet&lt;/big&gt;&lt;/u&gt; and it never worked the same?'''&lt;/big&gt;

           &lt;big&gt;&lt;big&gt;'''You Need This!'''&lt;/big&gt;&lt;/big&gt;

[http://gizmodo.com/347408/fujitsu-f705i-is-worlds-slimmest-waterproof-3g-cellphone]

Fujitsu has come out with the first 3G waterproof cell phone.  It's the Fujitsu F705i.

Features of this awesome phone include:
*waterproof up to 1 minute of submersion
*eight levels of ZOOM to make reading emails easier
*&quot;super clear voice&quot; automatically adjusts the volume of incoming calls to a comfortable and audible level based on the amount of ambient noise
*1.3 megapixel camera
*noise cancellation to drown out background noise

[http://blog.wired.com/gadgets/2008/01/fujitsus-ultrat.html] [http://www.pcworld.com/article/144127-8/10_cool_gadgets_you_cant_get_hereyet.html]

====mainly voice====

====more memory====

====keyboard/stylus====
Finger-operated touch-screens are excellent in the iphone which is easy to use.
[http://www.jeroenmulder.com/weblog/2007/06/iphone_keyboard_trying_too_hard.php]

====LCD screen====
The screen's resolution is 320 pixels and it is 3.5 inches diagonal. It has a special feature called multi-touch, where you can touch the screen to navigate. [http://www.intomobile.com/2007/03/11/apple-iphone-disappointing-lcd-screen.html]

It has a 160 ppi (pixels per inch) pixel density.
[http://www.intomobile.com/2007/03/11/apple-iphone-disappointing-lcd-screen.html]

The current iPod screen is excellent, with crisp 160ppi detail and a remarkable ability to let your mind read &quot;in between the pixels&quot; on album covers
[http://www.crazygadgetguru.com/posts/2007/6/5/iphone-lcd-screen-in-perspective.html]

====E-mail====
The iPhone also features an e-mail program that supports HTML e-mail, which enables the user to embed photos in an e-mail message. PDF, Microsoft Word, and Microsoft Excel attachments to mail messages can be viewed on the phone. Yahoo is currently the only e-mail provider.
[[w:IPhone]]

the iPhone syncs to your address book
[http://www.pcworld.com/article/id,137138/article.html]

====web====
====camera====
The iphone is equipped with a 2.0 megapixel camera. The images are clear in well lit areas; however, it lacks a flash and is not able to zoom. [http://www.apple.com/iphone/specs.html]

====music/video====
The iphone Supports a veriety of media. It is able to support such video formats as *.m4v, *.mp4,*.mov file and Music formats as AAC, Protected *AAC, *MP3, *MP3 *VBR, Audible formats 1, 2, and 3, Apple Lossless, *AIFF, and *WAV. [http://www.apple.com/iphone/specs.html]

====iTouch====
The NEW itouch is very similar to the iPhone. The only difference is that iTouch has a touch-screen with a built in wifi syste. This means that the iTouch is an iPod with WiFi capability. You could say that the iPhone is superior as it comes with the same features plus the fact that it can be used as a phone as well.

===== iPhone to Become Blu-Ray Player Remote  =====
[[Image:bluphone.jpg|400px|]]

According to NetBlender, iPhone and iPod touch users will be able to control their Blu-ray players using an application called BD Touch. The application will use the network capabilities of Blu-ray hardware and Apple's handheld devices to transfer data, allowing you to do many different things beyond controlling movie playback.

It will include a number of features that will be supported by the technology, such as automatically updating film collections held on an iPhone, including a feature which lets Blu-ray players/discs send a digital copy of a video to an iPhone.

[http://www.pcworld.com/article/id,144250/article.html?tk=nl_dnxnws]
[http://gizmodo.com/377290/iphone-to-become-blu+ray-player-remote]

==Connection==
===network===
====tree-and-branch====
[[Image:800px-HFC_Network_Diagram.png|frame|[[w:Image:HFC_Network_Diagram.png|Cable example]]]Broadcasting, where one central location delivers content simultaneously to many users (cable TV [[w:Hybrid_fibre-coaxial]], radio)
====switched-network====
users contact specific users as in a telephone network [[w:Public_switched_telephone_network]]
===file sharing===
====legal entanglements====
====hackers invade your computer====
====download virus====

==gaming==
===2nd life===
Second Life is an Internet-based virtual world which came to international attention via mainstream news media in late 2006 and early 2007. A downloadable client program called the Second Life Viewer enables its users, called &quot;Residents&quot;, to interact with each other, providing an advanced level of a social network service combined with general aspects of a metaverse. Residents can explore, meet other Residents, socialize, participate in individual and group activities, create and trade items (virtual property) and services from one another.[[w:Second_Life]]
A similar program to the one Second Life will be adding is called BrainGate.  It is a brain implant system designed to help those who have lost control of their limbs, or other bodily functions, such as patients with (ALS) or spinal cord injury. A computer chip is implanted into the brain, it monitors brain activity in the patient and converts the intention of the user into computer commands. Hair-thin electrodes sense the electro-magnetic signature of neurons firing in specific areas of the brain, the activity is translated into electrically charged signals and are then sent and decoded using a program, which can move either a robotic arm or a computer cursor.[[w:BrainGate]]

Second life is a program that can be used successfully with those who are paralyzed to stimulate mobility, it could also trace brain activity and arrest disease, it is capable of interpreting intention and commercial transations could be activated with the use of Linden dollars which can be later exchanged for US dollars.

Second Life can be used in a distance learning technology where virtual (but real) you can take classes; get a real filling of meeting other students, borrow books from libraries, do Tai Chi , go to the beach. This is very useful for people who are not able to go to the class or for people who don’t want to leave their homes.
http://www.kqed.org/quest/television/view/611
(http://www.pcworld.com/article/id,139969-c,technology/article.html)

===Xbox===
[[Image:Xboxarcadesku.jpg|thumb|200px|xbox360]]
Xbox the first game console developed by Microsoft It comes with a large hard drive, wireless controller and wired headset.
Xbox 360 is the latest version and has the best graphics of any video console on the market. This gaming unit is excellent for all role playing games, sports and learning software.
(http://www.xbox.com/en-CA/)
[http://www.xbox.com/en-US/hardware/compare101.htm?WT.svl=nav]
[[w:Xbox_360]]
====online video games====

Xbox has an online gaming platform called xbox live. People have to subscribe to play multiplayer with members from all around the world.  With the exception of single player specific games, all of Xbox360 titles now are online compatiable. XBOX live also allows the player to interact with others through communication devices. XBOX Live is convienient in the sense that there is no monthly fee, it is annual. [[w:Xbox#Multimedia]]
[http://www.xbox.com/en-US/live/?WT.svl=nav]

There are future talks of Microsoft merging the Xbox live with PC user platform with the &quot;GAMES for PC&quot; roll out.  However, this will give the mouse and keyboard combination an advantage over game pad users on virtually all games.

====media hub of house====
This is a system for extender play which is pre installed in all the x-box 360s. Includes movies, music, videos and pictures.
[http://www.xbox.com/en-GB/hardware/xbox360/benefits/mediacentre.htm]

===PS3===
[[File:PS3s and controllers at E3 2006.jpg|thumb|200px|PlayStation 3 at E3 2006]]

The PlayStation 3 is the third iteration of a home console produced by Sony.  The PlayStation 3 offers a unified online gaming service known as the PlayStation Network.  Other features include connectivity with the PlayStation Portable, Blu-ray disc and large hard-drive capacity for a console. The PlayStation 3 was first released on November 11, 2006 in Japan, November 17, 2006 in North America, and March 23, 2007 in Europe. [[w:PlayStation_3]]

====movie quality graphics====

The PS3 features a Full HD (up to 1080p) x 2 channels

[http://ps3.gamespy.com/articles/614/614972p3.html]

===PSP===

====Specs.====
[[Image:Psp1.png|thumb|200px|PSP]]
The PSP (Play Station Portable) is a handheld portable consel that consists of video, music, games and internet.

[[w:PlayStation_Portable#System_Software|Reference Wikisite]]

====functions====
The functions are:
-video capibilities: which you can buy (UMDs) and/or download (MPEGs)
-music: download (MP3s)
-access internet through WI-FI
-pictures (JPEGs)
-main function is to play games (UMDs- Universal Media Disc)
[[w:PlayStation_Portable]]

====advantages====
1.  Many different variations and accessories (head set, carrying case, cleaning cloth, wrist strap, head phones, etc.)
2.  New and exciting colours
3.  More games
4.  Download games
5.  Play Back Movies
6.  Access to Wireless Networks
7.  Share Games with Friends
8.  Slimmer and Lighter

====disadvantages====
1.  Screen is smaller
2.  Only 2 hours of game play
3.  You must buy your own memory cards
4.  Fairly heavy
5.  Design is not very attractive
6.  Can't hook it up to your T.V. or stereo

====cell interacts====

The ps3 cell is a 9 core processor, one of these cores is a PowerPC and acts as a controller.  The remaining 8 cores are called SPEs and these are very high performance vector processors.  Each SPE contains it's own block of high speed RAM and is capable of 32 GigaFlops.  The SPEs are independent processors and can act alone or can be set up to process a stream of data with different SPEs working on different stages.
[http://www.blachford.info/computer/Cell/Cell0_v2.html]

===Nintendo===
====simpler====
Nintendo Wii is fitted with a motion sensor so it allow players to simplify the game controls comparing to the traditional game pad. The games that are available on the Wii platform are often very intuitive and can be picked up by people of all ages.

This is a game system that kids/teens like to use.

====cheaper====
It cost less than $250 US [http://www.gamespot.com/ds/action/supermariobrosds/news.html?sid=6151827&amp;cpage=5]
The first Nintendo was marketed for $199 US.
[http://wiki.answers.com/Q/How_much_did_the_first_nintendo_cost]

====wireless motion ====
The Wireless Remote is motion sensing capability. It allows the user to interact with items on screen via movement and pointing through the use of accelerometer and optical sensor technology.
[http://www.wikipedia.org]

====sensor console====
Unlike a light gun that senses light from a television screen, the Wii Remote senses light from the consoles Sensor Bar (model number RVL-014), which allows consistent usage regardless of a television's type or size.
[http://www.wikipedia.org]

====Smart Downloads====
Japan has launched a download channel which allows Wii users to download game demos, view promos, and see screen shots free of charge. It downloads temporary information into  hand held internal memory and is deleted once the unit is turned off.  It is also smart because it recommends game choice based on your playing record, your age and gender preferences. [http://www.pcworld.com/article/id,140011-c,gameconsoles/article.html#]

Other benefits for Wii owners is that, they are able to view:Commercials, box art, game play footage screen shots, and interviews about various Nintendo products including prices and release dates for upcoming DS and Wii games. [http://www.pcworld.com/article/id,140011-c,gameconsoles/article.html]


===Wii===

[[Image:Wii_Wiimotea.png|thumb]]
====Description====
The Wii is the fifth home video game console released by nintendo.  The distinguishing feature of the Wii is it wireless controller (the wii remote) which can be used as a handheld pointing device and can detect acceleration in three dimensions.  There have been many speculations as to why the company chose Wii for the name, the official reason is “ Wii sounds like 'we', which emphasizes that the console is for everyone. Wii can easily be remembered by people around the world, no matter what language they speak. No confusion. No need to abbreviate. Just Wii.[11] ”.
The Wii remote is the main controler for the console.  It uses built-in accelerometers and infrared detection to sense its position in 3D space when its pointed at the LED lights within the sensor bar. It connects to the concole using bluetooth.

Even Microsoft has noticed Nintendos success with their new Bluetooth infused controllers. In late 2008 Microsoft plans to release a &quot;Wii&quot; like controller for their current gaming consol, the Xbox 360. http://www.pcworld.com/article/id,144223/article.html?tk=nl_dnxnws

[[w:Wii]]
They are a great alternative for kids and families as the games are generally rated E for everyone and it keeps kids active while they are gaming.

====Cost====
The Wii's price at the time of introduction was approximately $249.99, with the remote costing $39.99 and the individulal games costing $49.99. Prices have been $100 - $200 more than the asking price because of the high demand, but do expect to go down to the average of $250 soon.
[http://wii.ign.com/articles/732/732669p1.html]
[http://www.pcworld.com/article/id,139949-c,gameconsoles/article.html]

====Supply====
Wii is available for purchase at all major department stores.  There have been some speculations that Nintendo has been producing fewer units than needed to meet demand, thus increasing the price people will pay for it.  There are approximately 1.8 millions Wii units produced per month.
[http://www.pcworld.com/article/id,139949-c,gameconsoles/article.html]

====Demand====
Since first launching the Wii in November of last year, Nintendo has seen widespread shortages at retail. The demand has been so high that Nintendo says it is working at maximum capacity in producing 1.8 million Wii units per month, hopefully in time for Christmas.
[http://www.pcworld.com/article/id,139949-c,gameconsoles/article.html]

[[Category: Introduction to Computers]]

==Safety and Ergonomics==
Prolonged use of computers can be a health risk to your body.

===eye===

[[Image:Iris.eye.225px.jpg|frame|eye suffering from CVS]]The eye can also be a victim and symptom of extended use on the computer.  This eye condition also called &quot;Computer Vision Syndrome&quot; or CVS and is a member of the family referred to RSI or Repetitive Strain Injury.  Prolonged periods of continuous focusing on a computer screen for uninterrupted periods of time.  Aggravating the condition can be improper light or air moving past the eye from overhead vents or fans.  Symptoms of the condition can include headaches, blurred vision, neck pain, fatigue, eye strain, dry, irritated and difficulty on refocusing[http://www.allaboutvision.com/cvs/faqs.htm].

A couple of ways to treat the condition is to allow time to rest your eye, redirecting your focus away from the screen, also the purchase of over the counter eye drops are both easy solutions.  There is a catchphrase that has proven to be a well know solution among &quot;computerites&quot;.  It is called 20-20-20 ~ focus your eyes on an object 20 feet away for 20 seconds every 20 minutes.

===neck/back===
'''''Neck and back pain are common complaints. The cause is usually poor posture, which decreases blood flow to certain muscles. These muscles stiffen up and hurt.'''''&lt;ref&gt;http://www.slais.ubc.ca/COURSES/libr500/02-03-wt1/www/A_Davis/neckback.htm&lt;/ref&gt;

'''''Neck Pain:'''''
Often begins gradually as a result of fixed staring at a small area or glancing repeatedly from one to another (from the screen to a document on your desk for example).  If the head is held at an angle greater than 15 degrees (for example holding the phone between your neck or shoulder, or looking down at your keyboard) will cause greater muscular fatigue and pain will become apparent more rapidly &lt;ref&gt;Stigliani, Joan. The Computer User's Survival Guide. Sebastopol, CA: O'Reilly &amp; Associates, Inc. 1995.&lt;/ref&gt;

'''''Back Pain:'''''
Sitting is one of the hardest positions in which to maintain proper posture, and many computer users regularly feel back pain.  Spinal compression is one of the most common problems because sitting tends to tilt the pelvis backward, flattening the lumbar curve and resulting in uneven and increased pressure on spinal disks. &lt;ref&gt;Sellers, Don. Zap! How Your Computer Can Hurt You – And What You Can Do About It. Edited by Stephen F. Roth.  Berkeley: Peach Pit Press, 1994.&lt;/ref&gt;

'''''Prevention'''''
*Be sure you have a proper workstation set-up.
*Take active breaks, move around and do a few stretches.
*Shift positions every now and then.
*Try not to fall habitually into one computer position – even small changes help avoid overtaxing certain muscles.
*Try not to round your shoulders – this puts extra pressure on your upper spine.
*Stay active, get up and move around to circulate your blood.  Sitting still for too long can slow blood circulation and muscle fatigue can set in.&lt;ref&gt;Sellers, Don. Zap! How Your Computer Can Hurt You – And What You Can Do About It. Edited by Stephen F. Roth.  Berkeley: Peach Pit Press, 1994.&lt;/ref&gt;
]]

===wrist===
RSI (also know as cumulative trauma disorder) is a soft-tissue injury in which muscles, nerves, or tendons become irritated or inflamed. RSI is caused by repetitive motions, excessive force, and extremes of motion. Over time these motions can strain the soft tissues, reducing circulation. These stresses create tiny tears in the muscles and tendons, which become inflamed. In extreme cases it can cause permanent tissue damage and disability.

The computer keyboard is an ergonomic nightmare. It tends to force you into an unnatural palms-down (pronated) wrist-cocked position. This strains the delicate muscles and tendons of the fingers and wrists, reducing circulation.

[http://www.webreference.com/rsi.html]

Carpal tunnel syndrome as referred to by many different websites is one of the main injuries associated with the wrist. Daily exercises and breaks away from the repetitive typing of the computer can help alleviate some of the symptoms of carpal tunnel syndrome.

'''Symptoms'''
usually start gradually, with frequent burning, tingling, or itching numbness in the palm of the hand and the fingers, especially the thumb and the index and middle fingers. Some carpal tunnel sufferers say their fingers feel useless and swollen, even though little or no swelling is apparent. The symptoms often first appear in one or both hands during the night, since many people sleep with flexed wrists. A person with carpal tunnel syndrome may wake up feeling the need to &quot;shake out&quot; the hand or wrist. As symptoms worsen, people might feel tingling during the day. Decreased grip strength may make it difficult to form a fist, grasp small objects, or perform other manual tasks. In chronic and/or untreated cases, the muscles at the base of the thumb may waste away. Some people are unable to tell between hot and cold by touch.

'''Prevention'''
At the workplace, workers can do on-the-job conditioning, perform stretching exercises, take frequent rest breaks, wear splints to keep wrists straight, and use correct posture and wrist position. Wearing fingerless gloves can help keep hands warm and flexible. Workstations, tools and tool handles, and tasks can be redesigned to enable the worker's wrist to maintain a natural position during work. Jobs can be rotated among workers. Employers can develop programs in ergonomics, the process of adapting workplace conditions and job demands to the capabilities of workers. However, research has not conclusively shown that these workplace changes prevent the occurrence of carpal tunnel syndrome.

[http://www.ninds.nih.gov/disorders/carpal_tunnel/detail_carpal_tunnel.htm]

]]


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{{Introduction to computers/header}}

==Receive==
===Time shifting===
Time shifting refers to the saving of a program (audio/visual) to a storage medium for future use. Specific types of softwares are listed below. [[w:Time_shifting]]

====TiVo====
TiVo is a common brand of DVR (Digital Video Recorder) in the United States.  The DVR is a device which allows viewers to record television programs onto a hard disk, enabling the program to be watched at a different time than when it airs. The recording can also be done at the same time as viewing another program; this is called  &quot;time shifting&quot;. [[w:TiVo]] Tivo can be programed to automatically record specific shows, or even entire seasons of shows. Because it is set up with an account, it can be accessed anywhere in the world.

====VOD====

VOD, or Video on Demand is a system that allows viewers to watch videos or clips of videos over a network. It is part of an interactive television system. (Wikipedia 2007).

VOD has a hard drive around 200 hours and is able to save and play back video when you want at your convenience

Video on Demand has its advantage over pay per view because it allows you to start the movie you want as soon as you order. It also allows you to watch the movie as many times as you want in a 24 hr period.

The first commercial VOD service to take place was in Hong Kong in 1990. [[w:Video_on_demand]]

===space shifting===
Is similar to time shifting only in the way of storage.  Space shifted materials are usually converted into a different file type, which leads to legal arguments about copyright infringement.[[w:Space_shifting]]

====MobiTV====
Is a global television and digital radio provider for mobile phones, palm devices, PCs and Windows mobiles.  This service makes it possible for individuals to have access several popular TV and radio stations right from their personal phone!  (wikipedia, 2007) [http://www.mobitv.com/products/products.php]

====TU Media====
TU Media stands for &quot;TV for you&quot;. It was the frist mobile service company and is based in Seoul, South Korea. Currently there are about a half million subscribers[[w:TU_Media]]

===digital radio===
[[Image:digital radio.jpg|thumb|200px|Digital Radio]]
Digital radio is the next generation of radio. It has the capacity to provide listeners with more services, clearer reception and sound quality and a range of other features, including tuning by station name, record and rewind capability, text information, graphics, pictures and web pages. [http://www.abc.net.au/radio/digital/]

'''Digital radio offers:'''

''Better Sound and Reception''

The digital radio receiver locks on to the strongest signal it can find and ignores everything else.

''More Choice''

As digital radio transmission is far more efficient than analogue AM/FM, more stations can be broadcast within the same amount spectrum. This means it has the potential to offer new services to expand listeners' choice - specialist program formats, for example, such as sport or specialist music.

''Easier Tuning''

With digital radio there's no need to remember the frequencies of stations, as you can tune by station name.

''Rewind and Record''

Digital radio receivers are essentially computers that receive and decode a digital program stream into a format that you can hear (and see in the small screen). Like computers, they also have the capacity to store information. Digital radio receivers can therefore pause, re-wind for a few minutes or store audio (via a memory card) for later listening. [http://www.abc.net.au/radio/digital/]

====satellite====
Satellite radio is also called digital radio. This radio service is sent from satelites that are orbiting the earth. Only people that subscribe to this service have access to decoding their signals. Exx. Sirius radio and XM radio
Sirius Canada has approx. 300,000 paid subscribers while XM radio has around 270,000.  (wikipedia, 2007)
Approx. 4 million subscribers in the states. Subscription is approx  $9.95/month - $12.95/month. There are are 2 SDARS (  satellite digital audio radio service ) providers in the states.
Satellite radio allows the subscriber to access more content from different parts of the globe, providing greater variety in their music/talk show selection.

====CD quality====

The quality is as good as a CD!

====hundreds of channels====
Compared to a regular radio, digital radio has many more channels available.

====No Commercials====
No commercials as the radio is sponsored by the satellite provider and of course, the subscribers also share the burden of the cost of operation.

===HD radio===
Is a higher quality of media output compared to the traditional radio.  Using better compression and better delivery methods to a propriety equipment, The client will be able to enjoy a wider range of programs at close to CD quality.
How it operates (four steps):
1. Radio Stations
2. Digital Signal Layer
3. Combined Analog
4. Receivers Reduce Interface

====analog + 2 digital====
Just because you want the latest and greatest, it doesn't mean you have to abandon the old analog format completely.  The new HD radio receivers are often HYBRID which will allow the client to listen to more choices of stations.

====better quality====
The compression of music is increase thus allowing close to CD quality medias to be streamed to the radio.
====hundreds of channels====
1200 stations to choose from AM and FM frequency
====free====
YES! Like traditional radio stations, all the programming in HD Radio both in FM and AM are FREE as they are supported by commercial advertising.

[[w:Hd_radio]]

===internet radio===
[[Image:O0A1.jpg|frame|200px|Internet Radio]]

Internet radio is a free service provided so that you can access a wider range of music just as easily as digital radio. Internet radio can be streamed and played by a software media player in the computer, such as iTunes.
Internet radio offers listeners with a continuous stream of audio that the listener cannot alter. It also provides listeners with the ability to listen in on international stations.[ http://en.wikipedia.org/wiki/Internet_radio]
[http://www.pcmag.com/encyclopedia_term/0,2542,t=Internet+radio&amp;i=45248,00.asp]

====most popular====
The most popular internet radio providers are Yahoo, AOL &amp; MSN.

====hundreds of channels====
Currently there are more than 4,000 broadcasts available on the Internet.
[http://www.pcmag.com/encyclopedia_term/0,2542,t=Internet+radio&amp;i=45248,00.asp]

====free====
Most internet radio broadcasts are free, such as SHOUT, Yahoo and Pandora.
[http://www.google.com/search?sourceid=navclient&amp;ie=UTF-8&amp;rlz=1T4GGLJ_enCA241CA241&amp;q=free+internet+radio]

===MP3 players===

[[File:Sony-NWZ-S616F.jpg|thumb|200px|Sony NWZ-S616F MP3 player]]

An MP3 is a device used to listen to music (or recordings) portably. The most famous of which is an Apple iPod. MP3 is a format that allows audio recordings to be compressed so they are small enough to be sent over the internet or stored as a digital file.

====hard drive vs flash====
A hard drive is much more common than a flash drive as it allows for more downloads, although a flash drive is susceptible to a drive crash.

====sampling rate====
This is the amount of times, in KB/second that a song is recorded and converted into a digital value. The better quality the song (or sampling rate) the more space it will take on your MP3.

====video====
Some MP3 players, such as the Apple iPod offer music video downloads along with the song. This obviously takes a lot more space.

====listening habits====
An MP3 makes listening to music in any environment possible. Some examples are: while working, during a boring lecture, jogging or while being forced to listen to someone's horrible taste in music in the car.
(Selena &amp; Mardi)

It is important because users have access to hundreds of songs while they are on the go!!!!

MP3 Webiste link: [http://www.mp3.com/hardware.php]

===DTV===

====clearer====
Digital television (DTV) is a television system that uses digital signals, rather than analog signals used by analog (traditional) TV.DTV (digital television) is much clearer than analog TV. Picture and sound quality are noticeably better because digital transmissions are free of snow, ghosts, or static noises.

http://dtvfacts.com/

Less prone to interference than an analog TV

====not pure digital====

DTV takes analog broadcast signals, coverts them into digital signals and then coverts them back into analog signals for viewing.

====digital broadcast by 2006====

The Federal Communications Commission (FCC) has mandated that all TV stations must be able to broadcast DTV by 2006. Viewers will be able to get movie-quality pictures, CD-quality sound as well as receive different kinds of information services.

The expected problems are that the old analog tv's will not work after the switch to digital. Some governments have allocated money for people to purchase the converter boxes that will soon be necessary. They also plan to use some of that money for public education on the digital transition.[http://www.pcworld.com/article/id,144139/article.html?tk=nl_dnxnws]

===HDTV===
[[File:Projection-screen-home2.jpg|thumb|200px|Example of HDTV]]

HDTV is short for &quot;high definition television&quot;.  It has a wider screen and a higher resolution than standard resolution.  The term &quot;high definition&quot; was first used in the 1930's and 1940's to describe the television systems in their infancy.  The High Definition works with digital broadcasting signals and has a wider screen and higher resolution than standard resolution.[[Image:144px-HDTV_example_-_Fish_40x46_squares.svg.pngjpg]|thumb|200|] HDTV technology was introduced in the United States in the 1990s by the Digital HDTV Grand Alliance, a group of television companies and MIT.  Terrestrial Analog Broadcasting will be terminated in all full power stations and replaced by Digital broadcasting, as of February 17, 2009.

====Way More Clear====
HDTV makes its picture clearer by using a wider screen and a greater resolution

HDTV allows for smoother motion, more natural colors, surround sound, and allows several input devices to work together. [[w:HDTV#Standard_resolutions]]

====pure digital====
Pure Digital refers to a constant stream of digital view as opposed to streams of digital-analog-digital

====16x9 like movies (not 4x3 like TV)====
Standard televisions have a 4:3 width:height ratio, whereas in contrast a HDTV has a 16:9 width:height ratio.

====expensive ====
Brand new HDTV's can cost anywhere between $500- $2000, but it all depends on the style of the TV, a tube system will cost less than a projection system
[http://tv.about.com/od/highdefinitionhdtv/f/cost_of_HDTV.htm]

====New York Leads in HD Uptake====
New Yorkers are leading in numbers when it comes to the HD movement. A recent survey shows that 17.5 percent of New Yorkers get HD broadcasts, above the 11.3 percent national average. [http://www.pcworld.com/article/id,139124-page,1-c,hdtv/article.html]

It is important because it has a wider screen and resolution than a regular tv.

====Aigo USB Dongle====

This new Dongle toy is one of many HD (High Definition) transmitters being installed in taxis, buses and public places all over China.  It uses a chip which picks up China's new HDTV mobile broadcast network. The network was set up specifically for Beijing's 2008 Olympics.  Just plug it into a laptop's USB port and receive the HD wireless feed.

[http://images.pcworld.com/news/graphics/144127-aigoDongle.jpg click here to see picture]

===SDTV===
Standard-definition television or SDTV refers to television systems that have a resolution that meets standards but not considered either enhanced definition or high definition. The term is usually used in reference to digital television, in particular when broadcasting at the same (or similar) resolution as analog systems.
http://en.wikipedia.org/wiki/Standard-definition_television

The difference between SDTV and HDTV is that the signal on SDTV is more compressed than that of HDTV. As the digital signal is compressed, broadcasters can transmit five SDTV programs, whereas HDTV can only broadcast one. Multiple program broadcasting, called multicasting, was not previously available with analog transmissions. The picture definition of SDTV is also slightly lower than on HDTV.

====DVD clearer (less than HDTV)====
Standard definition is equal to that of a DVD.

Quality of media that does not match high-definition or enhanced definition, so it is refered to as standard. [[w:SDTV]]

====more stations====
Because an HDTV is so expensive, broadcasters will use SDTV instead.

It is important because it enables broadcasters to transmit more info within the bandwidth allowing them to multicast their products.

===Smell-Phones===

Do you need to freshen up your room? Usually, to acquire a pleasant smell, one must either light up a scented candle or obtain the object that produces the desired smell. But wouldn't it be easier to simply press a few buttons on your cell phone? NTT Communication has decided to attempt to create such a phone.

This may seem a bit far fetched, sending smells from cell phone to cell phone. But sounds and images and even videos have traveled with cell phones, why not smells? NTT Communication has found a reasonable, reachable, believable method of allowing smells to travel, too. In their new cell-phones, sixteen base smells are stored. To send a fragrance, one must simply select the scent they would like to send. Then, the sending cell phone gives a &quot;recipe&quot; to the receiving cell phone. This recipe instructs the phone to release certain amounts of certain base smells. These base smells will combine to give off a familiar fragrance, such as roses or vanilla. This is much like how a printer combines the primary colours to make thousands of colours.

Furthermore, a holder of these new cell phones can create new smells with the base smells. If they wish they can post these recipes on the internet for fellow scented cell phone holders to use.

==Create==
===digital camera===
[[Image:800px-Small_sipix_ubt.jpeg|thumb|200px|[[w:Image:Small_sipix_ubt.jpeg|Example of a digital camera]]]A digital camera is an electronic device used to capture and store pictures digitally, instead of using film like regular cameras.
[[w:Digital_camera]] It then can be downloaded onto a computer to be edited and/or printed.[http://www.webopedia.com/TERM/d/digital_camera.html]

Older types of cameras tend to use 35mm photographic tape to store images.  Most digital cameras are multi-functional, being able to record video, sound as well as pictures in digital format. [[w:Digital_camera]]

====point n’ shoot====
Point n' Shoot cameras are (compact cameras) designed primarily for simple operation (eg taking pictures on a trip). Most of them use autofocus or focus free lenses and automatic exposure as well. Point-and-shoots are by far the best selling type of camera. They are really simple to use.
http://en.wikipedia.org/wiki/Point-and-shoot_camera

====SLR====
SLR stands for single lens reflex in digital cameras. This particular design
allows one camera to accommodate a very wide range of lens focal lengths.
SLR has only one lens, and a mirror diverts the image from the lens into the viewfinder; that mirror then retracts when the picture is taken so that the image can be recorded on the light sensor.
http://reviews.cnet.com/4520-7603_7-6241014-1.html
http://en.wikipedia.org/wiki/Point-and-shoot_camera

====megapixels====
a megapixel refers to one million pixels.  Each pixel is a single point or &quot;sample&quot; in a graphic image.  With context to the digital camera, megapixels refers to the number of sensor elements or number of display elements. With a greater number of megapixels, the higher resolution photos a camera will be able to take, because the image will contain more data that translates to greater detail in the photo. Most digitals cameras today have approximately 5-10 MP. (http://www.kenrockwell.com/tech/mpmyth.htm)

====storage====

Flash memory card stores pictures instead of being stored on film.
Flash card come in different formats, such as: Secure Digital, Multimedia, Compact Flash, Memory Sick and Smart Media.
Most digital cameras come with a starter card that holds a handful of photos. You can get reusable cards with at least 1 Gigabyte of storage space holding hundreds images.

====battery life====
====LCD screens====
LCD stands for Liquid Crystal Display. Liquid crystals are lined up between two sheets of glass. They are chemically altered to transmit or block light. LCD is used in digital cameras, cellular phones, and digital clocks.

====3D LCD TVs Could Debut Within Two Years====
A Taiwanese research group has developed 3D (three-dimensional) television picture technology that can be used on LCD (liquid crystal display) TVs as big as 42 inches, and believes such TVs will be on sale within two years. Users would be required to wear a specific polarized set of glasses for a clear 3D picture on the 3D LCD TV.
[http://www.pcworld.com]

====video clips====
The digital camera also has the ability to record images in video format. Whereas before, a digital camera could only shoot about 30 seconds of video, today some digital cameras can shoot as much as 44minutes of video.

====transfer images====
The principal methods for transferring images from the digital camera:
1) By using a direct connection between your camera and computer usually with a USB cable.
2) Insert the memory card into your computer or a card reader.
3) Putting the camera in a cradle attached to the PC
4) Using a photo printer with a built in card reader.
5) Use a portable hard drive-a Tripper.
6) Use a portable CD burner
7) Use a photo lab or a photo-printing kiosk.
[[Image:Example.jpg]]

===PDA===
Stands for 'Personal Digital Assistants' which are basically a pocket sized computer that's used for calculations, calendars, reminders, e-mails etc. [[w:Personal_digital_assistant]]
====flash memory====
A memory storage device in which the data is retained when power has been turned off.( non-volatile memory )  Lead to the introduction of small, light-weight items including PDA's. [http://pdaden.com/guide02.htm]

====synchronizing====
The ability to coordinate data with a PC.  Ensures that information stored on a PDA is up to date and consistent with the host computer. [[w:Personal_digital_assistant#Synchronization]]

===future PDA===
Ex. Smart phones
====TV====
Personal Digital Assistant might be able to play TV shows
Both broadcasting and receiving moving pictures and sound over a distance.
[[w:Television]]

====photographs====
Rather than using a PDA as an organizer, it may also become a handheld device that takes and displays photos, or powerpoint slides
[[w:Photograph]]

====weather meter====
Handheld weather meter can be useful to predict future weather conditions when going swimming, planning to play golf, or even going on a picnic.  Also, a handheld weather meter may even detect wind chill perfect for snowboarders

====GPS====
Wrist watch models of GPS (geo postioning system), a satellite based navigation system allows clients to monitor heart rates when jogging.  PDAs and even regular cellphones also have GPS capability to allow clients to know where they are. GPS locators have already become popular with serious hiking buffs.

===Tablets===

A Tablet PC is a notebook or slate-shaped mobile computer. Its touchscreen or graphics tablet/screen hybrid technology allows the user to operate the computer with a stylus or digital pen, or a fingertip, instead of a keyboard or mouse.A user can enter text using handwriting recognition, an on-screen (virtual) keyboard, speech recognition, or standard keyboard. Shorthand-like entry methods, which enable pen-driven input at speeds comparable to touch-typing.

There are different types of Tablets:

=====Slates=====
Slates, which resemble writing slates, are tablet PCs without a dedicated keyboard. Keyboards can usually be attached via a wireless or USB connection.

=====Thin-client slates=====
Thin-client slates consist of a touchscreen and an integrated wireless connection device. These units by design have limited processing power which is chiefly involved with Input/Output data processing such as video display, network communications, audio encoding/decoding, and input capture (touchscreen input, bar code reading, magnetic stripe reading (credit card swipe). The unit transmits data via a secured wireless connection to a remote server for processing

=====Convertibles=====
Convertible notebooks have a base body with an attached keyboard. They more closely resemble modern notebooks/laptops, and are usually heavier and larger than slates.

=====Hybrids=====
Hybrids share the features of the slate and convertible by using a detachable keyboard which operates in a similar fashion to a convertible when attached.

This is not to be confused with slate models that have a detachable keyboard—detachable keyboards for pure slate models do not rotate around to allow the tablet to rest on it like a convertible.

[[w:Tablet_PC]]

===mini tablet===
The mini tablet is a more advanced version of the tablet PC. The technology has improved to allow it to do things such as store electronic books. It is also significantly smaller, it measures 6&quot; x 8&quot;. [http://courseware.mymrc.ca/webapps/portal/frameset.jsp?tab=courses&amp;url=/bin/common/course.pl?course_id=_49548_1]

===smartphone===

Sim Chips

A Sim Chip is a chip that enables the use of GSM Cell phones in other countries.
This can be beneficial for students, and those in any profession that involves travel.

http://wso.williams.edu/wiki/index.php/Williams_SIM_Chip_Library

A smartphone is a mobile phone offering advanced capabilities mobile phone.
[[w:Smartphone]]

===iphone===

I-phone - The I-Phone is a multi-media device which combines a cellular phone, camera, personal media player,and an internet-enabled computer with a virtual keyboard. It is designed and marketed by Apple inc.

However, this device does not fit the category of UMPC as the end user only have read function and not edit function.  The document suites in the device will only allow you to open without making edit.

iPhone includes an SMS application with a full QWERTY soft keyboard to easily send and receive SMS messages in multiple sessions

===new phone===
Cellphones are now a form of PDA, they double as music players, TVs and personal organizers. Cellphones  are no longer limited to communications. Future cell phones will have the ability to provide live video conversation services. [http://www.pcworld.com/article/144127-1/10_cool_gadgets_you_cant_get_hereyet.html]
[http://mobilitytoday.com/vbmcms/images/a900duo.jpg]

&lt;big&gt;'''Ever dropped your cell phone in the lake or pool or &lt;u&gt;&lt;big&gt;toilet&lt;/big&gt;&lt;/u&gt; and it never worked the same?'''&lt;/big&gt;

           &lt;big&gt;&lt;big&gt;'''You Need This!'''&lt;/big&gt;&lt;/big&gt;

[http://gizmodo.com/347408/fujitsu-f705i-is-worlds-slimmest-waterproof-3g-cellphone]

Fujitsu has come out with the first 3G waterproof cell phone.  It's the Fujitsu F705i.

Features of this awesome phone include:
*waterproof up to 1 minute of submersion
*eight levels of ZOOM to make reading emails easier
*&quot;super clear voice&quot; automatically adjusts the volume of incoming calls to a comfortable and audible level based on the amount of ambient noise
*1.3 megapixel camera
*noise cancellation to drown out background noise

[http://blog.wired.com/gadgets/2008/01/fujitsus-ultrat.html] [http://www.pcworld.com/article/144127-8/10_cool_gadgets_you_cant_get_hereyet.html]

====mainly voice====

====more memory====

====keyboard/stylus====
Finger-operated touch-screens are excellent in the iphone which is easy to use.
[http://www.jeroenmulder.com/weblog/2007/06/iphone_keyboard_trying_too_hard.php]

====LCD screen====
The screen's resolution is 320 pixels and it is 3.5 inches diagonal. It has a special feature called multi-touch, where you can touch the screen to navigate. [http://www.intomobile.com/2007/03/11/apple-iphone-disappointing-lcd-screen.html]

It has a 160 ppi (pixels per inch) pixel density.
[http://www.intomobile.com/2007/03/11/apple-iphone-disappointing-lcd-screen.html]

The current iPod screen is excellent, with crisp 160ppi detail and a remarkable ability to let your mind read &quot;in between the pixels&quot; on album covers
[http://www.crazygadgetguru.com/posts/2007/6/5/iphone-lcd-screen-in-perspective.html]

====E-mail====
The iPhone also features an e-mail program that supports HTML e-mail, which enables the user to embed photos in an e-mail message. PDF, Microsoft Word, and Microsoft Excel attachments to mail messages can be viewed on the phone. Yahoo is currently the only e-mail provider.
[[w:IPhone]]

the iPhone syncs to your address book
[http://www.pcworld.com/article/id,137138/article.html]

====web====
====camera====
The iphone is equipped with a 2.0 megapixel camera. The images are clear in well lit areas; however, it lacks a flash and is not able to zoom. [http://www.apple.com/iphone/specs.html]

====music/video====
The iphone Supports a veriety of media. It is able to support such video formats as *.m4v, *.mp4,*.mov file and Music formats as AAC, Protected *AAC, *MP3, *MP3 *VBR, Audible formats 1, 2, and 3, Apple Lossless, *AIFF, and *WAV. [http://www.apple.com/iphone/specs.html]

====iTouch====
The NEW itouch is very similar to the iPhone. The only difference is that iTouch has a touch-screen with a built in wifi syste. This means that the iTouch is an iPod with WiFi capability. You could say that the iPhone is superior as it comes with the same features plus the fact that it can be used as a phone as well.

===== iPhone to Become Blu-Ray Player Remote  =====
[[Image:bluphone.jpg|400px|]]

According to NetBlender, iPhone and iPod touch users will be able to control their Blu-ray players using an application called BD Touch. The application will use the network capabilities of Blu-ray hardware and Apple's handheld devices to transfer data, allowing you to do many different things beyond controlling movie playback.

It will include a number of features that will be supported by the technology, such as automatically updating film collections held on an iPhone, including a feature which lets Blu-ray players/discs send a digital copy of a video to an iPhone.

[http://www.pcworld.com/article/id,144250/article.html?tk=nl_dnxnws]
[http://gizmodo.com/377290/iphone-to-become-blu+ray-player-remote]

==Connection==
===network===
====tree-and-branch====
[[Image:800px-HFC_Network_Diagram.png|frame|500px|[[w:Image:HFC_Network_Diagram.png|Cable example]]]Broadcasting, where one central location delivers content simultaneously to many users (cable TV [[w:Hybrid_fibre-coaxial]], radio)
====switched-network====
users contact specific users as in a telephone network [[w:Public_switched_telephone_network]]
===file sharing===
====legal entanglements====
====hackers invade your computer====
====download virus====

==gaming==
===2nd life===
Second Life is an Internet-based virtual world which came to international attention via mainstream news media in late 2006 and early 2007. A downloadable client program called the Second Life Viewer enables its users, called &quot;Residents&quot;, to interact with each other, providing an advanced level of a social network service combined with general aspects of a metaverse. Residents can explore, meet other Residents, socialize, participate in individual and group activities, create and trade items (virtual property) and services from one another.[[w:Second_Life]]
A similar program to the one Second Life will be adding is called BrainGate.  It is a brain implant system designed to help those who have lost control of their limbs, or other bodily functions, such as patients with (ALS) or spinal cord injury. A computer chip is implanted into the brain, it monitors brain activity in the patient and converts the intention of the user into computer commands. Hair-thin electrodes sense the electro-magnetic signature of neurons firing in specific areas of the brain, the activity is translated into electrically charged signals and are then sent and decoded using a program, which can move either a robotic arm or a computer cursor.[[w:BrainGate]]

Second life is a program that can be used successfully with those who are paralyzed to stimulate mobility, it could also trace brain activity and arrest disease, it is capable of interpreting intention and commercial transations could be activated with the use of Linden dollars which can be later exchanged for US dollars.

Second Life can be used in a distance learning technology where virtual (but real) you can take classes; get a real filling of meeting other students, borrow books from libraries, do Tai Chi , go to the beach. This is very useful for people who are not able to go to the class or for people who don’t want to leave their homes.
http://www.kqed.org/quest/television/view/611
(http://www.pcworld.com/article/id,139969-c,technology/article.html)

===Xbox===
[[Image:Xboxarcadesku.jpg|thumb|200px|xbox360]]
Xbox the first game console developed by Microsoft It comes with a large hard drive, wireless controller and wired headset.
Xbox 360 is the latest version and has the best graphics of any video console on the market. This gaming unit is excellent for all role playing games, sports and learning software.
(http://www.xbox.com/en-CA/)
[http://www.xbox.com/en-US/hardware/compare101.htm?WT.svl=nav]
[[w:Xbox_360]]
====online video games====

Xbox has an online gaming platform called xbox live. People have to subscribe to play multiplayer with members from all around the world.  With the exception of single player specific games, all of Xbox360 titles now are online compatiable. XBOX live also allows the player to interact with others through communication devices. XBOX Live is convienient in the sense that there is no monthly fee, it is annual. [[w:Xbox#Multimedia]]
[http://www.xbox.com/en-US/live/?WT.svl=nav]

There are future talks of Microsoft merging the Xbox live with PC user platform with the &quot;GAMES for PC&quot; roll out.  However, this will give the mouse and keyboard combination an advantage over game pad users on virtually all games.

====media hub of house====
This is a system for extender play which is pre installed in all the x-box 360s. Includes movies, music, videos and pictures.
[http://www.xbox.com/en-GB/hardware/xbox360/benefits/mediacentre.htm]

===PS3===
[[File:PS3s and controllers at E3 2006.jpg|thumb|200px|PlayStation 3 at E3 2006]]

The PlayStation 3 is the third iteration of a home console produced by Sony.  The PlayStation 3 offers a unified online gaming service known as the PlayStation Network.  Other features include connectivity with the PlayStation Portable, Blu-ray disc and large hard-drive capacity for a console. The PlayStation 3 was first released on November 11, 2006 in Japan, November 17, 2006 in North America, and March 23, 2007 in Europe. [[w:PlayStation_3]]

====movie quality graphics====

The PS3 features a Full HD (up to 1080p) x 2 channels

[http://ps3.gamespy.com/articles/614/614972p3.html]

===PSP===

====Specs.====
[[Image:Psp1.png|thumb|200px|PSP]]
The PSP (Play Station Portable) is a handheld portable consel that consists of video, music, games and internet.

[[w:PlayStation_Portable#System_Software|Reference Wikisite]]

====functions====
The functions are:
-video capibilities: which you can buy (UMDs) and/or download (MPEGs)
-music: download (MP3s)
-access internet through WI-FI
-pictures (JPEGs)
-main function is to play games (UMDs- Universal Media Disc)
[[w:PlayStation_Portable]]

====advantages====
1.  Many different variations and accessories (head set, carrying case, cleaning cloth, wrist strap, head phones, etc.)
2.  New and exciting colours
3.  More games
4.  Download games
5.  Play Back Movies
6.  Access to Wireless Networks
7.  Share Games with Friends
8.  Slimmer and Lighter

====disadvantages====
1.  Screen is smaller
2.  Only 2 hours of game play
3.  You must buy your own memory cards
4.  Fairly heavy
5.  Design is not very attractive
6.  Can't hook it up to your T.V. or stereo

====cell interacts====

The ps3 cell is a 9 core processor, one of these cores is a PowerPC and acts as a controller.  The remaining 8 cores are called SPEs and these are very high performance vector processors.  Each SPE contains it's own block of high speed RAM and is capable of 32 GigaFlops.  The SPEs are independent processors and can act alone or can be set up to process a stream of data with different SPEs working on different stages.
[http://www.blachford.info/computer/Cell/Cell0_v2.html]

===Nintendo===
====simpler====
Nintendo Wii is fitted with a motion sensor so it allow players to simplify the game controls comparing to the traditional game pad. The games that are available on the Wii platform are often very intuitive and can be picked up by people of all ages.

This is a game system that kids/teens like to use.

====cheaper====
It cost less than $250 US [http://www.gamespot.com/ds/action/supermariobrosds/news.html?sid=6151827&amp;cpage=5]
The first Nintendo was marketed for $199 US.
[http://wiki.answers.com/Q/How_much_did_the_first_nintendo_cost]

====wireless motion ====
The Wireless Remote is motion sensing capability. It allows the user to interact with items on screen via movement and pointing through the use of accelerometer and optical sensor technology.
[http://www.wikipedia.org]

====sensor console====
Unlike a light gun that senses light from a television screen, the Wii Remote senses light from the consoles Sensor Bar (model number RVL-014), which allows consistent usage regardless of a television's type or size.
[http://www.wikipedia.org]

====Smart Downloads====
Japan has launched a download channel which allows Wii users to download game demos, view promos, and see screen shots free of charge. It downloads temporary information into  hand held internal memory and is deleted once the unit is turned off.  It is also smart because it recommends game choice based on your playing record, your age and gender preferences. [http://www.pcworld.com/article/id,140011-c,gameconsoles/article.html#]

Other benefits for Wii owners is that, they are able to view:Commercials, box art, game play footage screen shots, and interviews about various Nintendo products including prices and release dates for upcoming DS and Wii games. [http://www.pcworld.com/article/id,140011-c,gameconsoles/article.html]


===Wii===

[[Image:Wii_Wiimotea.png|thumb]]
====Description====
The Wii is the fifth home video game console released by nintendo.  The distinguishing feature of the Wii is it wireless controller (the wii remote) which can be used as a handheld pointing device and can detect acceleration in three dimensions.  There have been many speculations as to why the company chose Wii for the name, the official reason is “ Wii sounds like 'we', which emphasizes that the console is for everyone. Wii can easily be remembered by people around the world, no matter what language they speak. No confusion. No need to abbreviate. Just Wii.[11] ”.
The Wii remote is the main controler for the console.  It uses built-in accelerometers and infrared detection to sense its position in 3D space when its pointed at the LED lights within the sensor bar. It connects to the concole using bluetooth.

Even Microsoft has noticed Nintendos success with their new Bluetooth infused controllers. In late 2008 Microsoft plans to release a &quot;Wii&quot; like controller for their current gaming consol, the Xbox 360. http://www.pcworld.com/article/id,144223/article.html?tk=nl_dnxnws

[[w:Wii]]
They are a great alternative for kids and families as the games are generally rated E for everyone and it keeps kids active while they are gaming.

====Cost====
The Wii's price at the time of introduction was approximately $249.99, with the remote costing $39.99 and the individulal games costing $49.99. Prices have been $100 - $200 more than the asking price because of the high demand, but do expect to go down to the average of $250 soon.
[http://wii.ign.com/articles/732/732669p1.html]
[http://www.pcworld.com/article/id,139949-c,gameconsoles/article.html]

====Supply====
Wii is available for purchase at all major department stores.  There have been some speculations that Nintendo has been producing fewer units than needed to meet demand, thus increasing the price people will pay for it.  There are approximately 1.8 millions Wii units produced per month.
[http://www.pcworld.com/article/id,139949-c,gameconsoles/article.html]

====Demand====
Since first launching the Wii in November of last year, Nintendo has seen widespread shortages at retail. The demand has been so high that Nintendo says it is working at maximum capacity in producing 1.8 million Wii units per month, hopefully in time for Christmas.
[http://www.pcworld.com/article/id,139949-c,gameconsoles/article.html]

[[Category: Introduction to Computers]]

==Safety and Ergonomics==
Prolonged use of computers can be a health risk to your body.

===eye===

[[Image:Iris.eye.225px.jpg|frame|eye suffering from CVS]]The eye can also be a victim and symptom of extended use on the computer.  This eye condition also called &quot;Computer Vision Syndrome&quot; or CVS and is a member of the family referred to RSI or Repetitive Strain Injury.  Prolonged periods of continuous focusing on a computer screen for uninterrupted periods of time.  Aggravating the condition can be improper light or air moving past the eye from overhead vents or fans.  Symptoms of the condition can include headaches, blurred vision, neck pain, fatigue, eye strain, dry, irritated and difficulty on refocusing[http://www.allaboutvision.com/cvs/faqs.htm].

A couple of ways to treat the condition is to allow time to rest your eye, redirecting your focus away from the screen, also the purchase of over the counter eye drops are both easy solutions.  There is a catchphrase that has proven to be a well know solution among &quot;computerites&quot;.  It is called 20-20-20 ~ focus your eyes on an object 20 feet away for 20 seconds every 20 minutes.

===neck/back===
'''''Neck and back pain are common complaints. The cause is usually poor posture, which decreases blood flow to certain muscles. These muscles stiffen up and hurt.'''''&lt;ref&gt;http://www.slais.ubc.ca/COURSES/libr500/02-03-wt1/www/A_Davis/neckback.htm&lt;/ref&gt;

'''''Neck Pain:'''''
Often begins gradually as a result of fixed staring at a small area or glancing repeatedly from one to another (from the screen to a document on your desk for example).  If the head is held at an angle greater than 15 degrees (for example holding the phone between your neck or shoulder, or looking down at your keyboard) will cause greater muscular fatigue and pain will become apparent more rapidly &lt;ref&gt;Stigliani, Joan. The Computer User's Survival Guide. Sebastopol, CA: O'Reilly &amp; Associates, Inc. 1995.&lt;/ref&gt;

'''''Back Pain:'''''
Sitting is one of the hardest positions in which to maintain proper posture, and many computer users regularly feel back pain.  Spinal compression is one of the most common problems because sitting tends to tilt the pelvis backward, flattening the lumbar curve and resulting in uneven and increased pressure on spinal disks. &lt;ref&gt;Sellers, Don. Zap! How Your Computer Can Hurt You – And What You Can Do About It. Edited by Stephen F. Roth.  Berkeley: Peach Pit Press, 1994.&lt;/ref&gt;

'''''Prevention'''''
*Be sure you have a proper workstation set-up.
*Take active breaks, move around and do a few stretches.
*Shift positions every now and then.
*Try not to fall habitually into one computer position – even small changes help avoid overtaxing certain muscles.
*Try not to round your shoulders – this puts extra pressure on your upper spine.
*Stay active, get up and move around to circulate your blood.  Sitting still for too long can slow blood circulation and muscle fatigue can set in.&lt;ref&gt;Sellers, Don. Zap! How Your Computer Can Hurt You – And What You Can Do About It. Edited by Stephen F. Roth.  Berkeley: Peach Pit Press, 1994.&lt;/ref&gt;
]]

===wrist===
RSI (also know as cumulative trauma disorder) is a soft-tissue injury in which muscles, nerves, or tendons become irritated or inflamed. RSI is caused by repetitive motions, excessive force, and extremes of motion. Over time these motions can strain the soft tissues, reducing circulation. These stresses create tiny tears in the muscles and tendons, which become inflamed. In extreme cases it can cause permanent tissue damage and disability.

The computer keyboard is an ergonomic nightmare. It tends to force you into an unnatural palms-down (pronated) wrist-cocked position. This strains the delicate muscles and tendons of the fingers and wrists, reducing circulation.

[http://www.webreference.com/rsi.html]

Carpal tunnel syndrome as referred to by many different websites is one of the main injuries associated with the wrist. Daily exercises and breaks away from the repetitive typing of the computer can help alleviate some of the symptoms of carpal tunnel syndrome.

'''Symptoms'''
usually start gradually, with frequent burning, tingling, or itching numbness in the palm of the hand and the fingers, especially the thumb and the index and middle fingers. Some carpal tunnel sufferers say their fingers feel useless and swollen, even though little or no swelling is apparent. The symptoms often first appear in one or both hands during the night, since many people sleep with flexed wrists. A person with carpal tunnel syndrome may wake up feeling the need to &quot;shake out&quot; the hand or wrist. As symptoms worsen, people might feel tingling during the day. Decreased grip strength may make it difficult to form a fist, grasp small objects, or perform other manual tasks. In chronic and/or untreated cases, the muscles at the base of the thumb may waste away. Some people are unable to tell between hot and cold by touch.

'''Prevention'''
At the workplace, workers can do on-the-job conditioning, perform stretching exercises, take frequent rest breaks, wear splints to keep wrists straight, and use correct posture and wrist position. Wearing fingerless gloves can help keep hands warm and flexible. Workstations, tools and tool handles, and tasks can be redesigned to enable the worker's wrist to maintain a natural position during work. Jobs can be rotated among workers. Employers can develop programs in ergonomics, the process of adapting workplace conditions and job demands to the capabilities of workers. However, research has not conclusively shown that these workplace changes prevent the occurrence of carpal tunnel syndrome.

[http://www.ninds.nih.gov/disorders/carpal_tunnel/detail_carpal_tunnel.htm]

]]


&lt;big&gt;Course Navigation&lt;/big&gt;
{| class=&quot;wikitable&quot;
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  <page>
    <title>Web 2.0</title>
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      <text bytes="30521" sha1="3zdvmg9aks7mme1oz2mo1z4kq3dw4oe" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

==Internet content: ownership and sharing==
&quot;Web 2.0&quot; is a term that can be used to refer to a qualitatively new and different pattern of internet behavior: a shift from an older era of restricted and expensive technologies for creation and internet-based sharing of digital media files to a new era of increasingly accessible and inexpensive technologies. As more and more people become empowered to participate on the internet as content producers, new patterns of content ownership and sharing have come into existence. The traditional model was that expensive digital content was protected by copyright, copies were sold and derivative works were possible only via rare and expensive special licensing agreements.

An alternative approach to digital media began with the [[w:Open-source software|Open-source software]] industry. Recognizing that software innovation is promoted by making software &quot;open&quot; to a distributed community of developers, some software developers began to experiment with new strategies for licensing software. In 2001, Wikipedia was launched with contents licensed under the GFDL and the [[w:Creative Commons|Creative Commons]] licenses began to be developed. A growing [[w:Free Culture movement|Free Culture movement]] supports the licensing of digital media files so as to facilitate file sharing and re-use of media for the creation of new works. In the collaborative environment of Web 2.0, sharing intellectual property, without the intermediate step of requesting permission directly from the owner, allows easier access to materials and fosters greater creativity.  However, owners of intellectual property must consider whether the Free Culture Movement adds value or takes away value from their work.  While some intellectual property might gain value from easier access, other intellectual property like artists' works might lose value.

**

==See also==
*[[Wikiversity:Interactive learning resources|Rich Learning Resources Platform]]
*[[Twitter]]
*[[Wikipedia:List of free software for Web 2.0 Services|List of free software for Web 2.0 Services]]
*[[Social_Media|Social Media]]

===Wikipedia===
* [[w:Web 2.0 for development|Web 2.0 for development]]

==External links==
*[http://www.martinblueprint.co.uk/wikka/wikka.php?wakka=HomePage Web 2.0 research] - by M. Dimartino Marriott.

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
[[Category:Web 2.0]]
Concepts
Social software has emerged as a major component of the Web 2.0 movement. The idea dates as far back as the 1960s and JCR Licklider’s thoughts on using networked computing to connect people in order to boost their knowledge and their ability to learn. The Internet technologies of the subsequent generation have been profoundly social, as listservs, Usenet groups, discussion software, groupware, and Web-based communities have linked people around the world. During the past few years, a group of Web projects and services became perceived as especially connective, receiving the rubric of “social software”: blogs, wikis, trackback, podcasting, videoblogs, and enough social networking tools like MySpace and Facebook to give rise to an abbreviation mocking their very prevalence: YASN (Yet Another Social Network). Consider the differences between these and static or database-driven Web pages. Wikis are all about user modification; CNN’s front page is decisively not. It is true that blogs are Web pages, but their reverse-chronological structure implies a different rhetorical purpose than a Web page, which has no inherent timeliness. That altered rhetoric helped shape a different audience, the blogging public, with its emergent social practices of blogrolling, extensive hyperlinking, and discussion threads attached not to pages but to content chunks within them. Reading and searching this world is significantly different from searching the entire Web world. Still, social software does not indicate a sharp break with the old but, rather, the gradual emergence of a new type of practice.

==References==
* &quot;Understanding Web 2.0&quot;, San Murugesan, IEEE IT Professional, 2007 
* &quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications&quot;, San Murugesan (Editor), Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}
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      <text bytes="30448" sha1="fxr9709mylgy9pn7fse9twz2lfdk6gk" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

==Internet content: ownership and sharing==
&quot;Web 2.0&quot; is a term that can be used to refer to a qualitatively new and different pattern of internet behavior: a shift from an older era of restricted and expensive technologies for creation and internet-based sharing of digital media files to a new era of increasingly accessible and inexpensive technologies. As more and more people become empowered to participate on the internet as content producers, new patterns of content ownership and sharing have come into existence. The traditional model was that expensive digital content was protected by copyright, copies were sold and derivative works were possible only via rare and expensive special licensing agreements.

An alternative approach to digital media began with the [[w:Open-source software|Open-source software]] industry. Recognizing that software innovation is promoted by making software &quot;open&quot; to a distributed community of developers, some software developers began to experiment with new strategies for licensing software. In 2001, Wikipedia was launched with contents licensed under the GFDL and the [[w:Creative Commons|Creative Commons]] licenses began to be developed. A growing [[w:Free Culture movement|Free Culture movement]] supports the licensing of digital media files so as to facilitate file sharing and re-use of media for the creation of new works. In the collaborative environment of Web 2.0, sharing intellectual property, without the intermediate step of requesting permission directly from the owner, allows easier access to materials and fosters greater creativity.  However, owners of intellectual property must consider whether the Free Culture Movement adds value or takes away value from their work.  While some intellectual property might gain value from easier access, other intellectual property like artists' works might lose value.

**

==See also==
*[[Wikiversity:Interactive learning resources|Rich Learning Resources Platform]]
*[[Twitter]]
*[[Wikipedia:List of free software for Web 2.0 Services|List of free software for Web 2.0 Services]]
*[[Social_Media|Social Media]]

==External links==
*[http://www.martinblueprint.co.uk/wikka/wikka.php?wakka=HomePage Web 2.0 research] - by M. Dimartino Marriott.

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
[[Category:Web 2.0]]
Concepts
Social software has emerged as a major component of the Web 2.0 movement. The idea dates as far back as the 1960s and JCR Licklider’s thoughts on using networked computing to connect people in order to boost their knowledge and their ability to learn. The Internet technologies of the subsequent generation have been profoundly social, as listservs, Usenet groups, discussion software, groupware, and Web-based communities have linked people around the world. During the past few years, a group of Web projects and services became perceived as especially connective, receiving the rubric of “social software”: blogs, wikis, trackback, podcasting, videoblogs, and enough social networking tools like MySpace and Facebook to give rise to an abbreviation mocking their very prevalence: YASN (Yet Another Social Network). Consider the differences between these and static or database-driven Web pages. Wikis are all about user modification; CNN’s front page is decisively not. It is true that blogs are Web pages, but their reverse-chronological structure implies a different rhetorical purpose than a Web page, which has no inherent timeliness. That altered rhetoric helped shape a different audience, the blogging public, with its emergent social practices of blogrolling, extensive hyperlinking, and discussion threads attached not to pages but to content chunks within them. Reading and searching this world is significantly different from searching the entire Web world. Still, social software does not indicate a sharp break with the old but, rather, the gradual emergence of a new type of practice.

==References==
* &quot;Understanding Web 2.0&quot;, San Murugesan, IEEE IT Professional, 2007 
* &quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications&quot;, San Murugesan (Editor), Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}
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      <text bytes="28697" sha1="fm9w3p4lrcojpd8hwtjo4nzmxf2b6nh" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

==Internet content: ownership and sharing==
&quot;Web 2.0&quot; is a term that can be used to refer to a qualitatively new and different pattern of internet behavior: a shift from an older era of restricted and expensive technologies for creation and internet-based sharing of digital media files to a new era of increasingly accessible and inexpensive technologies. As more and more people become empowered to participate on the internet as content producers, new patterns of content ownership and sharing have come into existence. The traditional model was that expensive digital content was protected by copyright, copies were sold and derivative works were possible only via rare and expensive special licensing agreements.

An alternative approach to digital media began with the [[w:Open-source software|Open-source software]] industry. Recognizing that software innovation is promoted by making software &quot;open&quot; to a distributed community of developers, some software developers began to experiment with new strategies for licensing software. In 2001, Wikipedia was launched with contents licensed under the GFDL and the [[w:Creative Commons|Creative Commons]] licenses began to be developed. A growing [[w:Free Culture movement|Free Culture movement]] supports the licensing of digital media files so as to facilitate file sharing and re-use of media for the creation of new works. In the collaborative environment of Web 2.0, sharing intellectual property, without the intermediate step of requesting permission directly from the owner, allows easier access to materials and fosters greater creativity.  However, owners of intellectual property must consider whether the Free Culture Movement adds value or takes away value from their work.  While some intellectual property might gain value from easier access, other intellectual property like artists' works might lose value.

**

==See also==
*[[Wikiversity:Interactive learning resources|Rich Learning Resources Platform]]
*[[Twitter]]
*[[Wikipedia:List of free software for Web 2.0 Services|List of free software for Web 2.0 Services]]
*[[Social_Media|Social Media]]

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
[[Category:Web 2.0]]

==References==
* &quot;Understanding Web 2.0&quot;, San Murugesan, IEEE IT Professional, 2007 
* &quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications&quot;, San Murugesan (Editor), Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}
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      <text bytes="28380" sha1="7yzychcn8t0h3t9u7b9rak22zun0zlr" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

==Internet content: ownership and sharing==
&quot;Web 2.0&quot; is a term that can be used to refer to a qualitatively new and different pattern of internet behavior: a shift from an older era of restricted and expensive technologies for creation and internet-based sharing of digital media files to a new era of increasingly accessible and inexpensive technologies. As more and more people become empowered to participate on the internet as content producers, new patterns of content ownership and sharing have come into existence. The traditional model was that expensive digital content was protected by copyright, copies were sold and derivative works were possible only via rare and expensive special licensing agreements.

An alternative approach to digital media began with the [[w:Open-source software|Open-source software]] industry. Recognizing that software innovation is promoted by making software &quot;open&quot; to a distributed community of developers, some software developers began to experiment with new strategies for licensing software. In 2001, Wikipedia was launched with contents licensed under the GFDL and the [[w:Creative Commons|Creative Commons]] licenses began to be developed. A growing [[w:Free Culture movement|Free Culture movement]] supports the licensing of digital media files so as to facilitate file sharing and re-use of media for the creation of new works. In the collaborative environment of Web 2.0, sharing intellectual property, without the intermediate step of requesting permission directly from the owner, allows easier access to materials and fosters greater creativity.  However, owners of intellectual property must consider whether the Free Culture Movement adds value or takes away value from their work.  While some intellectual property might gain value from easier access, other intellectual property like artists' works might lose value.

**

==See also==
*[[Wikiversity:Interactive learning resources|Rich Learning Resources Platform]]
*[[Twitter]]
*[[Wikipedia:List of free software for Web 2.0 Services|List of free software for Web 2.0 Services]]
*[[Social_Media|Social Media]]

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
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      <text bytes="28140" sha1="k9muf96kyz9fnm645kuffob4jaweffr" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

==Internet content: ownership and sharing==
&quot;Web 2.0&quot; is a term that can be used to refer to a qualitatively new and different pattern of internet behavior: a shift from an older era of restricted and expensive technologies for creation and internet-based sharing of digital media files to a new era of increasingly accessible and inexpensive technologies. As more and more people become empowered to participate on the internet as content producers, new patterns of content ownership and sharing have come into existence. The traditional model was that expensive digital content was protected by copyright, copies were sold and derivative works were possible only via rare and expensive special licensing agreements.

An alternative approach to digital media began with the [[w:Open-source software|Open-source software]] industry. Recognizing that software innovation is promoted by making software &quot;open&quot; to a distributed community of developers, some software developers began to experiment with new strategies for licensing software. In 2001, Wikipedia was launched with contents licensed under the GFDL and the [[w:Creative Commons|Creative Commons]] licenses began to be developed. A growing [[w:Free Culture movement|Free Culture movement]] supports the licensing of digital media files so as to facilitate file sharing and re-use of media for the creation of new works. In the collaborative environment of Web 2.0, sharing intellectual property, without the intermediate step of requesting permission directly from the owner, allows easier access to materials and fosters greater creativity.  However, owners of intellectual property must consider whether the Free Culture Movement adds value or takes away value from their work.  While some intellectual property might gain value from easier access, other intellectual property like artists' works might lose value.

**

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
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      <text bytes="26259" sha1="qg2txre99wegwog35rrzdm3jr8ti57p" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
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Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
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Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
[[Category:Web 2.0]]</text>
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      <text bytes="32915" sha1="elz8vh0fehfho2rz0ijnbhbwa6lmy06" xml:space="preserve">[[Image:Web 2.0 report.png|thumb|right|320px|[[Media:Web 2.0 copyleft.ogg|Click here to watch the video]] about Web 2.0 and copyleft media ([[w:Wikipedia:Media help (Ogg)|Help]] with Ogg video file play). This video explores Web 2.0 and copyleft licensing of media files. [[Image:Web 2.0 copyleft.ogg|Click here to download]] the video. 4 minutes 24 seconds play time, 7.73 MB file size.]]
Welcome to the Wikiversity learning project  '''Web 2.0'''. Participants  explore tools for accessing, evaluating, transforming and creating  internet content, including media such as digital audio and video, while actively participating in multiple course-related social networks.

==Expectations==
Web 2.0 is, by design, immersive, interactive, and collaborative. This course is based on experiential learning. In other words, to really increase your knowledge regarding Web 2.0, you must personally use and utilize Web 2.0 tools and services. This holds true whether you are interested in self-paced independent study or using this course as a guide to teach others.  It is essential that you stay active and contribute to online communities and knowledge, and reflect on your experiences. Start a [[Wikipedia:Blog|blog]], [http://www.wordpress.com Wordpress], [http://www.facebook.com Facebook] page, or similar forum that can contain your personal reflection. As you go through the course, try to write a reflective post on each new thing that you learn.

==What is Web 2.0?==
[[Image:Web 1.0 elements.png|thumb|right|300px|Key elements and connectivity of Web 1.0. Click image to enlarge. The early internet was characterized by slow internet connectivity ([[w:Dial-up access|dialup]]) and few content creation opportunities for most internet users (content consumers).]]
During the 1990s, the [[w:World Wide Web|World Wide Web]] provided a way for people to use a network of computers to efficiently exchange files. In general, content for the Web was created by a relatively small group of individuals or small &quot;content development groups.&quot; Once created, the content ([[w:HTML|HTML pages]] and [[w:Digital imaging|media files]]) was uploaded to servers and then downloaded by &quot;content consumers&quot; who used a [[w:Web browser|web browser]] to display webpages. The average person was not involved with creation of Web content. This period of time is now referred to as Web 1.0.

The evolution of the Web has led to what is called &quot;Web 2.0&quot;. What is new about Web 2.0 is the gradual and continuing increase in technologies that allow more people to participate in Web content creation. These facilitating technologies include advances at the level of the computer hardware available to most people and at the level of software that makes it easier for people to create Web content.

&quot;Web 2.0 is both a usage and a technology paradigm. It is a collection of technologies, business strategies, and social trends. Web 2.0 is more dynamic and interactive than its predecessor, Web 1.0, letting users access content from a Web site and contribute to the contents of that Web site. Web 2.0 enables users to keep up with a site's most recent content edits even without visiting the actual Web page. It also lets developers create new Web applications that draw on data, information or services available on the Internet to more precisely manufacture their programs to fit their desired demographic.&quot;&lt;ref&gt;[http://www.computer.org/portal/web/buildyourcareer/fa009 Understanding Web 2.0 - IEEE Computer Society]&lt;/ref&gt; 

Web 2.0 is an umbrella term encompassing several new Web technologies. These technologies will be outlined later. &quot;It harnesses the Web in a more interactive and collaborative manner, emphasizing peers' social interaction and collective intelligence, and presents new opportunities for leveraging the Web and engaging its users more effectively.&quot;

===Objectives===
[[Wikipedia:Web 2.0|*Define Web 2.0.]]

*Where Web 2.0 started and understand the direction it is going.

*Discover new Web 2.0 tools and applications, and apply them to daily lives and jobs.

*Explore and contribute to a variety of online communities, including adding content, posting comments, correcting false information, and learning how to be collaborative.

*Anticipate future web trends: Web 3.0? Web 4.0? Web 10-million-point-oh?

*Contribute knowledge gained using Web 2.0 tools in professional/personal lives.

*Learn how others are using Web 2.0 tools and weigh pros/cons.

*Learn how to troubleshoot hardware/software issues.

*Become familiar with leading tools in specific categories.

*Become aware of ethics and privacy in Web 2.0 and its applications.

*Understand Web 2.0 and be able to educate others on the topic.

*Develop discipline in approaching the complexities of multiple tasks, time lines, tools, logins, URLs in an unstructured environment.

===Hardware===
During the past 15 years, increasing numbers of people have obtained access to computer technology and the Internet. Internet data traffic passed voice data traffic at about the turn of the century and Web traffic now greatly surpasses voice data transfer&lt;ref&gt;[http://www.cs.columbia.edu/~hgs/internet/traffic.html Long-Term Traffic Statistics]&lt;/ref&gt;. At about the same time, home internet access reached about 50% of homes in the USA. In this century, use of [[w:Broadband|high-speed]] internet connections has increased rapidly with over 50 million U.S. residential broadband connections achieved in 2006. The emergence of smartphone technology is still changing the landscape of access to the Web, as people can easily carry Internet ready devices with them at all times. 
A recent survey shows a 15% increase in Smartphone sales than the previous year. Smartphones compromise about 12 percent of the mobile phone market, a number that is only expected to increase in coming years.&lt;ref&gt;[http://news.cnet.com/8301-13579_3-9906697-37.html]&lt;/ref&gt;
[http://wp.nmc.org/horizon2010/ The 2010 Horizon Report] predicts that, in the next 12 months or less, [http://wp.nmc.org/horizon2010/chapters/mobile-computing/ mobile computing] will enter into mainstream use for teaching, learning, or creative inquiry.  Devices from smart phones to netbooks are portable tools for productivity, learning, and communication, offering an increasing range of activities fully supported by applications designed especially for mobiles.

A 2012 study by Edison Research and Arbitron found that there was a 50% increase in smartphone ownership from 2011 to 2012 and that about 50% of cell phone owners own a smartphone. Smartphone owners are more likely to access social networking sites and public domains like youtube at least 5X a day from their phones. 60% of smartphone owners report that they keep their phone at an arms lengths away at all times. Smartphone users spend almost the same amount of time on their smartphones as they do watching tv. The smartphone has enabled users to be connected and using web 2.0 applications on the go. This number will only grow as well all become connected with mobile phones, computers and tablets. [4]

The study also reported that 3 in 10 smartphone owners own a tablet. Tablets have the capability to browse the internet, create and share presentations, videos conference with clients, stay connected with corporate email, download books, games and videos, watch movies, share photos and much more. [5]

As of January 2014, Pew Research Center reports that:

* 90% of American adults have a cell phone
* 58% of American adults have a smartphone
* 32% of American adults own an e-reader
* 42% of American adults own a tablet computer&lt;ref&gt;http://www.pewinternet.org/fact-sheets/mobile-technology-fact-sheet/&lt;/ref&gt;

===Software===
Web 2.0 is characterized by software that supports easy Web content creation in the form of [[Web Writing/Blogs|blogs]], [[wiki]]s, digital media uploading websites, and new types of online [[social networking website]]s. Software advances make it easier for more people to participate as Web content creators. Websites where users are participants in Web content creation have brought an increasingly robust social nature to Web 2.0 that has built upon the spirit of simpler online communities that formed in the first decade of the World Wide Web. Another important trend influencing Web 2.0 is an increase in options for openness in the creation of software and the growing phenomenon of the [[w:Free Culture movement|Free Culture Movement]].

===Early software allowing computer network users to add content to servers===
The first widely used type of software that allowed computer users to upload files to servers was software that allowed users to make use of [[Introduction to Programming|computer programming languages]] running on [[w:Mainframe computer|mainframe computers]]. One of the first types of nonprogramming-related software that allowed users of computer networks to upload content of their own design was [[w:E-mail|email]] software. Computer networks with email systems pre-dated the Internet, but their use only became wide-spread in combination with the Internet and the World Wide Web. See [[w:Web-based email]].

Some of the earliest online communities grew up around &quot;[[w:Bulletin board system|bulletin board systems]].&quot; &quot;Message board&quot; systems that were centers of social interaction in Bulletin board systems evolved into [[w:Internet forum|Internet forums]]. Some internet forums are email-based [[w:Mailing list|mailing lists]] ([http://lists.wikimedia.org/pipermail/wikiversity-l/ example]) while others function independent of email or use a mixture of email and non-email-based methods for adding &quot;[[w:Post count|posts]]&quot; to the &quot;community discussion&quot; ([http://discussions.apple.com example]).

One of the most influential online discussion forums has been [[w:Usenet|Usenet]]. Two serious problems for Usenet are spam and use of Usenet servers for pornography and illegal file transfers. Some Usenet groups have moderators who screen for off-topic postings. In terms of bandwidth use, most Usenet traffic is not text-based discussion, but rather digital media files such as illegally shared software, music, and movies&lt;ref&gt;[http://arstechnica.com/news.ars/post/20060224-6253.html MPAA turns attention to USENET, takes on Torrentspy, Isohunt, others] by Ryan Paul February 24, 2006 in ''Ars Technica''.&lt;/ref&gt;. See: [[w:Warez]].

[[w:Internet Relay Chat|Internet Relay Chat]] (IRC) provides another example of a text-based messaging system that spawned internet-based communities pre-dating the growth of the World Wide Web in the late 1990s. With more powerful personal computers and higher speed internet connections, [[w:Voice chat|voice chat]] and [[w:iChat|video chat]] have become increasingly popular. Such &quot;live&quot; channels for communication are useful for allowing members of internet-based communities to communicate effectively and support community cohesion. See: [[Wikiversity:Chat]].

In addition to online discussion systems (above), &quot;Web 1.0&quot; included a [[w:Web hosting service|Web hosting service]] industry that provided users with server space for their own HTML pages and media files. Some web hosting companies attempted to develop a &quot;community model&quot; in which users with similar interests could &quot;congregate&quot; and interact online. One of the more famous examples is [[w:GeoCities|GeoCities]]. Geocities users could construct personal webpages and participate in topical discussion groups. Similar Web 2.0 websites host wikis (see [[w:Comparison of wiki farms|wiki farm]]) and [[w:Weblog software|blogs]].

====Cloud Technologies====

The term &quot;cloud&quot; is used as a metaphor  for the Internet, based on the cloud drawing used in the past to represent the telephone network,  and later to depict the Internet in computer network diagrams as an abstraction  of the underlying infrastructure it represents.  Typical cloud computing providers deliver common business applications online that are accessed from another Web service or software like a Web browser, while the software and data are stored on servers. A key element of cloud computing is customization and the creation of a user-defined experience.

Most cloud computing infrastructures consist of services delivered through common centers and built on servers. Clouds often appear as single points of access for all consumers' computing needs. Commercial offerings are generally expected to meet quality of service requirements of customers, and typically include [[Wikipedia:Service level agreement|service level agreements]] (SLAs). The major cloud service providers include Microsoft, Salesforce, Dropbox, Amazon and Google.

====Archives====
While many &quot;Web 1.0&quot; online discussion systems featured message archive systems, the content of the online discussions was generally of transient relevance to the active participants. With the growth of the World Wide Web, some online discussion systems either made use of associated web pages or integrated into an HTML-based interface. A comprehensive approach to archiving Web content is the [[w:Internet Archive|Internet Archive]] project, but many websites routinely exclude themselves from the archive. Even when an archive system exists for user-uploaded computer network content, the nature of email, online discussion forums, and personal websites makes much of the content quickly dated and irrelevant.

===Evolving technologies===
[[Image:Web 2.0 elements.png|thumb|right|300px|Web 2.0 is characterized by hardware and software that facilitate internet content creation and sharing.]]
Blogs, wikis, media uploading websites, and social networking sites are four examples of newer technologies that support broader participation in the process of content creation for the internet. Blogs are particularly omnipresent among businesses that adopt a web site to connect and engage with customers. Given the role that social media plays in helping a majority of sites get found on the Internet, corporate blogs often serve as a creative channel to readers with an affinity to extraordinary content.&lt;ref&gt;[http://www.microsoft.com/business/en-sg/Content/Pages/article.aspx?cbcid=107&amp;listid=31a9054a-6493-495d-af28-f943d1ee4075 A Simple Guide in Building A Good Website and Brand]&lt;/ref&gt; 

====Blogs====
&quot;Blog&quot; is an abbreviated version of &quot;weblog,&quot; which is a term used to describe websites that maintain an ongoing chronicle of information. A blog features diary-type commentary and links to articles on other websites, usually presented as a list of entries in reverse chronological order. Blogs range from the personal to the political, and can focus on one narrow subject or a whole range of subjects.&lt;ref&gt;[http://codex.wordpress.org/Introduction_to_Blogging]&lt;/ref&gt;
Usenet discussion group contributors and personal website authors were among the first bloggers (see [[w:Blog#History|Wikipedia]]). Starting in the late 90s, websites and software devoted to blogging became available via the World Wide Web. In the early part of this century, blogging increased in popularity and is now an integral feature of many online communities and social networking sites such as [[w:MySpace|MySpace]], [[Wikipedia:Blogspot|Blogspot]] or [https://www.tumblr.com/ Tumblr]. See also blogs here at [[:Category:Blogs|Wikiversity]]. 
Remember that blogs are public, not the same as magazine articles, books, or personal journals hidden between the mattress and the box spring. Blogs are generally communities networked by subjects, interests, and niches. When a user enters the blogosphere, whether as a blogger or a blog reader, they are joining a community (or communities) of people who usually encourage a high degree of interaction. Think of it this way: when you read a magazine article, it is a one-way communication. Knowledge is only transferred from the writer to the reader. Blogs are a bit different. With blogs, this transfer of knowledge from writer to reader still occurs, but a blog affords the reader to then become a writer as well. There is a comment field where the reader can leave feedback, share additional information, or ask questions. And bloggers encourage this. Often they want to start and maintain open lines of communication with their followers. Conversations also exist '''between''' blogs. Bloggers routinely link to other bloggers in their communities through [[blogrolls]] and [[in-post]] references. This not only broadens conversations, it also raises reader awareness about other resources.

If you decide to blog, keep in mind that you are entering a community. To increase your popularity In that community, be sure to post regularly and comment on other blogs.  Link back to other bloggers you wish to form a stronger relationship with.

=====Helpful blogging resources to get you started=====
*[http://www.bloggingbasics101.com/ Blogging Basics 101]
* [http://www.smashingmagazine.com/2009/08/09/10-harsh-truths-about-corporate-blogging/ 10 Harsh Truths About Corporate Blogging]
* [http://www.copyblogger.com/successful-bloggers/ Secrets of Successful Bloggers]
* [http://www.copyblogger.com/blogging-sins/ Seven Deadly Sins of Blogging]

====Wikis====
Traditionally, the ability to edit a particular [[Wikiversity:Web page|webpage]] is severely restricted, often to just one person. [[w:Wiki|Wiki]] technology was first used in 1995, by [[w:Ward_Cunningham|Ward Cunningham]] and introduced a simple way for many people to collaboratively edit a website's webpages. Wiki websites achieve functionality as an online community by providing user pages (where participants can describe their personal interests) and an assortment of forum and discussion pages where wiki participants (&quot;editors&quot;) can participate in community discussions. [[w:Main Page|Wikipedia]] was started in 2001 and became widely known by 2006, particularly among school age internet users. By mid-2007, Wikipedia had become a top 10 website and as many as 6 percent of internet users make use of Wikipedia&lt;ref&gt;[http://www.alexa.com/data/details/traffic_details?url=en.wikipedia.org%2Fwiki%2FMain_Page Alexa traffic data]&lt;/ref&gt;. Many smaller wiki websites exist, some facilitated by [[w:Wiki farm|Wiki farms]], other wikis are run independently by individuals or organizations. Wiki is a Hawaiian word meaning &quot;fast&quot; or &quot;quick.&quot;

====Media sharing websites====
Due to low bandwidth connections (dialup) available in the early internet, image files were the dominant media file format during the 1990s. Digital audio for CDs and larger hard drives made audio files an increasingly popular file format during the 1990s. DVD use did not surpass video tape until 2003. Digital cameras and personal computers with optical disk drives became increasingly common in the early years of this century. Image sharing websites such as [[w:Flickr|Flickr]] and video sharing websites such as [[w:YouTube|YouTube]] allow users to upload and share their pictures and video. Broadband internet, larger hard drives, and faster CPUs in personal computers now allow more individuals to work with digital video files. Websites such as YouTube provide user interfaces that include support for text-based special-interest discussion groups as well as video blogs.

====Social Networking websites====
There are a variety of social networking websites, including [http://www.facebook.com Facebook], [http://www.myspace.com Myspace], [http://www.linkedin.com/ LinkedIn], [http://www.orkut.com/ Orkut], [http://www.ping.com/ Ping], [https://plus.google.com/ Google+], and [http://twitter.com/ Twitter]. These sites facilitate online communication through a variety of media. The interactive, interlinking environment supports the creation of personal and business webpages where information, photos and videos are shared. In February 2010, Google released [http://www.google.com/buzz Google Buzz], a service for sharing thoughts, multimedia, and social media website feeds using the existing email service, [[w:Gmail|Gmail]]. Google Buzz is an open environment that adheres to open standards, meaning that developers will be able to create applications for buzz across many platforms. Google buzz was retired in 2011.  In its place, Google launched [https://plus.google.com/ Google+] in 2011. Google+ integrates social services such as Google Profiles, and other services like Circles and Hangouts.  Google+ is available as a website and on mobile devices.  Social bookmarking sites, such as [http://delicious.com/ Delicious], [http://digg.com/ Digg], [http://www.reddit.com/ Reddit], and [http://www.stumbleupon.com/ StumbleUpon] allow people to discover, organize and share content on the web and to access their own favorites from any personal computer.

==Web 2.0 Research==
For an overview on some of the recent research work and the resulting application on Web 2.0, refer to:

'''&quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications'''&quot;, '''San Murugesan (Editor),''' http://www.igi-global.com/reference/details.asp?id=34850, Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}

'''&quot;Why Web 2.0 is Good for Learning and for Research: Principles and Prototypes'''&quot;,http://wwwconference.org/www2008/papers/pdf/p705-ullrichA.pdf, WWW 2008, The International World Wide Web Conferences CommitteeApril 21–25, 2008, Beijing, China, ACM 978-1-60558-085-2/08/04.

==Activities==
*Find a Web 2.0 technology that you have not previously used and describe your experience on a Wikiversity page.  Some ideas are:
**Start a blog [[Web Writing/Blogs|blog]] and post several entries each week.
***Follow at least five blogs and comment on them regularly
***Upload at least five vlogs (video logs). Here you can find instructions on how to create a [http://weblogs.about.com/od/creatingablog/ht/CreateaVlog.htm vlog].
***Choose three Web 2.0 tools that you haven’t used before and start using them for at least a week.  Then post a link to your profile and share your experiences on your blog.  If you have items you can embed such as photos, videos, or audio then embed them to your blog as well.
**Create and upload some digital media, such as photos or video.
***Upload photos to a photo sharing website.  Then share those photos with others.
***Create a slideshow of your photos and embed them in your blog, [http://www.ning.com Ning], [http://www.facebook.com Facebook], [http://www.myspace.com MySpace], etc.
**Create a glossary and have students add the latest Web 2.0 tools both name and definition. 
**Create an account at a [[wiki]] website and edit or add content.
**Join a social networking website and connect with family, friends, and/or coworkers.
*** Sign up to follow 5 or more friends/classmates in social networks such as Twitter, Flickr, and blogs.
*Participate in web content creation at a project such as [http://wherearethejoneses.wikidot.com/ Where are the Joneses]. Describe your experience on your blog.
*Create a Wikimedia Foundation user name, view the tutorials on Wikipedia, then find one article (on a topic you are knowledgeable about) to edit and improve on Wikipedia.
*What data exist documenting participation in blogging, wiki editing, creation and sharing of digital media?
**[http://technorati.com/weblog/2006/11/161.html technorati stats 2006]
**[http://wm.sieheauch.de/?cat=4 wikimetrics] - a blog with some useful information
*Visit 10 or more Web 2.0  sites like Flicker, Twitter, Facebook, register and place your profile. Have a system for organizing this information - perhaps starting out with a bookmarking site like Delicious can keep disorganization at bay. Consider creating one log-in that can be used at various sites. Some sites won't allow a log-in that begins with a number, so explore and share your findings with others on your blog.
**Explore a social network aggregation platform and register at least 5 social networking accounts/profiles with it. Streamline all your social networking activities into a RSS news feed.
**Which sites do you visit most often?  Why?
**What do the good sites have in common?
* Link 5 or more of your sites to each other.
* Sign up for an RSS feed in an area of interest. Follow the feed for 3 days and write about the experience.
*Explore an online video sharing site, such as YouTube or Google Videos. Create an account and examine the features of the site.  Consider what makes this a Web 2.0 technology.  Upload a video of your choice. Watch this [http://www.youtube.com/watch?v=_O7iUiftbKU video] if you need assistance. Write about your experience here or on your blog.
*Join a [http://www.ning.com Ning] network (a &quot;do it yourself&quot; site built from scratch to create a social network of your own) and get a feel for how it works.  Explore all of the features and settings.  Then start your own network and invite others to join.  Be sure to include blog entries, photos, videos, links, Web 2.0 profiles, and other content.  Be sure to connect to other Web 2.0 tools with your Ning network.  This helps people to reach out and connect to users with the same interests, and produce a happy environment.
*See a link you want to save, or have many links that you want to share with others? Try social bookmarking-you can store, organize, share, and network with other users. Sites like [http://www.digg.com Digg], [http://www.pinterest.com Pinterest], [http://www.stumbleupon.com StumbleUpon],[http://www.delicious.com Delicious] are great places to start.
*Use several Web 2.0 tools that are similar.  Then compare and contrast their features.  Which ones did you like the most?  Why?  Which ones did you like the least?  Why?  Post your experiences to your blog.  Some examples of tools are:
**[[w:Social networking|Social networking]]
**[[w:Social bookmarking|Social bookmarking]] 
**[[w:List_of_video_sharing_websites|Video Sharing]]
**Audio Sharing
**[[w:Photo sharing|Photo sharing]]
*Watch the following video produced by Karl Fisch and Scott McLeod to see the future of Education, Learning, and the Role of the Internet and Web 2.0 in the educational process: 
**Did You Know 2.0: https://www.youtube.com/watch?v=pMcfrLYDm2U

==Internet content: ownership and sharing==
&quot;Web 2.0&quot; is a term that can be used to refer to a qualitatively new and different pattern of internet behavior: a shift from an older era of restricted and expensive technologies for creation and internet-based sharing of digital media files to a new era of increasingly accessible and inexpensive technologies. As more and more people become empowered to participate on the internet as content producers, new patterns of content ownership and sharing have come into existence. The traditional model was that expensive digital content was protected by copyright, copies were sold and derivative works were possible only via rare and expensive special licensing agreements.

An alternative approach to digital media began with the [[w:Open-source software|Open-source software]] industry. Recognizing that software innovation is promoted by making software &quot;open&quot; to a distributed community of developers, some software developers began to experiment with new strategies for licensing software. In 2001, Wikipedia was launched with contents licensed under the GFDL and the [[w:Creative Commons|Creative Commons]] licenses began to be developed. A growing [[w:Free Culture movement|Free Culture movement]] supports the licensing of digital media files so as to facilitate file sharing and re-use of media for the creation of new works. In the collaborative environment of Web 2.0, sharing intellectual property, without the intermediate step of requesting permission directly from the owner, allows easier access to materials and fosters greater creativity.  However, owners of intellectual property must consider whether the Free Culture Movement adds value or takes away value from their work.  While some intellectual property might gain value from easier access, other intellectual property like artists' works might lose value.

==Tools for digital media file creation==
Wikiversity [[learning resource]]s for digital media file creation and editing.
* Text
**[[Operating a word processing application|ICAU1129A - Operate a word processing application]]
**[[Word processing challenges]]
*[http://web2.sys-con.com/ Web 2.0 Journal]

* Website Creation
**[http://www.adobe.com/products/dreamweaver.html Adobe Dreamweaver]
**[http://textwrangler.en.uptodown.com/mac Textwrangler]
**[http://www.onblastblog.com/ OnBlastBlog]

* Blogs
**[http://www.wordpress.org/ WordPress]
**[https://www.tumblr.com/ Tumblr]
**[https://www.www.blogger.com/ Blogger]
**[http://www.weebly.com/ Weebly]
**[http://www.typepad.com/ Typepad]

*Images
**[[GNU Image Manipulation Program (GIMP)]]
**[[Review:Inkscape]]
**[https://pixlr.com Pixlr]
**[[Adobe Photoshop]]
**[[Paint.net]]
**[[2D Animation process]]
**[http://www.photoscape.org/ps/main/index.php PhotoScape]
**[[Sumo Paint]] - Online photo and painting manipulation application
**[[Artweaver]] - Photo manipulaton application (Free version and for fee)
**[http://www.aviary.com/ Aviary]- An online suite of graphic and audio programs

*Audio
**[[Audacity]] - Free sound editor for recording and editing
**[[Wiki Campus Radio]] - Live internet chat and podcasts
**[[Podcasting]] - includes video podcasting
**[[:Category:Apple GarageBand|Apple GarageBand]] - Macintosh software for manipulating digital audio
**[http://www.avid.com/US/products/pro-tools-software/ Pro Tools]

*Video
**[[Landscape rendering]] - computer-generated landscape scenes
**[[DAZ Studio/Human figures|Rendering models of living organisms]] - computer-generated characters
**[[Filmmaking Basics/3D Storyboard|Lesson:3D Storyboard]] - at [[Filmmaking|Narrative film production, Wikiversity Film School]]
**[[Digital Puppet Animation|Course:Digital Puppet Animation]] - 3D computer-generated animation 
**[[Video software|Apple iMovie]] - Video editing software
**[[Quicktime Player/Recorder 10.0]]- screen/video recording
**[http://www.apple.com/final-cut-pro/ Final Cut Pro]

*Animation
**[[GoAnimate]] - a cloud-based tool for creating short cartoons, including 'scenes', 'props', and text-to-voice software to create dialogue.

See additional links at [[Portal:Internet audio and video|Topic:Internet audio and video]].&lt;br&gt;
See also: [[Template:Web 2.0]]; 
*[http://www.go2web20.net/ Web 2.0 Tools and Applications]

==See also==
*[[Wikiversity:Interactive learning resources|Rich Learning Resources Platform]]
*[[Twitter]]
*[[Wikipedia:List of free software for Web 2.0 Services|List of free software for Web 2.0 Services]]
*[[Social_Media|Social Media]]

===Wikipedia===
* [[w:Web 2.0 for development|Web 2.0 for development]]

==External links==
*[http://www.martinblueprint.co.uk/wikka/wikka.php?wakka=HomePage Web 2.0 research] - by M. Dimartino Marriott.

[[Category:Information technology]]
[[Category:Web_Technology]]
[[Category:Secondary research]]
[[Category:Web 2.0]]
Concepts
Social software has emerged as a major component of the Web 2.0 movement. The idea dates as far back as the 1960s and JCR Licklider’s thoughts on using networked computing to connect people in order to boost their knowledge and their ability to learn. The Internet technologies of the subsequent generation have been profoundly social, as listservs, Usenet groups, discussion software, groupware, and Web-based communities have linked people around the world. During the past few years, a group of Web projects and services became perceived as especially connective, receiving the rubric of “social software”: blogs, wikis, trackback, podcasting, videoblogs, and enough social networking tools like MySpace and Facebook to give rise to an abbreviation mocking their very prevalence: YASN (Yet Another Social Network). Consider the differences between these and static or database-driven Web pages. Wikis are all about user modification; CNN’s front page is decisively not. It is true that blogs are Web pages, but their reverse-chronological structure implies a different rhetorical purpose than a Web page, which has no inherent timeliness. That altered rhetoric helped shape a different audience, the blogging public, with its emergent social practices of blogrolling, extensive hyperlinking, and discussion threads attached not to pages but to content chunks within them. Reading and searching this world is significantly different from searching the entire Web world. Still, social software does not indicate a sharp break with the old but, rather, the gradual emergence of a new type of practice.

==References==
* &quot;Understanding Web 2.0&quot;, San Murugesan, IEEE IT Professional, 2007 
* &quot;Handbook of Research on Web 2.0, 3.0, and X.0: Technologies, Business, and Social Applications&quot;, San Murugesan (Editor), Information Science Research, Hershey – New York, October 2009, {{ISBN|978-1-60566-384-5}}
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==Grammar==

Just like in English, Swedish verbs have two inflected tenses: present and past. 
'''Verb forms: example ''arbeta'''''
{| class=&quot;wikitable&quot; border=&quot;1&quot;
!Form
!English
!Swedish
!Formation
|-
|Imperative
|''----!''
| ----!
|Stem
|-
|Infinitive
|''(to) ----''
|(att) ----
|stem + ''-a,'' or no change
|-
|Present
|''(I) ----''
|(jag) ''----'''''r'''
|stem + ''-r, -er''
|-
|Past
|''(I) ----ed''
|Jag ''----'''''de'''
|stem + ''-de, -te''
|-
|Supine
|''(I have/had) ----ed''
|Jag har ''----'''''t'''
|stem + ''-t, -tt, -it''
|}
'''1st conjugation tenses: example ''arbeta'''''
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Tense
! English
! Swedish
! Form of verb
|-
|Imperative
|''Work!''
|Arbeta!
|stem
|-
| Present
| ''I work''
| Jag arbeta'''r'''
| stem + ''-r''
|-
| Past
| ''I worked''
| Jag arbeta'''de'''
| stem + ''-de''
|-
| Simple perfect
| ''I have worked''
| Jag har arbeta'''t'''
| ''har +'' supine
|-
|Past perfect
|''I had worked''
|Jag hade arbeta'''t'''
|''hade'' + supine
|-
| Simple future
| ''I will work''
| Jag ska arbeta
| ''ska'' + infinitive
|}

'''2nd conjugation: example ''bränna'''''
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Tense
! English
! Swedish
! Form of verb
|-
|Imperative
|''Burn!''
|Bränn!
|Stem
|-
| Present
| ''I burn''
| Jag bränn'''er'''
| stem + ''-er''
|-
| Past
| ''I burned''
| Jag brän'''de'''
| stem + ''-de, -te''
|-
| Simple perfect
| ''I have burned''
| Jag har brän'''t'''
| ''har'' + supine
|-
|Past perfect
|''I had burned''
|Jag hade brän'''t'''
|''hade'' + supine
|-
| Simple future
| ''I will burn''
| Jag ska bränn'''a'''
| ''ska'' + infinitive
|}
'''3rd conjugation: example ''tro'''''
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Tense
! English
! Swedish
! Form of verb
|-
|Imperative
|Believe!
|Tro!
|Stem
|-
| Present
| ''I believe''
| Jag tro'''r'''
| stem + ''-r''
|-
| Past
| ''I believed''
| Jag tro'''dde'''
| stem + ''-dde''
|-
| Simple perfect
| ''I have believed''
| Jag har tro'''tt'''
| ''har'' + supine
|-
|Past perfect
|''I had believed''
|Jag hade tro'''tt'''
|''hade'' + supine
|-
| Simple future
| ''I will believe''
| Jag ska tro
| ''ska'' + infinitive
|}


'''Irregular verbs'''

Like English, Swedish also has irregular verbs (past tense is shown through a vowel change). Most verbs that are irregular in English are also irregular when translated to Swedish. 

 {| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Form
! English
! Swedish
! Formation
|-
|Imperative
|''eat!''
|äta!
|Stem
|-
| Infinitive
| ''(to) eat''
| (att) äta!
| stem + ''-a,'' or no change
|-
| Present
| ''(I) eat''
| (jag) äter
| stem + ''-r, -er''
|-
| Simple past
| ''ate''
| åt
| vowel change
|-
|Past participle
|''eaten''
|ätit
|stem + ''-t, -tt, -it''
|}

{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Tense
! English
! Swedish
|-
| Imperative
| ''Walk!''
| Gå!
|-
| Present
| ''I walk''
| Jag gå'''r'''
|-
| Past
| ''I walked''
| Jag gick
|-
| Simple perfect
| ''I have walked''
| Jag har gått
|-
| Past perfect
| ''I had walked''
| Jag hade gått
|-
|Simple future
|''I will walk''
|Jag ska gå
|}

Some irregular verbs combine a vowel change with word endings:
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
! Tense
! English
! Swedish
! Formation
|-
|Imperative
|''Find!''
|Finn!
|Stem
|-
| Present
| ''I find''
| Jag finn'''er'''
| stem + ''-er''
|-
| Past
| ''I found''
| Jag fann
| Stem + vowel change
|-
| Simple perfect
| ''I have found''
| Jag har funn'''it'''
| ''har'' + vowel change + ''-it''
|-
|Past perfect
|''I had found''
|Jag hade funn'''it'''
|''hade'' + vowel change + ''-it''
|-
| Simple future
| ''I will find''
| Jag ska finna
| ''ska'' + infinitive
|}


''The irregular verbs '''har/hade''' and '''ska/skulle''' are used in complex verb tenses in much the same way as '''have/had''' and '''will/would''' are used in English.''

'''har &amp; hade'''
&lt;table&gt;
&lt;tr&gt;&lt;TD&gt;(att) ha&lt;/TD&gt;&lt;TD&gt;''to have''&lt;/TD&gt;&lt;/tr&gt;   
&lt;tr&gt;&lt;TD&gt;jag hade&lt;/TD&gt;&lt;TD&gt;''I had''&lt;/TD&gt;&lt;/tr&gt; 
&lt;tr&gt;&lt;TD&gt;jag har haft&lt;/TD&gt;&lt;TD&gt;''I have had''&lt;/TD&gt;&lt;/tr&gt;

&lt;tr&gt;&lt;TD&gt;jag har&lt;/TD&gt;&lt;TD&gt;''I have''&lt;/TD&gt;&lt;/tr&gt;
&lt;/table&gt;

'''ska &amp; skulle'''
&lt;table&gt;
&lt;tr&gt;&lt;TD&gt;jag ska&lt;/TD&gt;&lt;TD&gt;''I will''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;jag skulle&lt;/TD&gt;&lt;TD&gt;''I would''&lt;/TD&gt;&lt;/tr&gt;
&lt;/table&gt;


''Note that Swedish does not have a correlate to the English continuous/progressive forms. In '''extremely''' formal language, you could use a noun form to express this concept: Jag är i arbete''
{| class=&quot;wikitable&quot;
|+
!English
!Swedish
|-
|''I work''
|Jag arbetar
|-
|''I am working''
|Jag arbetar
|-
|''I worked''
|Jag arbetade
|-
|''I was working''
|Jag arbetade
|}

== Participles ==

=== Past participles ===
In English, the supine and past participle forms of verbs are the same. In Swedish however, the past participle form is separate from the supine, and is only used for adjectivialising the verb. For instance:
{|
&lt;TD&gt;''det '''gångna''' året''.
&lt;/TD&gt;&lt;TD&gt;''the '''past''' year''&lt;/TD&gt;
|}
'''Regular verb past participle:'''

If the following noun is in the singular common gender, then the past participle is formed by adding -d to the stem. If it is in the singular neuter gender, it is instead formed with -t. All plural forms of the regular past participle are formed with -de.

'''Irregular verb past participle when verb stem ends with a:'''

If the following noun is in the singular common gender, then the past participle is formed with -en. If it is in the singular neuter gender, it is formed with -et instead. All plural forms are formed with -na.

'''Irregular verb part participles when verb stem ends with any other vowel:'''

If the following noun is in the singular common gender, then the past participle is formed with -ngen. If it is in the singular neuter gender, it is formed with -nget instead. All plural forms are formed with -ngna.

=== Present participles ===
In Swedish, present participles are formed with -ande, unless the verb stem ends with any vowel other than a, in which case it is formed with -ende. Like in English, present participles can be used adjectivially, adverbially/to make another verb in the sentence more specific, and as a noun. ''Unlike'' in English (as mentioned previously), present participles '''cannot''' be used as continuous/progressive forms.

'''Adjectivially'''&lt;table&gt;
&lt;tr&gt;&lt;TD&gt;Klara har ett '''lysande''' leende&lt;/TD&gt;&lt;TD&gt;''Klara has a '''beeming''' smile''&lt;/TD&gt;&lt;/tr&gt;&lt;/table&gt;'''Adverbially'''&lt;table&gt;
&lt;tr&gt;&lt;TD&gt;Carl kom '''hoppande'''&lt;/TD&gt;&lt;TD&gt;''Carl came '''hopping'''''&lt;/TD&gt;&lt;/tr&gt;&lt;/table&gt;
{|
&lt;TD&gt;Jag ringer '''angående''' din annons
&lt;/TD&gt;&lt;TD&gt;''I'm calling '''in regards to''' your ad''&lt;/TD&gt;
|}
'''Nounally'''&lt;table&gt;
&lt;tr&gt;&lt;TD&gt;ett '''meddelande'''&lt;/TD&gt;&lt;TD&gt;''a '''message'''''&lt;/TD&gt;&lt;/tr&gt;&lt;/table&gt;

==Example sentences==
&lt;table&gt;
&lt;tr&gt;&lt;TD&gt;Jag arbetade igår.&lt;/TD&gt;&lt;TD&gt;''I worked yesterday.''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;Det har gått ett år.&lt;/TD&gt;&lt;TD&gt;''One year has passed.''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;Ni ska arbeta i morgon.&lt;/TD&gt;&lt;TD&gt;''You will work tomorrow.''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;De gick och arbetade i parken.&lt;/TD&gt;&lt;TD&gt;''They went for work in the park.'' (or ''They walked and worked in the park.'')&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;De arbetade och gick i parken.&lt;/TD&gt;&lt;TD&gt;''They worked and walked in the park.''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;Vi skulle ha gått dit.&lt;/TD&gt;&lt;TD&gt;''We should have gone there.''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;Jag hade tur.&lt;/TD&gt;&lt;TD&gt;''I was lucky.''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;Jag arbetar (nu).&lt;/TD&gt;&lt;TD&gt;''I am working (right now).''&lt;/TD&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;TD&gt;Jag arbetar två dagar i veckan.&lt;/TD&gt;&lt;TD&gt;''I work two days a week.''&lt;/TD&gt;&lt;/tr&gt;
&lt;/table&gt;

==Example text==
'''Kalles väg till arbetet''' 

Det var morgon. Kalle gick ut på gatan. Han skulle gå vägen till arbetet. Den här vägen hade han gått många gånger förut. &quot;När jag har arbetat klart ska jag gå hem igen.&quot;, sa Kalle. &quot;Fast då går jag nog en annan väg!&quot;

'''Kalle's way to his work.'''

''It was morning. Kalle entered the street. He would walk the road to his work. This road he had walked many times before. &quot;When I've finished work I will walk home again.&quot;, Kalle said. &quot;But then I'll probably walk another route!&quot;''

==Exercises==
''Please translate into English:''

1. De har gått en annan väg.

2. Erik och Lina gick ut på puben.

3. Det var många i parken.

4. Hon har varit duktig.

5. Jag arbetade förut.

''Please translate into Swedish:''

6. Per, you are lucky!

7. He has another car.

8. She is exiting now.

9. Once upon a time mummy was lucky.  

10. What is the meaning of this?


[[../Answers to exercises/]]

==Glossary==
&lt;table&gt;
&lt;tr&gt;&lt;td&gt;annan&lt;/td&gt;&lt;td&gt;''another, other''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;den här/det här&lt;/td&gt;&lt;td&gt;''this''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;då&lt;/td&gt;&lt;td&gt;''then''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;fast&lt;/td&gt;&lt;td&gt;''but (in spoken language)''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;förut&lt;/td&gt;&lt;td&gt;''previously''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;att arbeta klart&lt;/td&gt;&lt;td&gt;''to finish work''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;ett arbete&lt;/td&gt;&lt;td&gt;''a work''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;en gata&lt;/td&gt;&lt;td&gt;''a street''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) gå&lt;/td&gt;&lt;td&gt;''to walk''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) gå dit&lt;/td&gt;&lt;td&gt;''to walk there''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) gå ut&lt;/td&gt;&lt;td&gt;''to exit''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) gå ut på&lt;/td&gt;&lt;td&gt;''to enter [a street], to go to [the pub], to mean something''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;en gång&lt;/td&gt;&lt;td&gt;''one time''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) ha&lt;/td&gt;&lt;td&gt;''to have''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) ha tur&lt;/td&gt;&lt;td&gt;''to be lucky''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;ett hem&lt;/td&gt;&lt;td&gt;''a home''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;en morgon&lt;/td&gt;&lt;td&gt;''a morning''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;nu&lt;/td&gt;&lt;td&gt;''now''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;många&lt;/td&gt;&lt;td&gt;''many''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;nog&lt;/td&gt;&lt;td&gt;''probably, enough''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;när&lt;/td&gt;&lt;td&gt;''when''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;en pub&lt;/td&gt;&lt;td&gt;''a pub''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;(att) säga&lt;/td&gt;&lt;td&gt;''to say''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;vad&lt;/td&gt;&lt;td&gt;''what''&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td&gt;en väg&lt;/td&gt;&lt;td&gt;''a road, way, route''&lt;/td&gt;&lt;/tr&gt;
&lt;/table&gt;
[[Category:Language introductions]]
[[Category:Swedish]]
[[Category:Verbs]]
[[Category:Grammatical tenses]]</text>
      <sha1>im7yy88st8x2fc7gjedbocq4qet1k74</sha1>
    </revision>
  </page>
  <page>
    <title>Czech Language/Family Members</title>
    <ns>0</ns>
    <id>51132</id>
    <revision>
      <id>2692700</id>
      <parentid>2614585</parentid>
      <timestamp>2024-12-19T20:57:38Z</timestamp>
      <contributor>
        <ip>108.178.83.254</ip>
      </contributor>
      <comment>/* Grouped by relation (cs-en) */</comment>
      <origin>2692700</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2063" sha1="cqralbpcrq48uxqjo9kwqhbs78ooay9" xml:space="preserve">[[Image:Cs_family_tree.svg|thumb|right|340px|''Top to bottom, left to right: grandmother, grandfather; mother, father; sister, brother, husband; son, daughter.'']]

== Vocab List ==

=== Grouped by relation (cs-en) ===
''rodina'' = family

* '''''prarodiče'' = grandparents''' 
** ''babička'' = grandmother
*** ''prababička'' = great-grandmother
** ''dědeček'' = grandfather
*** ''pradědeček'' = great-grandfather
* '''''rodiče'' = parents'''
** ''matka'' = mother
*** ''máma, maminka'' = mum, mama
** ''otec'' = father
*** ''táta, tatínek'' = dad, daddy
* '''''strýc''/''teta'' = uncle/aunt'''
* '''''synovec''/''neteř'' = nephew/niece'''
* '''''bratranec''/''sestřenice'' = cousin'''
* '''spouses &amp; in-laws'''
** ''manžel'' = husband
** ''manželka'' = wife
** ''tchán'' = father-in-law
** ''tchýně'' = mother-in-law
** ''švagr'' = brother-in-law
** ''švagrová'' = sister-in-law
** ''zeť'' = son-in-law
** ''snacha'' = daughter-in-law
* '''''sourozenci'' = siblings'''
** ''bratr'' = brother
*** ''brácha'' = bro
*** ''bratříček'' = little bro
** ''sestra'' = sister
*** ''ségra'' = sis
*** ''sestřička'' = little sis
* '''''děti'' = children'''
** ''dcera'' = daughter
** ''syn'' = son
* '''''vnoučata'' = grandchildren'''
** ''vnuk'' = grandson
** ''vnučka'' = granddaughter

=== Alphabetical (en-cs) ===
* aunt = ''teta''
* brother = ''bratr''
* brother-in-law = ''švagr''
* cousin (m/f) = ''bratranec''/''sestřenice''
* daughter = ''dcera''
* daughter-in-law = ''snacha''
* father = ''otec''
* father-in-law = ''tchán''
* grandaughter = ''vnučka''
* grandfather = ''dědeček''
* grandmother = ''babička''
* grandson = ''vnuk''
* great-grandfather = ''pradědeček''
* great-grandmother = ''prababička''
* husband = ''manžel''
* mother = ''matka''
* mother-in-law = ''tchýně''
* nephew = ''synovec''
* niece = ''neteř''
* sister = ''sestra''
* sister-in-law = ''švagrová''
* son = ''syn''
* son-in-law = ''zeť''
* uncle = ''strýc''
* wife = ''manželka''


[[Category:Czech_Vocab_Resources]]
[[Category:Family]]</text>
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  </page>
  <page>
    <title>Educational Media Awareness Campaign/Human anatomy/POTD 3</title>
    <ns>0</ns>
    <id>55798</id>
    <revision>
      <id>2692643</id>
      <parentid>2172861</parentid>
      <timestamp>2024-12-19T17:27:05Z</timestamp>
      <contributor>
        <username>Ternera</username>
        <id>2468111</id>
      </contributor>
      <comment>png --&gt; svg ([[Commons:Commons:GlobalReplace|GlobalReplace v0.6.5]])</comment>
      <origin>2692643</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="506" sha1="7tlenkbz1m5a5f9akyti2v4m823fq87" xml:space="preserve">{{Educational Media Awareness Campaign/POTD|The Heart|Heart numlabels.svg||b1=1. Right atrium&lt;br&gt;2. Left atrium&lt;br&gt;3. Superior Vena Cava&lt;br&gt;4. Aorta&lt;br&gt;5. Pulmonary Artery&lt;br&gt;6. Pulmonary Vein&lt;br&gt;7. Mitral valve&lt;br&gt;|b2=8. Aortic valve&lt;br&gt;9. Left ventricle&lt;br&gt;10. Right ventricle&lt;br&gt;11. Inferior Vena Cava&lt;br&gt;12. Tricuspid Valve&lt;br&gt;13. Pulmonary Valve&lt;br&gt;&amp;nbsp;&lt;br&gt;|[[:commons:Category:Medicine|Medical images]] - [[:commons:Category:Anatomy|Anatomy images]] - [[:commons:Category:Biology|Biology images]]}}</text>
      <sha1>7tlenkbz1m5a5f9akyti2v4m823fq87</sha1>
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  </page>
  <page>
    <title>Wikiversity:VRT</title>
    <ns>4</ns>
    <id>65173</id>
    <revision>
      <id>2692690</id>
      <parentid>2303855</parentid>
      <timestamp>2024-12-19T20:42:16Z</timestamp>
      <contributor>
        <username>Nintendofan885</username>
        <id>2887676</id>
      </contributor>
      <minor />
      <comment>/* See also */ change more</comment>
      <origin>2692690</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1025" sha1="3pu3x40atkufi88h9x3o5d1v6exoqfa" xml:space="preserve">Within Wikiversity, the '''Volunteer Response Team''' ('''VRT''') refers to the people and software that organize, handle, and respond to e-mails sent to the [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF) in regards to Wikiversity issues. The WMF utilises VRTS software to process the e-mail that it receives. This provides an organized way for multiple people to categorize and respond to email. 

VRT can be reached through email at info-en(AT)wikimedia.org or info(AT)wikiversity.org

==Wikiversity participants who have VRT access==
(This is a partial list - please add yourself if you have access)
*[[User:Mikael Häggström|Mikael Häggström]]

==Wikiversity participants who have applied for VRT access==

==See also==
*[[Wikiversity:Copyright issues]] - email contact + procedure
*[[meta:VRT]]
*[[w:Wikipedia:VRT|Wikipedia:VRT]]

* [https://secure.wikimedia.org/otrs/index.pl VRT login] (no public access)
* [https://vrt-wiki.wikimedia.org VRT Wiki] (no public access)

[[Category:Wikiversity administration]]</text>
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      <comment>/* See also */ interface</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1029" sha1="nhce19zwpf342jx7w1byn3colal3208" xml:space="preserve">Within Wikiversity, the '''Volunteer Response Team''' ('''VRT''') refers to the people and software that organize, handle, and respond to e-mails sent to the [[w:Wikimedia Foundation|Wikimedia Foundation]] (WMF) in regards to Wikiversity issues. The WMF utilises VRTS software to process the e-mail that it receives. This provides an organized way for multiple people to categorize and respond to email. 

VRT can be reached through email at info-en(AT)wikimedia.org or info(AT)wikiversity.org

==Wikiversity participants who have VRT access==
(This is a partial list - please add yourself if you have access)
*[[User:Mikael Häggström|Mikael Häggström]]

==Wikiversity participants who have applied for VRT access==

==See also==
*[[Wikiversity:Copyright issues]] - email contact + procedure
*[[meta:VRT]]
*[[w:Wikipedia:VRT|Wikipedia:VRT]]

* [https://secure.wikimedia.org/otrs/index.pl VRT interface] (no public access)
* [https://vrt-wiki.wikimedia.org VRT wiki] (no public access)

[[Category:Wikiversity administration]]</text>
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  <page>
    <title>Talk:Doctor of Philosophy</title>
    <ns>1</ns>
    <id>106391</id>
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      <text bytes="20059" sha1="8ltelui05xdyr98o7nf1zagv9len1mc" xml:space="preserve">==Participants==
# [[User:Leighblackall|Leighblackall]] 22:54, 21 January 2011 (UTC)
# [[User:Peterrawsthorne|Peterrawsthorne]] 16:19, 23 January 2011 (UTC)
# [[User:Steelemaley|Steelemaley]] 18:31, 1 June 2011 (UTC)
# Back in November 2010 I posted a message about my project on open PhD and now I see a group of people networking together to get this done. My congratulations for starting this group and I would like to join you in this endevour. I look forward to hearing from you. [[User:Opriter/PhD]] August 14 2011 -- ''Good to hear from you Opriter, I hope you found your way into the email forum... over time, I think our group will slowly grow. [[User:Leighblackall|Leighblackall]] ([[User talk:Leighblackall|talk]]) 12:23, 13 July 2012 (UTC)''
# Your name

==Communications==
* [https://groups.google.com/group/open-and-networked-phds?hl=en email forum]
* [http://twitter.com/#/list/leighblackall/openphd Twitter list] (I'm a bit unsure on twitterlists, I've created one under my username space.. is there a way to create one in its own name space that anyone can join? And it might be better to call it onphd for open and networked doctor of philosophy) [[User:Leighblackall|Leighblackall]] 23:26, 21 January 2011 (UTC)

==Peer assessment, badges, institution assessment==
*On 18 April 2012, #### introduced himself by email to Peter Rawsthorne and Leigh Blackall, explaining that he was interested in making an effort toward an #onPhD
*Leigh encouraged #### to [https://groups.google.com/forum/?fromgroups#!forum/open-and-networked-phds join the email forum] and mentioned that Charles Darwin University has a PhD-by-publication program that assesses work created prior to enrolling. Peter explained his intension to looking into Badges, as a possible way of furthering his interests in empowering peer to peer assessment.
*On 11 July 2012 #### made contact with Leigh and Peter again, citing [https://plus.google.com/u/0/113557555125803834811/posts/M2P9iWWxyZz a Google+ link Leigh had posted] that contained a link to [http://www.cdu.edu.au/research/office/documents/howtoapply_PhD_by_publication.doc the CDU PhD-by-publication process]. #### Was following up the suggestion of finding a CDU based supervisor if the CDU process was to be possible, and asking Leigh to find out how much assessment of a PhD thesis would cost. 
*Leigh replied asking #### to provide a link to a web page of publications or works in progress that he would like to present for consideration. Peter challenged Leigh and #### to instead focus on peer review and badging before going to the institutions. Leigh and #### agreed.
*On 13 July 2012 (UTC+10) Leigh [https://twitter.com/leighblackall/status/223163994431688704 contacted Thomas Steele-Maley on Twitter] to check if he also had work ready to present. This conversation went to email, including Peter, so that Thomas could catch up on the conversation that had lead to an agreement to focus on peer to peer badging. Thomas posted to Twitter an endorsement of the peer to peer badging concept, using the #onPhD tag. Peter suggested that Wikiversity become an open badge issuer, and asked what the criteria for onPhD badge might be. Leigh suggested &quot;original, formatted, situated, open data(?), reviewed, reviews actioned, iterative, all process documented&quot; and that the group work up the criteria on [http://en.wikiversity.org/wiki/Doctor_of_Philosophy the WIkiversity page for PhD].  Peter updated the Wikiversity page with info about open badges. [https://twitter.com/steelemaley/status/223590587243311104 Thomas brought Mary Anne Reilly and Robert Greco into the Twitter exchange], suggesting to Mary Anne that she would make a good mentor. Mary Anne seemed to like the peer to peer badging concept, and Robert suggested contact with [http://www.mightymatthern.com/?page_id=130 Matt Hern] and [http://www.social-ecology.org/ The Institute for Social Ecology]. [https://twitter.com/leighblackall/status/223677461832007680 Leigh suggested a get together] on email or Google Hangout. American time zones went into night.
*[https://groups.google.com/forum/?hl=en&amp;fromgroups#!topic/open-and-networked-phds/3Gn58dOPkg4 Leigh posted to the onPhD email forum], an update on the conversation, and a link to these notes
--[[User:Leighblackall|Leighblackall]] ([[User talk:Leighblackall|talk]]) 12:18, 13 July 2012 (UTC)

==Folksonomy==
* tag = '''''onphd''''' Use it in delicious, youtube, blog posts, etc, and we can each subscribe to the RSS feeds for that tag and track each other's work at a distance. - Leigh, why didn't you suggest '''''openphd''''' as the tag? -- [[User:Peterrawsthorne|Peterrawsthorne]] 16:33, 24 January 2011 (UTC)
: Hi Pete, because I stupidly polluted that tag field with bookmarks to do with my PhD, rather than openphd generally, and after monitoring it for nearly a year, it seems no one is using it. I thought we may as well start a fresh, with onphd standing for open and networked - where the networked part is VERY important IMO [[User:Leighblackall|Leighblackall]] 04:19, 25 January 2011 (UTC)
::Ok, the open and networked makes complete sense. Actually, I see the networked part more important than the open part... but then it would become '''''nophd'''''.  Then of course, isn't that what we are working on, a '''''nophd'''''. [[User:Peterrawsthorne|Peterrawsthorne]] 08:27, 25 January 2011 (UTC)
::'''I've changed my mind!''' I think I'm going to start using '''''nophd''''' as my hashtag for a Networked and Open PhD. It makes more sense, from 1. a lifelong learning perspective and 2. how the PhD is a brand &quot;owned&quot; by academia. I know this now more than ever, what I am perusing has nothing to do with the academic credential of a PhD. And I have no desire to have the PhD for reputation or employment. I do not want to be an academic or a researcher. I want to follow my bliss and life-long learn (this base attachment is essential for completion). I want to share this with those who care to follow, I want to empower people with skills and knowledge in how to be a life-long learner for themselves within this connected world. I want to work on '''&quot;NO PhD&quot;'''. Sorry for the rant, just emotionally connecting to the idea - [[User:Peterrawsthorne|Peter Rawsthorne]] 20:52, 18 February 2011 (UTC)
:::I'm with you Pete. Its also a bit like the ''unconference'' trend in internet circles. Instead of unphd, we're saying no phd. I especially react negatively towards the credential inflation (not to mention, conflict of interest) that some Universities partake in by requiring their staff to all have PhDs.. which is what has compelled me into this space. I see that sort of policy as only diminishing the value of a PhD, as people scramble to get one not because that truly want or respect the process, but because they want to preserve their income. Completely understandable - its the policy at fault, not the individuals. So, my own decision to attempt nophd, is in many ways to reject that, and preserve the integrity of the PhD. Doing it not for job security, but for genuine interest and desire to learn the skills of a researcher. [[User:Leighblackall|Leighblackall]] 00:26, 9 April 2011 (UTC)
::::I can here the chorus and possible rancor around the point I am going to make but I really do not like the tag nophd as it seems to be an oxymoron that I still do not understand--and am open to deliberate fully.  My biggest issues revolve around positives and negatives and the power of symbols: onphd or Open and Networked PhD tells the story aptly...What does Open mean to all here and why is it an issue?....I do not seek the official knowledge of institutions and their social schooling and hierarchical colonial....but if we are to use the PhD at all I would rather associate this with the tag onphd which captures the spirit of my work and helps to redesign the terminology....Leigh mentions an interest in preserving the PhD, I agree wholeheartedly and I see this group providing space for the new and next in PhD....folksonomy is important and for this we have acronyms and thus this conversation is of import. Looking forward to the deliberation, I offer that onphd is better for this community...[[User:Steelemaley|Steelemaley]] 18:31, 1 June 2011 (UTC)
:::::Steelemaley, I think I went away and thought about your points, and forgot to post back. Over time I've come to agree with you, and #onPhD has similar cheeky meaning to #noPhD.. we're switching 'on'. I see it has caught on in Twitter a little, so I'm happy to keep using that tag. [[User:Leighblackall|Leighblackall]] ([[User talk:Leighblackall|talk]]) 12:21, 13 July 2012 (UTC)
:::::Steelemaley &amp; Leigh, I've grown... I've let go of my disruptive tendencies. I'm going with the #onphd, I'm turning 'on' and tuning in ;) The Open and Networked PhD is a powerful acronym and doesn't have the negative slant of the nophd. So we continue with #onphd, and when we want to plant our tongue deeply in our cheek we have the #nophd as the anti-tag ([[User:Peterrawsthorne|Peter Rawsthorne]] ([[User talk:Peterrawsthorne|talk]]) 15:30, 13 July 2012 (UTC))

==Leigh and Thomas Skype conversation 31 May 2011==
*General Discussion: Appropriation of terms and praxis, ethics and the critique of learning with internet based tools.
*How is technology critiqued and what impact does this have on research in the field? Thomas suggested Mander, J.[http://www.amazon.com/Absence-Sacred-Failure-Technology-Survival/dp/0871565099 In absence of the sacred] and [http://www.amazon.com/Arguments-Elimination-Television-Jerry-Mander/dp/0688082742/ref=pd_sim_b_1 Four Arguments for Elimination of Television] as sources. and Leigh suggested Bowers: False Promises of Constructivism http://leighblackall.blogspot.com/2011/05/summary-of-chet-bowers-false-promises.html
* the discussion turned to Anarchism and Thomas brought up [http://pinboard.in/search/?query=colinward+&amp;all=Search+All Colin Ward]and the ideas/praxis of &quot;mutual aid&quot; see [https://pinboard.in/search/?query=kropotkin+&amp;mine=Search+Mine Kropotkin]
*On Design Based Research (Thomas will generate book list), Leigh suggested Teemu Leinonen (Helsinki) practices design based research
*On polity v local knowledge and sharing much discussion about the role of the UN in education and culture Thomas suggested UESCO and specifically that  [http://portal.unesco.org/en/ev.php-URL_ID=35389&amp;URL_DO=DO_TOPIC&amp;URL_SECTION=201.html One Man Biosphere Project]represents a culturally relative UN project. Leigh spoke of  [http://freeuniversitysf.org/2011/02/15/this-campus-of-the-heart/ Free University of San Francisco] and the discussion of learning and culture in this movement and suggested more examples in Ubiquitous Learning paper... Thomas brought up the salience of human self determination and suggested the work of [http://pinboard.in/u:steelemaley/t:taiaiake/ Taiaiake Alfred] Leigh underscore the import that any polity &quot;can be reigned in at any moment&quot; if interfering with human self-determination, research for human collective future. Thomas suggested reading Boulding 1998 [http://www.amazon.com/Building-Global-Civic-Culture-Interdependent/dp/0815624875ivic Culture]
*On developing ethical frameworks/design for NO/ONPhD Leigh wanted to explore permaculture principles for ethical research and praxis in on/nophd. Thomas brought up walking Historians/sociologists/anthropologists which Leigh likened to Participatory Action Researchers. Leigh suggested we approach on/nophd as autoenthographic researchers, asking NO/ON PhD participants to write position statements focused on ethics and and/as an introduction
*Leigh is going to create a page on WV PhD - as relates Wikipedia, is to make sure that the range of approaches to PhD are documented, Thomas is LOOKING FORWARD to reading this or having a tag to research.
So good to meet.

[[User:Steelemaley|Steelemaley]] 18:31, 1 June 2011 (UTC)

==Badging ONPhD==
I think we need to work out how to set up a Hangout as an open invite, and first in gets a seat at the table.

Peter Rawsthorne I met and agreed that, with regard to Badging an Open and Networked PhD, we need a process for people to demonstrate eligibility as a candidate (Joelle, I think we need your help here). A PhD Candidate is someone who has been accepted by a university. Many people put this acceptance on their CVs and the like, that they are a PhD Candidate (meaning a PhD project is in progress). We think ONPhDs need an equivalent to this, a process that can enable them to declare they are prepared to undertake an ONPhD project. We thing P2PU is a good venue for this. Peter and I are going to start a P2PU &quot;course&quot;, based on a range of application forms and processes we have studied in formal universities, and hopefully Joelle's close guidance. The aim is to develop a P2PU &quot;course&quot; that serves as a guide for getting people prepared and articulate for setting themselves up for an ON PhD. The course would include things like:

* Identifying and listing past research projects and publications - or equivalent
* An expression of intent to undergo an ON PhD that demonstrates an understanding for the criteria for an ON PhD (documented online, openly, iteratively, etc (needs development)
* An outline of the intended project for the ON PhD, and evidence that the candidate knows the present situation and context for their project (acknowledges prior work, has the seeds of a question and its justification - such as understanding the projects epistemological and ontological bases) 
* Evidence of ready offers of supervision from qualified researchers. Letters supporting their plan, evidence of reviews given on prior work, etc
* Evidence of a preparedness to follow ethical boundaries in research work, with a process that obtains formal clearance deemed ideal

Peter is going to start the P2PU course, I'll follow him in when it's set up.
[[User:Leighblackall|Leighblackall]] ([[User talk:Leighblackall|talk]]) 02:39, 24 August 2012 (UTC)

==NoPhD==
Perhaps by analogy to [[w:en:NoSQL]]?  Anyway I think that presentation is more clear.  Also, note that it has &quot;worked&quot; quite well in the past, for people like Dr Richard Stallman (who now has almost as many honorary doctorates as cats have lives)!  I'm quite close to the end of a &quot;real&quot; Ph. D. so I don't want to get too distracted helping with this project, but I do have a lot of resources I could share, including [http://www.amazon.co.uk/Unwritten-Rules-Research-Study-Skills/dp/0335237029 this book] and [http://manainkblog.typepad.com/faultlines/files/BoudLee2005.pdf this paper].  I did something vaguely similar for a &quot;NoMA&quot;, which worked out OK, but I do think it would have benefitted from a bit more of a networked aspect. [[User:Arided|Arided]] ([[User talk:Arided|talk]]) 09:26, 12 November 2012 (UTC)

== Imitation of a real system ==

It seems to me, that openPh.D. is proposed as a copy of a real Ph.D. and my question here is, if we need it? I am wondering why a doctorate consists of entrance exams, passing different subjects, writing a dissertation thesis, ending exams, visiting another institution, and publishing an article or presentation at the conference. I guess the goal is to learn and demonstrate proficiency in a certain field and learn and demonstrate skills in doing research. But the question is to what degree? What should know about the Ph.D. and how does it differ from the knowledge of a docent (with academic title Doc.) or a professor (Prof.)? I am thinking about it, because the current system, at least in the Czech Republic, seems rigid, dysfunctional, and unclear, discouraging several promising scientists. Just comprehension, only 30% of Ph.D. of students complete their doctorate in the Czech Republic within the stipulated time (3 years), and I suspect that about 60 % do not complete it at all. So, to shed light on the reason why I would look for a new and better model of a doctorate with the use of modern technologies, I will present my negative experiences with the system in the Czech Republic:

&lt;nowiki&gt;#&lt;/nowiki&gt;The biggest problem is, you '''don't have a supervisor.''' For my master's degree, I wanted to do ethnobotany or simply sociological research, but the supervisor who dealt with this repeatedly rejected me, so I ended up with another one and did laboratory research in a completely different field. Just after completing my master's degree, I didn't continue to Ph.D. because of the lack of money. I had to take care of my apartment and the money provided by the University, where not enough. During my studies, I wanted to study second Master's in research of Agriculture education, but I was not able to find a supervisor. Years after I still had of finding a supervisor, but I got one unfortunately without a knowledge of the topic. So I have changed University, but I cannot enroll in Ph.D. because I haven't found a supervisor for the topic I wanted to study (information science - information behavior, or artwork documentation). So I am doing research in the field, but without being backed by the university and someone asks me, why I am not doing Ph.D. - This would take me to a question, why Ph.D. candidate needs a supervisor? Probably to help with methodology, explain how to do a pier review, provide tools or finances and help to publish the first article. But I had an experience, that you may find yourself in a position, where a supervisor is providing nothing of this. So then the question is, what are the other ways, you can get these skills and opportunities?

&lt;nowiki&gt;#&lt;/nowiki&gt;And secondly, concerning professional knowledge, I would say that it is possible to recharge it with a suitable offer of courses. Why would a person have to take subject exams at university and not be able to choose anywhere else? The problem is, for example, the '''low quality of teaching''' that I am experiencing during my current studies, '''the obsolescence of information in the given field,''' or the low professionalism and experience of teachers and researchers. So why not solve it with some open courses, or just an exam that verifies the student's knowledge and how he learns it is up to him? Because today you already de facto have to learn the subject yourself in your way and the school is not able to save you time and provide quality teaching.

&lt;nowiki&gt;#&lt;/nowiki&gt;Third '''problem was finances'''. On the first attempt, I cannot enroll at all. On the second attempt, the supervisor was calling the research to be extended if the doctorate has been prolonged officially from 3 to 4 years, which increased the budget of the project, but the department was not providing a grant.

So, if I had to sum it up, I think that the requirements for doctors should be defined and then simple ways to meet those requirements should be suggested. It will solve. In the European Union, a semester-long stay at a research workplace in another country is now required, but it might be possible to negotiate here, because, for example, in terms of financial support, the Union also offers support for people who are not students at any school. And if we didn't want to copy even this, we can create a new standard for Ph.D., which will not strive to be included in the group of other Ph.D.s, but will gradually build its authority. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:04, 26 April 2023 (UTC)

:@[[User:Juandev|Juandev]]: I only recall one user doing this, and unfortunately, I can't remember who it was. They would be the one to work with on the proposal. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:06, 28 April 2023 (UTC)

== Proposal in general ==

FYI I am [https://zenodo.org/records/14528438 proposing it differently]. Major difference would be that the certification is still needed. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:30, 19 December 2024 (UTC)</text>
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    <title>Understanding Arithmetic Circuits</title>
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      <timestamp>2024-12-19T14:23:15Z</timestamp>
      <contributor>
        <username>Young1lim</username>
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      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|A]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20241216.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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      <timestamp>2024-12-19T14:29:58Z</timestamp>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20211108.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20221130.pdf|A]]||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20241217.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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    <title>Complex analysis in plain view</title>
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      <timestamp>2024-12-19T14:23:32Z</timestamp>
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        <username>Young1lim</username>
        <id>21186</id>
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      <comment>/* Geometric Series Examples */</comment>
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      <text bytes="5241" sha1="13jwmgcjghfnxsgkey1cjcx00pcgdl7" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

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

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

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

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

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

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

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

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

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

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

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

====Geometric Series Examples====

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

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

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

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

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

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20241216.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|C.pdf]]) 

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

* Double Pole Case

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

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

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

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

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

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      <text bytes="853" sha1="3cx9t3anrdy6lxzsa3wd9fo4zlvw9mr" xml:space="preserve">&lt;!-- Add your name to the list using * ~~~~ N.Snipe --&gt;
Adopt this portal by adding your name [[{{titleparts|2}}/Participants|here]]:
*Dr Kunal Ashok Chaudhari Since 20 September 2012‎ with adding content to [[Draft:Medicine|Medicine]] --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 22:15, 9 May 2018 (UTC)
*Dr Kunal Ashok Chaudhari 29 January 2024
*Dr Andres Alberto Diaz Jaramillo 11 Marzo 2024
* Homzy Chaputula - 23 April 2020
* Mwanja Moses-since 29 April 2022
* Jakobina Kapweya - 26 May 2022
* Evaristo Mateyo - 29 June 2022
* Osama mohamed - 11 August 2022
* Adham wael - 11 August 2022
* Utkrisht Awasthi - 31 August 2023
*Lerato Koza -25 October 2023
* Arpita Singh - 24 December 2023
*Steven Smith- 24 December 2023
*Kevin Christopher - 19 December 2024</text>
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    <title>Was 9/11 an inside job?</title>
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      <text bytes="16956" sha1="9w54zeqzoxfz7en1tl1qggkp3fdz4cb" xml:space="preserve">{{Wikidebate}}

{{History}}

The [[Wikipedia:September 11 attacks|September 11, 2001 attacks]] in New York City and Washington D.C. forever changed the landscape of American culture and geopolitics around the globe. Is the official explanation of who conducted the attacks accurate and reliable? Or is it possible that some government agents had inside information of the attack or even planned the attack itself?

== 9/11 was an inside job ==

=== Pro ===

* {{Argument for}}  There is evidence of insider trading shortly prior to the attacks,&lt;ref&gt;{{Cite journal|last=Poteshman|first=Allen M.|date=2006|title=Unusual Option Market Activity and the Terrorist Attacks of September 11, 2001|url=https://www.jstor.org/stable/10.1086/503645|journal=The Journal of Business|volume=79|issue=4|pages=1703–1726|doi=10.1086/503645|issn=0021-9398}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|date=2021-03-31|title=9/11 conspiracy theories|url=https://en.wikipedia.org/w/index.php?title=9/11_conspiracy_theories&amp;oldid=1015278683|journal=Wikipedia|language=en}}&lt;/ref&gt; which suggests that many people anticipated the attack. It's hard to imagine that such information would reach the ears of traders yet remain unknown to US intelligence and defense. From a cursory search, the 500+ page 9/11 commission report only seems to mention possible insider trading in a one-paragraph endnote to chapter five, which asserts that this is coincidence and states that the (unnamed) traders had no connection with the attacks. A significant portion of these trades were made through Alex Brown inc. whose former president A. B. Krongard was appointed executive director of the CIA on March 16, 2001.&lt;ref&gt;{{Cite web|url=https://www.independent.co.uk/news/business/news/mystery-terror-insider-dealers-9237061.html|title=Mystery of terror 'insider dealers'|date=2014-04-04|website=The Independent|language=en|access-date=2021-05-31}}&lt;/ref&gt; This was not mentioned in the 9/11 commission report. [https://en.wikipedia.org/wiki/Larry_Silverstein#World_Trade_Center Larry Silverstein] bought the WTC in January 2001 and insured it, and spent a lot of time at the buildings in the subsequent months. Summarizing the relevant parts of that Wiki page: &quot;Silverstein has said in interviews that he usually spent his mornings in breakfast meetings at Windows on the World on top of the World Trade Center North Tower, and with new tenants in the building. However, the morning of September 11, 2001, his wife insisted that he attend a medical appointment. Due to the appointment, he escaped almost certain death&quot;. Apparently not satisfied with the payout he was offered, &quot;Following the September 11, 2001, attacks, Silverstein sought to collect double the face amount (~$7.1 billion) on the basis that the two separate airplane strikes into two separate buildings constituted two occurrences within the meaning of the policies&quot;. He sued the insurance companies and eventually won over four billion dollars. The primary motive was probably to establish casus belli for wars in the middle east. 
* {{Argument for}} Besides the twin towers, one more building in the World Trade Center complex collapsed, namely the 7 World Trade Center. This building was not hit by any plane, nor did it receive much more debris damage than any of the other buildings surrounding the twin towers. It did catch on fire and burn for some hours, but not nearly enough to make the building collapse as completely and as fast as it did, at nearly free-fall speed for the first few seconds of its collapse.&lt;ref&gt;{{Citation|last=WTC911demolition|title=WTC Building 7 Collapse - 23 angles|date=2011-10-02|url=https://www.youtube.com/watch?v=JnLcUxV1dPo&amp;t=2m41s|accessdate=2019-06-16}}&lt;/ref&gt; Prior to 9/11 no steel-framed high rise building had ever collapsed because of fire. But if the fire didn't cause the collapse, then the only remaining explanation is a controlled demolition, which would implicate high-level government officials. As extra support, some suspicious &quot;drills&quot; were reported that day on the building. {{Citation needed}}
** {{Objection}} The fires did not knock out all of the supports at once. They weakened the supports around column 79, causing floors 8 to 14 to collapse in the inside of the building. This then led to column 79 failing, causing the east penthouse to collapse, damaging surrounding columns. This set off a chain reaction of columns failing from the east side to the west side. This meant that the full weight of the building was loaded onto the perimeter support, which buckled between floors 7 and 17 eight seconds after the east penthouse collapsed, causing the remaining exterior of the building to collapse as a single unit.&lt;ref&gt;{{Cite web|url=https://www.nist.gov/engineering-laboratory/final-reports-nist-world-trade-center-disaster-investigation|title=Final Reports from the NIST World Trade Center Disaster Investigation|last=Thompson|first=Kristy D.|date=2011-06-30|website=NIST|language=en|access-date=2019-06-16}}&lt;/ref&gt; So the collapse was not instant and can be explained by fires. You then say that this is unheard of and no example of a similar collapse can be provided. That's absolutely correct. It was completely unheard of for a high rise to have its lower floors on fire for so many hours without firefighters stopping the fire. But it happened on 9/11.
** {{Objection}} There was no evidence of the explosives required for a controlled demolition.
*** {{Objection}} The microscopic iron spheres found at ground zero were a direct byproduct of thermitic material, such as thermite, being used to weaken the structure before its collapse. This coupled with the red hot steel pouring from windows, &quot;as if it had melted in a foundry&quot;, quoted from well documented news footage of a NYPD firefighter, is definitive proof enough there was some sort of control going on behind the chaos.
**** {{Objection}} A random quotation from a random firefighter is not definitive proof of anything, except that in the midst of chaotic panic the firefighter said something that is vague enough to support any conclusion one wants to support.
***** {{Objection}} The firefighter is interviewed formally by a journalist so these records should still exist.
***** {{Objection}} Slothful Induction/Personal Incredulity fallacy.
****** {{Objection}} Fallacy fallacy.
* {{Argument for}} Husley, L. 2019. [http://ine.uaf.edu/wtc7 A Structural Reevaluation of the Collapse of World Trade Center 7]. Institute of Northern Engineering: ''&quot;...The principal conclusion of our study is that fire did not cause the collapse of WTC 7 on 9/11, contrary to the conclusions of NIST and private engineering firms that studied the collapse. The secondary conclusion of our study is that the collapse of WTC 7 was a global failure involving the near-simultaneous failure of every column in the building....&quot;''
** {{Objection}} No tests were conducted for the presence of explosives residue and operational protocol (not to mention basic common sense) stipulates that these tests should have occurred. Without either being inside the building or having video footage of the interior of the building at the time, there is no reliable way for you to reach the conclusions you have reached, making the specificity of your claims outright preposterous. You are certainly correct that there was a chain reaction; however, this chain reaction could have been caused by the building's resistance to gravity being compromised by explosives.
* {{Argument for}} The September 11 attacks were a thinly-veiled excuse to invade Iraq in order to establish American dominance, get oil, and vindicate the first Bush Presidency's defense of Kuwait in the 1990s during the first Persian Gulf War.
** {{Objection}} The USA imports little oil from the Middle East, with around 50% coming from North America and less than 15% from the Persian Gulf. It's not cheap oil if it requires shipping across the 12,000 miles between the two locations, and so oil is not a good motivation for the attack.
*** {{Objection}} However, you don't actually need to ship the oil to American soil. You could just sell it as the Iraqis used to. Once you control the oil, you could run the regional market and make millions and maybe billions of dollars in sales. You could have a even bigger influence over the global oil market. There are many good reasons where shipping isn't required.
* {{Argument for}} Osama Bin Laden had been an ally of the CIA before, during the Afghan-Soviet war, so it would be plausible that he had a connection with the US government in order to mount the attack via his terrorist cell and give the government an excuse for invasion in the middle east. The invasion produced many exclusive oil contracts afterwards, including those of Halliburton, an enterprise related to Dick Cheney and the Bush administration. Besides, it took quite a while for the US to find and kill Osama, not until the Obama administration.

=== Con ===

* {{Argument against}} Any possible motive for the government to do this would not need to be executed in this way.
** {{Objection}} Citizens experiencing fear or anger are easier to manipulate, and this was used as a tool to herd the populace into further wars.
* {{Argument against}} The events of 9/11 can be explained far more simply as terrorist attacks than a complex conspiracy with unclear motivation.
** {{Objection}} How bout when the CIA was going to plot a war against Cuba by shooting down a plane and killing innocent Americans? It's not the first time something like this could have happened
** {{Objection}} Many things can be explained in simpler terms, doesn’t mean it’s correct
*** {{Objection}} Occam's Razor
** {{Objection}} Simple explanations is what simple people crave. The motivation was simple and clear. Use this &quot;terrorist&quot; attack to expand the Patriot act and make citizens ok with being spied on for the greater good of the country. Assert dominance in a region  that doesn't like you or your banking system.
* {{Argument against}} There is no evidence of members of the conspiracy, even though this would require the perfect silence of a large amount of individuals at different levels of government, when the government has a difficult time keeping far less scandalous secrets hidden. Such theories strain credulity and there is simply no reason to accept them other than the desire to believe them.
** {{Objection}} &quot;Absence of evidence&quot; is no evidence at all.&lt;ref name=&quot;:0&quot;&gt; {{cite web | title =Absence of evidence &amp; Argumentum ad Ignorantiam wiki page | url=https://en.wikipedia.org/w/index.php?title=Absence_of_evidence }} &lt;/ref&gt;
** {{Objection}} They also have less reason to keep &quot;far less scandalous&quot; secrets hidden. Why would they resort to strong-arm tactics and censorship to cover up minor scandals? That makes no sense.
** {{Objection}} You haven't provided an argument here so much as an abuse of language. What the US government claimed happened was a complex conspiracy involving an international terrorist organisation. The US government has so far been unable to prove this is what happened, making it a theory. Somehow, many people in America (and worldwide) seem to have been convinced that critically questioning the US government's theory about an al-Qaeda conspiracy makes one a conspiracy theorist. The very logic of language itself reveals the lunacy in slandering those who dispute the US government's 9/11 narrative with egregious smear attacks like, &quot;conspiracy theorist&quot; or &quot;twofer&quot;. These are just simple lawyer's tricks, purpose-built to attack the credibility of the opposition.
* {{Argument against}} Soon after the event, al-Qaeda claimed responsibility for the attack.
** {{Objection}} Al-Qaeda are probably not above claiming responsibility for something they didn't actually do if it furthers their organisation's goals. If 9/11 were an inside job, al-Qaeda would probably claim responsibility anyway either unprompted or at the suggestion of whatever secretive cabal really did it.
** {{Objection}} This is simply false. Whoever perpetrated the attack tried to fabricate evidence implicating the leader of al-Qaeda in the form of a video tape. This video tape has been declared fraudulent by numerous scholars, including Professor Bruce Lawrence who, according to the Daily Mirror (UK), is the world's &quot;foremost authority on (Osama) bin Laden&quot;. The actual Osama bin Laden gave two public interviews after 9/11 in which his identity was confirmed. He categorically denied all involvement in both. The solitary link the US government has between al-Qaeda and the 9/11 attacks is the written confession of a man claiming to be Khalid Sheikh Mohammed, who provided his confession after being waterboarded a reported 183 times at Guantanamo Bay. Moreover, Khalid Sheikh Mohammed was reported by the Asian Press to have died during an ISI raid in Karachi in 2002. There is literally no proof the man being held in US custody is even Khalid Sheikh Mohammed in the first place.
*** {{Objection}} Professor Bruce Lawrence, the gentleman who you claimed to have made a comment about the allegedly &quot;fraudulent&quot; video from al-Qaeda, is a Humanities Professor of Religion at Duke University and publishes books about the Islamic faith. The only public comment he has ever made even slightly related to the 9/11 attacks is that he believes that Islam has no connection with terrorism. Besides, The Daily Mirror is a British tabloid that has published stories that were later revealed to be a hoax.
* {{Argument against}} It would be impossible to keep a conspiracy this large secret. Someone would have leaked reliable and verifiable information at some point and every journalist in the world would be keen to break the biggest story in the 21st century.
** {{Objection}} What about the fact that these people who destroyed the buildings in NYC and DC, were trained in Canada? The reason I bring this argument into vision is there is a thing such as &quot;Domestic Terrorism&quot; . I believe and have investigated much of this theory. The attacks came from greed and the way the greed could be covered-up, is by making people look in a different direction. How do we even know if what was told to us by Al-Queda to be true, many of us don't know their language and we would be counting on the Translates for the truth,right?
** {{Objection}} &quot;Absence of evidence&quot; is no evidence at all.&lt;ref name=&quot;:0&quot; /&gt;
** {{Objection}} When there is a leak of even the most minor sort, the groundwork has already been laid to discredit their mental well being. Short of Dick Cheney himself, there isn't any credibility in low level or anonymous leaks.
** {{Objection}} The conspiracy that official story says committed the 9/11 attacks was a single al-Qaeda cell, why should the conspiracy that really did it necessarily be much bigger to the point that discovery is inevitable?
** {{Objection}} They sign NDAs and it's probably difficult to &quot;leak&quot; information anonymously due to the advanced SIGINT capabilities of various governmental agencies e.g. the NSA. Even if you could, there's no guarantee that any given journalist isn't actually an informant. Snowden probably lives in constant fear of extradition and cannot return to his home country on pain of lifelong imprisonment. Russia could turn him over on a whim if he gets on their bad side too. It's hardly a way to live. His situation gives us information about the mindset, culture, and motivations of the people involved in such organizations. They can do no such thing as &quot;leak information&quot;, or rather, they cannot be relied upon to do so.
* {{Argument against}} Fact 1. No plane wreckage on the lawn of pentagon.  2. The second plane was not a jet liner. 3. No building anywhere in the world has fell due to plane going into it  &amp; there has been many around the world. This was a setup by the US government to go into Iraq and Afghanistan to control the oil. Saddam was giving oil for food program through the UN. The US could not put fuel prices up while Saddam was giving away oil. This was a oil war.
* {{Argument against}} Why would the government plot an attack against citizens, which would harm the significant population? And even if it wanted to do it, it shouldn't have been executed like that.
** {{Objection}} Did you really ask this question? The greed of humanity will always prevail in the face of things like morality in decision making. If we can press a button killing thousands over a cause they’re not necessarily fighting for, we can kill a few thousand of our own over big money and the securement of our governments placement in one of the most lucrative businesses in the world at the time of attack.

== See also ==
* [[Are all conspiracy theories nonsense?]]
* [[Is capitalism sustainable?]]
* [[Did the United States need to use atomic weapons to end World War II?]]

== Notes and references ==

{{Reflist}}

[[Category:September 11 attacks]]
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{{Wikipedia2|Children's Sleep Habits Questionnaire}}

The '''Children's Sleep Habits Questionnaire''' ('''CSHQ''') is a psychological questionnaire designed to measure sleep behaviors in children and adolescents ages 4–12. The 52-question test is filled out by the parent, and the parent is asked to rate the frequency of the described sleep behaviors that their child has engaged in over the last week. It takes approximately 10 to 15 minutes to complete the questionnaire.&lt;ref name=CSHQ&gt; Owens, JA; Spirito, A; McGuinn, M. (2000). &quot;Children’s Sleep Habits Questionnaire (CSHQ)&quot;. &lt;/ref&gt;

CSHQ can identify behaviorally based and medically based sleep problems in children with and without neurodevelopment diagnoses.&lt;ref name=Jones&gt;{{cite journal|last1=Goodlin-Jones|first1=BL|last2=Sitnick|first2=SL|last3=Tang|first3=K|last4=Liu|first4=J|last5=Anders|first5=TF|title=The Children's Sleep Habits Questionnaire in toddlers and preschool children.|journal=Journal of Developmental and Behavioral Pediatrics  |date=April 2008|volume=29|issue=2|pages=82–88|doi=10.1097/dbp.0b013e318163c39a}}&lt;/ref&gt; &lt;ref name=Owens&gt;{{cite journal|last1=Owens|first1=JA|last2=Spirito|first2=A|last3=McGuinn|first3=M|title=The Children's Sleep Habits Questionnaire (CSHQ): psychometric properties of a survey instrument for school-aged children.|journal=Sleep|date=15 December 2000|volume=23|issue=8|pages=1043–51|doi=10.1093/sleep/23.8.1d}}&lt;/ref&gt; 

== Scoring and interpretation ==

Each question on the test addresses the frequency of the following behavioral characteristics pertaining to sleep: bedtime resistance, sleep-onset delay, sleep duration, sleep anxiety, night waking, parasomnia, sleep disordered breathing, and daytime sleepiness. Parents are also asked if each behavioral characteristic is a current issue for the child. 

Only 33 of the 52 questions on the CSHQ are scored, for a maximum total score of 99. Each question is scored on a 1-3 scale, from “rarely (0-1 times a week),” “sometimes (2-4 times),” and “usually (5-7 times).” 6 of the 33 questions scored have the reverse scaling, so “rarely” would correspond to 3 points, and so on. 

A total score of 41 and above suggests significant sleep problems for the child.

==References==
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{{Wikipedia2|Positive and Negative Syndrome Scale}}

The '''Positive and Negative Syndrome Scale''' ('''PANSS''') is a medical scale used for measuring symptom severity of patients with schizophrenia. It is widely used in the study of antipsychotic therapy. The scale is the &quot;gold standard&quot; for evaluating the effects of psychopharmacological treatments.&lt;ref&gt;{{Cite journal |last1=Liechti |first1=Stacy |last2=Capodilupo |first2=Gianna |last3=Opler |first3=Douglas J. |last4=Opler |first4=Mark |last5=Yang |first5=Lawrence H. |date=2017-12-01 |title=A Developmental History of the Positive and Negative Syndrome Scale (PANSS) |journal=Innovations in Clinical Neuroscience |volume=14 |issue=11–12 |pages=12–17 |issn=2158-8333 |pmc=5788246 |pmid=29410932}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Opler|first1=Mark G.A.|last2=Yavorsky|first2=Christian|last3=Daniel|first3=David G.|date=2017-12-01|title=Positive and Negative Syndrome Scale (PANSS) Training|journal=Innovations in Clinical Neuroscience|volume=14|issue=11–12|pages=77–81|issn=2158-8333|pmc=5788255|pmid=29410941}}&lt;/ref&gt;

The name refers to the two types of symptoms in schizophrenia, as defined by the American Psychiatric Association (APA): positive symptoms, which refer to an excess or distortion of normal functions (e.g., hallucinations and delusions), and negative symptoms, which represent a diminution or loss of normal functions. Some of these functions which may be lost include normal thoughts, actions, ability to tell fantasies from reality, and the ability to properly express emotions.&lt;ref&gt;{{Cite web|url=https://www.webmd.com/schizophrenia/mental-health-schizophrenia|title=Mental Health and Schizophrenia|website=WebMD|language=en|access-date=2019-07-29}}&lt;/ref&gt;

The PANSS is a relatively brief interview, requiring 45 to 50 minutes to administer.&lt;ref name=&quot;Kay1987&quot;&gt;{{cite journal |vauthors=Kay SR, Fiszbein A, Opler LA |title=The positive and negative syndrome scale (PANSS) for schizophrenia. |journal=Schizophr Bull |volume=13 |issue= 2 |pages=261–76 |year=1987 |pmid=3616518 |doi=10.1093/schbul/13.2.261|doi-access=free }}&lt;/ref&gt; The interviewer must be trained to a standardized level of reliability to conduct the assessment.&lt;ref&gt;{{citation |url=https://books.google.com/books?id=TUvKyhgOpxQC&amp;pg=PA395 |title=A Guide to Assessments that Work |author=John Hunsley |author2=Eric J. Mash |publisher=Oxford University Press US |year= 2008 |isbn=978-0-19-531064-1 }}&lt;/ref&gt;

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{{Wikipedia2|Positive and Negative Syndrome Scale}}

The '''Positive and Negative Syndrome Scale''' ('''PANSS''') is a medical scale used for measuring symptom severity of patients with schizophrenia. It is widely used in the study of antipsychotic therapy. The scale is the &quot;gold standard&quot; for evaluating the effects of psychopharmacological treatments.&lt;ref&gt;{{Cite journal |last1=Liechti |first1=Stacy |last2=Capodilupo |first2=Gianna |last3=Opler |first3=Douglas J. |last4=Opler |first4=Mark |last5=Yang |first5=Lawrence H. |date=2017-12-01 |title=A Developmental History of the Positive and Negative Syndrome Scale (PANSS) |journal=Innovations in Clinical Neuroscience |volume=14 |issue=11–12 |pages=12–17 |issn=2158-8333 |pmc=5788246 |pmid=29410932}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last1=Opler|first1=Mark G.A.|last2=Yavorsky|first2=Christian|last3=Daniel|first3=David G.|date=2017-12-01|title=Positive and Negative Syndrome Scale (PANSS) Training|journal=Innovations in Clinical Neuroscience|volume=14|issue=11–12|pages=77–81|issn=2158-8333|pmc=5788255|pmid=29410941}}&lt;/ref&gt;

The name refers to the two types of symptoms in schizophrenia, as defined by the American Psychiatric Association (APA): positive symptoms, which refer to an excess or distortion of normal functions (e.g., hallucinations and delusions), and negative symptoms, which represent a diminution or loss of normal functions. Some of these functions which may be lost include normal thoughts, actions, ability to tell fantasies from reality, and the ability to properly express emotions.&lt;ref&gt;{{Cite web|url=https://www.webmd.com/schizophrenia/mental-health-schizophrenia|title=Mental Health and Schizophrenia|website=WebMD|language=en|access-date=2019-07-29}}&lt;/ref&gt;

The PANSS is a relatively brief interview, requiring 45 to 50 minutes to administer.&lt;ref name=&quot;Kay1987&quot;&gt;{{cite journal |vauthors=Kay SR, Fiszbein A, Opler LA |title=The positive and negative syndrome scale (PANSS) for schizophrenia. |journal=Schizophr Bull |volume=13 |issue= 2 |pages=261–76 |year=1987 |pmid=3616518 |doi=10.1093/schbul/13.2.261|doi-access=free }}&lt;/ref&gt; The interviewer must be trained to a standardized level of reliability to conduct the assessment.&lt;ref&gt;{{citation |url=https://books.google.com/books?id=TUvKyhgOpxQC&amp;pg=PA395 |title=A Guide to Assessments that Work |author=John Hunsley |author2=Eric J. Mash |publisher=Oxford University Press US |year= 2008 |isbn=978-0-19-531064-1 }}&lt;/ref&gt;

==Scoring and Interpretation==
The patient is rated from 1 to 7 on 30 different symptoms based on the interview as well as reports of family members or primary care hospital workers. 

The Positive Scale has 7 items with a minimum score of 7 and a maximum score of 49. This subscale assesses delusions, conceptual disorganization, hallucinations, excitement, grandiosity, suspiciousness/persecution, and hostility.

The Negative Scale has 7 items with a minimum score of 7 and a maximum score of 49. This subscale assesses blunted affect, emotional withdrawal, poor rapport, passive/apathetic social withdrawal, difficulty in abstract thinking, lack of spontaneity and flow of conversation, and stereotyped thinking.

The General Psychopathology Scale has 16 items with minimum score of 16 and a maximum score of 112. This subscale assesses somatic concern, anxiety, guilt feelings, tension, mannerisms and posturing, depression, motor retardation, uncooperativeness, unusual thought content, disorientation, poor attention, lack of judgment and insight, disturbance of volition, poor impulse control, preoccupation, and active social avoidance.

The PANSS has a total minimum score of 30 and maximum score 120. 1 is given as the lowest score for each item.

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{{Wikipedia2|Dimensional Obsessive-Compulsive Scale}}

The '''Dimensional Obsessive-Compulsive Scale''' ('''DOCS''') is a 20-item self-report instrument that assesses the severity of Obsessive–compulsive disorder(OCD) symptoms along four empirically supported theme-based dimensions: (a) contamination, (b) responsibility for harm and mistakes, (c) incompleteness/symmetry, and (d) unacceptable (taboo) thoughts.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|title = Assessment of obsessive-compulsive symptom dimensions: Development and evaluation of the Dimensional Obsessive-Compulsive Scale.|journal = Psychological Assessment|pages = 180–198|volume = 22|issue = 1|doi = 10.1037/a0018260|first1 = Jonathan S.|last1 = Abramowitz|first2 = Brett J.|last2 = Deacon|first3 = Bunmi O.|last3 = Olatunji|first4 = Michael G.|last4 = Wheaton|first5 = Noah C.|last5 = Berman|first6 = Diane|last6 = Losardo|first7 = Kiara R.|last7 = Timpano|first8 = Patrick B.|last8 = McGrath|first9 = Bradley C.|last9 = Riemann|pmid=20230164|year=2010| s2cid=7206349 }}&lt;/ref&gt;

The DOCS was developed in 2010 by a team of experts on OCD led by Jonathan Abramowitz to improve upon existing OCD measures and advance the assessment and understanding of OCD. The DOCS contains four subscales (corresponding to the four symptom dimensions) that have been shown to have good reliability, validity, diagnostic sensitivity, and sensitivity to treatment effects in a variety of settings cross-culturally and in different languages. As such, the DOCS meets the needs of clinicians and researchers who wish to measure current OCD symptoms or assess changes in symptoms over time (e.g., over the course of treatment).&lt;ref&gt;{{Cite journal|title = Assessing obsessive-compulsive disorder (OCD): A review of self-report measures|journal = Journal of Obsessive-Compulsive and Related Disorders|pages = 312–324|volume = 1|issue = 4|doi = 10.1016/j.jocrd.2012.08.001|first1 = Mathilde K.|last1 = Overduin|first2 = Adrian|last2 = Furnham|year = 2012}}&lt;/ref&gt;

==Scoring and Interpretation==
Each of the four DOCS subscales begins with a general description and broad inclusive examples of the obsessions and compulsions within the particular symptom dimension. Respondents are next asked to consider any obsessions and compulsions within that symptom dimension that they have experienced within the last month and rate on a scale from 0 (no symptoms) to 4 (extreme symptoms) about (1) the time occupied by obsessions and compulsions, (2) avoidance behavior, (3) associated distress, (4) functional interference, and (5) difficulty disregarding the obsessions and refraining from the compulsions. 

The DOCS subscales assesses the severity of the patient's own symptoms, rather than pre-defined symptoms as in most OCD measures. Within each subscale, the five item scores are summed to produce a subscale score (range = 0-20). The four subscale scores can be summed to produce an overall DOCS total score (range = 0-80).

A DOCS total score of 18 optimally distinguishes between someone with OCD and someone without a psychiatric diagnosis; while a score of 21 optimally distinguishes between someone with OCD and someone with an anxiety disorder.&lt;ref name=&quot;:0&quot; /&gt; As of this time, there are no empirically derived cutoff scores for mild, moderate, or severe OCD symptoms.

==References==
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{{Wikipedia2|The Mood and Feelings Questionnaire}}

The '''Mood and Feelings Questionnaire (MFQ)''' is a survey that measures depression in children and adolescents ages 6-18. The questionnaire was created by researchers at Duke University as part of the Great Smokey Mountain epidemiological project in Western North Carolina. MFQ has six versions, containing short (13 item) and long (33 item) forms of each of the following: a youth self-report, a version that a parent would complete, and a self-report version for adults. It takes 5-10 minutes to administer the questionnaire and is used by  clinicians in community samples of ages 6-18.&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last1=Angold|first1=A|last2=Costello|first2=EJ|date=1988|title=Scales to assess child and adolescent depression: checklists, screens, and nets|journal=Journal of the American Academy of Child and Adolescent Psychiatry|volume=27|issue=6|pages=726–737|doi=10.1097/00004583-198811000-00011|pmid=3058677}}&lt;/ref&gt;

== Scoring and interpretation ==
The Mood and Feelings Questionnaire has several tests, one short and one long, with the short questionnaire including 13 questions and the long questionnaire consisting of 33 questions.  Scoring of the questionnaire works by summing the point values allocated to each question.&lt;ref name=&quot;:0&quot;/&gt; The responses and their allocated point values are as follows:

&quot;not true&quot; = 0 points

&quot;somewhat true&quot; = 1 point

&quot;true&quot; = 2 points

Scores on the short MFQ range from 0 to 26, whereas scores on the long MFQ range from 0 to 66. Higher score indicates more severe depressive symptoms.  Scores larger than 12 on the short version or larger than 27 on the long version are suggestive of likely depression and warrant further clinical assessment.&lt;ref name=&quot;:0&quot; /&gt;

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      <text bytes="46091" sha1="68sy14ocg8r78ygwxh5rzypqisylllf" xml:space="preserve">'''Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs at the Planck scale'''

An orbital simulation program is described that emulates gravitational and atomic orbitals as the sum of individual particle-particle orbital pair rotations at the [[w:Planck_units |Planck scale]]. The simulation is dimensionless, the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]], however it can translate to the Planck units for comparison to real world orbits.

[[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point (unit of mass) accordingly, we can 'design' the required orbits. No other parameters are used. The 26 points orbit each other resulting in 325 point-point orbitals.]]
For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''&lt;sub&gt;P&lt;/sub&gt; (for example, a 1kg satellite would be divided into 1kg/''m''&lt;sub&gt;P&lt;/sub&gt; = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=3. Gravitational orbits emerge from Planck scale n-body rotating orbital pairs |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.11496.93445/17}}&lt;/ref&gt;. 

Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... 

The basic simulation uses only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs, information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). 

For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together and a 'photon' is added in a series of steps. As the electron continues to orbit the nucleus during the transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although the spiral path is semi-classical, it exhibits the quantum states, and suggests that quantization could have geometrical origins.  


=== Theory ===
In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is an oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model.

If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10&lt;sup&gt;23&lt;/sup&gt; electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity becomes a mass to mass interaction. 

Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map particle to particle orbitals for which both particles are in the wave-state, the H atom is a well-researched particle-to-particle orbital pair (electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the photon-orbital model&lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;, in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). 

It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically cancels into integers which correspond with the orbital energy levels where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this quantization has a geometrical origin.

Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period ''t'' measured between these integer radius can be used to solve the transition frequencies ''f'' via the formula &lt;math&gt;f/c = t \lambda_H/(n_f^2-n_i^2)&lt;/math&gt;. We also note a 'transition signature' where the duration of period between different energy levels (''n'' shells) is not consistent, and this signature appears in both the experimental and the simulated results. This deviation from the expected results is most pronounced in the ''n'' = 1 to ''n'' = 2 transition (where the transition period is comparatively longer).
 
In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The gravitational orbits that we observe are the time averaging sum of the underlying multiple gravitational orbitals. 


=== N-body orbitals ===
[[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]]

The simulation universe is a 4-axis hypersphere expanding in increments &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}&lt;/ref&gt; with 3-axis (the [[v:Black-hole_(Planck)  |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'.

The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m&lt;sub&gt;P&lt;/sub&gt;]], [[w:Planck length |Planck length l&lt;sub&gt;p&lt;/sub&gt;]], [[w:Planck time |Planck time t&lt;sub&gt;p&lt;/sub&gt;]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''&lt;sub&gt;p&lt;/sub&gt;/''t''&lt;sub&gt;p&lt;/sub&gt;). These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. 

Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). 

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}}&lt;/math&gt; 

As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation.

Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt; is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence.


[[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]]
==== 2 body orbits ('''x, y''' plane) ====
For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. 

The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units

:&lt;math&gt;r_s = \frac{2 l_p M}{m_P}&lt;/math&gt;

As the simulation itself is dimensionless, we can remove the dimensioned length component &lt;math&gt;2 l_p&lt;/math&gt;, and as each point is analogous to 1 unit of Planck mass &lt;math&gt;m_P&lt;/math&gt;, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius &lt;math&gt;r_s&lt;/math&gt; = number of central points = total points - 1 (the orbiting point).


After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. 

Key:

1. &lt;math&gt;r_s&lt;/math&gt; = '''i'''; number of center mass points (the orbited object). 

2. '''j&lt;sub&gt;max&lt;/sub&gt;''' = radius to mass co-efficient.

3. '''j''' = number of points, including virtual (for simple 2 body orbits with only 1 orbiting point, '''j''' = '''i''' + 1 ).

4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 

5. '''r&lt;sub&gt;α&lt;/sub&gt;''' = a radius constant, here r&lt;sub&gt;α&lt;/sub&gt; = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals.

:&lt;math&gt;r_{orbital} = {r_{\alpha}}^2 \;*\; r_{wavelength} &lt;/math&gt;


==== Orbital formulas (2-D plane)====
Outer = orbiting point, inner = orbited center

:&lt;math&gt;r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2&lt;/math&gt;, orbital radius

:&lt;math&gt;r_{barycenter} = \frac{r_{outer}}{j}&lt;/math&gt;, barycenter

:&lt;math&gt;v_{outer} = \frac{i}{j_{max} r_{\alpha}} &lt;/math&gt;, orbiting point velocity  

:&lt;math&gt;v_{inner} = \frac{1}{j_{max} r_{\alpha}}&lt;/math&gt;, orbited point(s) velocity  

:&lt;math&gt;t_{outer} = \frac{2 \pi r_{outer}}{v_{outer}} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 &lt;/math&gt;, orbiting point period

:&lt;math&gt;l_{outer} = 2 \pi (r_{outer} - r_{barycenter})&lt;/math&gt;, distance travelled 


Simulation data:

:period &lt;math&gt;t_{sim}&lt;/math&gt;

:length &lt;math&gt;l_{sim}&lt;/math&gt;

:radius &lt;math&gt;r_{sim} = \frac{l_{sim}}{2 \pi}&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = \frac{l_{sim}}{t_{sim}}&lt;/math&gt; 

:barycenter &lt;math&gt;b_{sim} = \frac{x_{max} + x_{min}}{2}&lt;/math&gt;


[[File:gravity-orbit-hyperbolic-spiral.jpg|thumb|right|576px|Object leaving a gravitational circular orbit (j&lt;sub&gt;max&lt;/sub&gt; = j) with constant outward motion follows the same [[v:Fine-structure_constant_(spiral) |alpha hyperbolic spiral]] as an ionizing electron]]
For example; 8 mass points (28 orbitals) divided into ''j'' = 8 (total points), ''i'' = ''j'' - 1 (7 center mass points). After 1 complete orbit, actual period '''t''' and distance travelled '''l''' are noted and compared with the above formulas. 

1) ''j''&lt;sub&gt;max&lt;/sub&gt; = i+1 = 8 

:period &lt;math&gt;t = 74465.0516,\; t_{outer} = 74471.6125&lt;/math&gt; 

:length &lt;math&gt;l = l_{sim} = 3935.7664,\; l_{outer} = 3936.1032&lt;/math&gt; 

:radius &lt;math&gt;r_{sim} = 626.3951&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = 1/18.920137&lt;/math&gt;

:barycenter &lt;math&gt;b_{sim} = 89.5241,\; r_{barycenter} = 89.4929&lt;/math&gt;


2) ''j''&lt;sub&gt;max&lt;/sub&gt; = 32*i+1 = 225 

:period &lt;math&gt;t = 1656793370.3483,\; t_{outer} = 1656793381.3051&lt;/math&gt; 

:length &lt;math&gt;l = l_{sim} = 3113519.1259,\; l_{outer} = 3113519.1385&lt;/math&gt; 

:radius &lt;math&gt;r_{sim} = 495531.959&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = 1/532.128856&lt;/math&gt;

:barycenter &lt;math&gt;b_{sim} = 70790.283, \;r_{barycenter} = 70790.280&lt;/math&gt;


3) Moon orbit. 

From the [[w:standard gravitational parameter |standard gravitational parameters]], the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising ''i'' = 81 points, ''j'' = i + 1.

:&lt;math&gt;\frac{3.986004418\;x10^{14}}{4.9048695\;x10^{12}} = 81.2663&lt;/math&gt;

:&lt;math&gt;r_{earth-moon}&lt;/math&gt; = 384400km

:&lt;math&gt;M_{earth}&lt;/math&gt; = 0.597378 10&lt;sup&gt;25&lt;/sup&gt;kg


Solving &lt;math&gt;j_{max}&lt;/math&gt;

:&lt;math&gt;r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2 = \frac{2 r_{earth-moon} m_P}{M_{earth} l_p}&lt;/math&gt;

:&lt;math&gt;j_{max} = 1440443&lt;/math&gt;


Gives

:&lt;math&gt;t_{outer} =  4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 (\frac{l_p}{c}) = 0.8643\; 10^{-26}&lt;/math&gt;s

:&lt;math&gt;t_{outer} \frac{M_{earth}} {m_P } = 2371844&lt;/math&gt;s (27.452 days)

:&lt;math&gt;v_{Moon} = (c) \frac{i}{j_{max}{r_{\alpha}}} = 1018.3m/s&lt;/math&gt;

:&lt;math&gt;v_{Earth} = (c) \frac{1}{j_{max} r_{\alpha}}  = 12.57m/s&lt;/math&gt;

:&lt;math&gt;r_{barycenter} = \frac{r_{earth-moon}}{j} = 4688km&lt;/math&gt;


==== Gravitational coupling constant ====
In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α&lt;sub&gt;G&lt;/sub&gt;'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio;

:&lt;math&gt;\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}&lt;/math&gt;

For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α&lt;sub&gt;G&lt;/sub&gt; then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] &lt;ref&gt;Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}&lt;/ref&gt;). 

:&lt;math&gt;{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}&lt;/math&gt;

As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α&lt;sub&gt;G&lt;/sub&gt; is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10&lt;sup&gt;42&lt;/sup&gt; units of Planck time (&lt;math&gt;t_p = 10^{-42}s&lt;/math&gt;), this occurs about once every 3 minutes.

:&lt;math&gt;\frac{{\alpha_G}^{-1}}{t_p}&lt;/math&gt;

Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak.  

If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f&lt;sub&gt;e&lt;/sub&gt;, then the point will require 10&lt;sup&gt;23&lt;/sup&gt; electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time).

:&lt;math&gt;f_e = \frac{m_P}{m_e} = 10^{23}&lt;/math&gt;

For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have &lt;math&gt;1kg/m_P = 45940509&lt;/math&gt; particles in the point-state. The earth (A) will have &lt;math&gt;5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}&lt;/math&gt; particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite;

:&lt;math&gt;N_{orbitals} =  (\frac{m_A}{m_P})(\frac{m_B}{m_P})  = 0.1261\; x10^{41}&lt;/math&gt;
 

Examples:

:&lt;math&gt;i = \frac{M_{earth}}{m_P} = 0.27444  \;x10^{33}&lt;/math&gt; (earth as the center mass)

:&lt;math&gt;i 2 l_p = 0.00887&lt;/math&gt; (earth Schwarzschild radius)

:&lt;math&gt;s = \frac{1kg}{m_P} = 45940509&lt;/math&gt; (1kg orbiting satellite)

:&lt;math&gt;j = N_{orbitals} = i*s = 0.1261  \;x10^{41}&lt;/math&gt;


1) 1kg satellite at earth surface orbit 

:&lt;math&gt;r_{o} = 6371000 km&lt;/math&gt; (earth surface)

:&lt;math&gt;j_{max} = \frac{j}{r_a}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 2289.41&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 7909.7924&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374&lt;/math&gt; s 


2) 1kg satellite at a synchronous orbit radius

:&lt;math&gt;r_o = 42164.17 km&lt;/math&gt; 

:&lt;math&gt;j_{max} =  \frac{j}{r_a} \sqrt{\frac{r_{o}}{i l_p}}  = 0.74256\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 5889.674&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 3074.66&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165&lt;/math&gt; s 


3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). 

:&lt;math&gt;E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J&lt;/math&gt; (energy per orbital)

:&lt;math&gt;N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}&lt;/math&gt; (number of orbitals)

:&lt;math&gt;E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg&lt;/math&gt;


4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs.

:&lt;math&gt;N_{sun} = \frac{M_{sun}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{planet} = \frac{M_{planet}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{orbitals} = N_{sun}N_{planet} &lt;/math&gt;

:&lt;math&gt;n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} &lt;/math&gt;

:&lt;math&gt;L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} &lt;/math&gt;

The orbital angular momentum of the planets;

 mercury = .9153 x10&lt;sup&gt;39&lt;/sup&gt;  
 venus    = .1844 x10&lt;sup&gt;41&lt;/sup&gt;  
 earth    = .2662 x10&lt;sup&gt;41&lt;/sup&gt; 
 mars     = .3530 x10&lt;sup&gt;40&lt;/sup&gt; 
 jupiter   = .1929 x10&lt;sup&gt;44&lt;/sup&gt;   
 pluto   = .365 x10&lt;sup&gt;39&lt;/sup&gt;   


Orbital angular momentum combined with orbit velocity cancels ''n&lt;sub&gt;g&lt;/sub&gt;'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly.

:&lt;math&gt;L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} &lt;/math&gt;


[[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2&lt;sup&gt;32&lt;/sup&gt; steps they have clumped to form 1 large mass and 2 orbiting masses.]]
==== Freely moving points ====
The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted.   

In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2&lt;sup&gt;32&lt;/sup&gt; steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. 


[[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]]
==== Orbital trajectory (circular vs. straight) ====
Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. 

In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit.

The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). 

The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other.


==== Precession ====
semi-minor axis: &lt;math&gt;b = \alpha l^2 \lambda_A&lt;/math&gt;

semi-major axis: &lt;math&gt;a = \alpha n^2 \lambda_A&lt;/math&gt;

radius of curvature :&lt;math&gt;L = \frac{b^2}{a} = \frac{a l^4 \lambda_A}{n^2}&lt;/math&gt;

:&lt;math&gt;\frac{3 \lambda_A}{2 L} = \frac{3 n^2}{2 \alpha l^4}&lt;/math&gt;


arc secs per 100 years (drift):

:&lt;math&gt;T_{earth}&lt;/math&gt; = 365.25 days 

drift = &lt;math&gt;\frac{3 n^2}{2 \alpha l^4} 1296000 \frac{100 T_{earth}}{T_{planet}}&lt;/math&gt; 

 Mercury (eccentricity = 0.205630)
 T = 87.9691 days
 a = 57909050 km (''n'' = 378.2734) 
 b = 56671523 km (''l'' = 374.2096)
 drift = 42.98

 Venus (eccentricity = 0.006772) 
 T = 224.701 days
 a = 108208000 km (''n'' = 517.085) 
 b = 108205519 km (''l'' = 517.079)
 drift = 8.6247

 Earth (eccentricity = 0.0167)
 T = 365.25 days
 a = 149598000 km (''n'' = 607.989) 
 b = 149577138 km (''l'' = 607.946)
 drift = 3.8388

 Mars (eccentricity = 0.0934)
 T = 686.980 days
 a = 227939366 km (''n'' = 750.485) 
 b = 226942967 km (''l'' = 748.843)
 drift = 1.351


[[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]]
==== Hyper-sphere orbit ====
{{main|Relativity (Planck)}}

Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt;. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. 

While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B&lt;sup&gt;1&lt;/sup&gt; to B&lt;sup&gt;11&lt;/sup&gt;) around the ''A'' time-line (vertical expansion) axis ('''t&lt;sub&gt;d&lt;/sub&gt;''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. 


For object '''B'''

:&lt;math&gt;t_d = \sqrt{t^2 - {t_0}^2} = t \sqrt{1 - v_{outer}^2}&lt;/math&gt; 

For object '''A'''

:&lt;math&gt;t_d = t \sqrt{1 - v_{inner}^2}&lt;/math&gt;


==== Planck force ====

:&lt;math&gt;F_p = \frac{m_P c^2}{l_p}&lt;/math&gt;

:&lt;math&gt;M_a = \frac{m_P \lambda_a}{2 l_p} ,\;m_b = \frac{m_P \lambda_b}{2 l_p}&lt;/math&gt;

:&lt;math&gt;F_g = \frac{M_a m_b G}{R^2} = \frac{\lambda_a \lambda_b F_p}{4 R_g^2}  = \frac{\lambda_a \lambda_b F_p}{4 \alpha^2 n^4 (\lambda_a + \lambda_b)^2} &lt;/math&gt;

a) &lt;math&gt;M_a = m_b&lt;/math&gt;

:&lt;math&gt;F_g = \frac{F_p}{{(4 \alpha n^2)}^2} &lt;/math&gt;

b) &lt;math&gt;M_a &gt;&gt; m_b&lt;/math&gt;

:&lt;math&gt;F_g = \frac{\lambda_b F_p}{{(2 \alpha n^2)}^2 \lambda_a} = \frac{m_b c^2}{2 \alpha^2 n^4 \lambda_a} = m_b a_g&lt;/math&gt;


=== Atomic orbitals ===
[[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]]
In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times. This means that we can treat the atomic orbital as a simple 2-body orbit with the electron as the orbiting point. Although this approach can only map the electron point-state (and so offers no direct information regarding the electron as a wave), during electron transition between ''n''-shell orbitals, we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]], this is significant because periodically the spiral angle components converge reducing to integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r'').

As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin. &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;.


==== Simulation ====
The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational and atomic orbits is only in the angle of rotation &lt;math&gt;\beta&lt;/math&gt;' which has an additional &lt;math&gt;r_{\alpha}&lt;/math&gt; term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately.

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}&lt;/math&gt; 


===== Rydberg atom =====
For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength), at the ''n'' = 1 orbital, 1 complete rotation becomes (on a 2-D plane);

:&lt;math&gt;t_{ref} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha \lambda_{atom}&lt;/math&gt; 


Adding a relativistic term

:&lt;math&gt;t_{rel} = 2\pi 4\alpha^2 \sqrt{1 - \frac{1}{4\alpha^2}}&lt;/math&gt;

&lt;math&gt;1t_{rel}&lt;/math&gt; = 471961.214...

&lt;math&gt;4t_{rel}&lt;/math&gt; = 1887844.85912...

&lt;math&gt;9t_{rel}&lt;/math&gt; = 4247650.93303...

&lt;math&gt;16t_{rel}&lt;/math&gt; = 7551379.43650...


===== H atom =====
Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]).

H(1s-2s) =  2466 061 413 187.035 kHz &lt;ref&gt;http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf&lt;/ref&gt;  

H(1s-3s) =  2922 743 278 665.79 kHz &lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/33243883/&lt;/ref&gt;  

H(1s-4s) =  3082 581 563 822.63 kHz &lt;ref&gt;https://codata.org/&lt;/ref&gt;   

H(1s-∞s) =  3288 086 857 127.60 kHz &lt;ref&gt;https://codata.org/ (109678.77174307cm-1)&lt;/ref&gt; (''n'' = ∞)


R = 10973731.568157 &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt; ([[w:Rydberg constant |Rydberg constant]])

α =137.035999177 (inverse fine structure constant &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt;


The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. 

&lt;math&gt;r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}&lt;/math&gt;


Dividing (dimensioned) wavelength (&lt;math&gt;r_{wavelength}&lt;/math&gt;) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The &lt;math&gt;(n^2 - 1)&lt;/math&gt; term gives the number of orbital wavelengths in the transition phase;

:&lt;math&gt;h_{(1s-ns)} = (n^2 - 1) \frac{\lambda_H }{H(1s-ns)}&lt;/math&gt;

&lt;math&gt;h_{(1s-2s)}&lt;/math&gt; = 1887839.82626...

&lt;math&gt;h_{(1s-3s)}&lt;/math&gt; = 4247634.04874...

&lt;math&gt;h_{(1s-4s)}&lt;/math&gt; = 7551347.55306...


===== Simulation atom =====
The following example simulates an electron transition, the electron begins at radius &lt;math&gt;r = r_{orbital}&lt;/math&gt; and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model). 

The period &lt;math&gt;t_{sim}&lt;/math&gt; and length &lt;math&gt;l_{sim}&lt;/math&gt; are measured at integer &lt;math&gt;n^2 r&lt;/math&gt; (''n'' &gt; 1) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...) (the angles converge to give integer values at these radius), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles. These (integer radius) values can then be used to calculate the transition frequencies.


In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''&lt;sub&gt;sim&lt;/sub&gt; = period and ''l''&lt;sub&gt;sim&lt;/sub&gt; = distance travelled by the electron (&lt;math&gt;l_{orbital} = l_{sim}&lt;/math&gt; at ''n''=1), the radius coefficient ''r''&lt;sub&gt;n&lt;/sub&gt; = radius divided by &lt;math&gt;r_{orbital}&lt;/math&gt;. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static.
[[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]]


:&lt;math&gt;j_{atom} = 250&lt;/math&gt; (atomic mass)

:&lt;math&gt;i_{nucleus} = j_{atom} -1 = 249&lt;/math&gt; (relative nucleus mass)

:&lt;math&gt;r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2&lt;/math&gt; = 2.0160965

:&lt;math&gt;r_{orbital} = 2 \alpha \;*\; r_{wavelength} &lt;/math&gt; (radius) = 552.5556

:&lt;math&gt;t_n = \frac{t_{sim}}{r_{wavelength}}&lt;/math&gt; 

:&lt;math&gt;l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}&lt;/math&gt; 

:&lt;math&gt;r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}&lt;/math&gt; 

:&lt;math&gt;r_n = \frac{r_b}{r_{orbital}}&lt;/math&gt; 


{| class=&quot;wikitable&quot;
|+Electron transition (mass = 250; ''n''=1 to ''n''=5)
! ''r''&lt;sub&gt;n&lt;/sub&gt;
! ''t''&lt;sub&gt;sim&lt;/sub&gt;
! ''l''&lt;sub&gt;n&lt;/sub&gt;
! angle
! ''x'', ''y'' (electron) 
! ''x'', ''y'' (nucleus) 
! ''x'', ''y'' (barycenter) 
|-1
| 1
| 471957.072
| 0.9999897
| 360
| 550.334,  0.0036
| -2.2102,  -0.00002 
| -0.00004,  -0.00001
|-
| 4
| 1887867.293
| 2.000012
| 359.952489
| 2202.8558,  0.0001
| -7.9565,  -1.9475
| 0.8868,  -1.9397
|-
| 9 
| 4247689.502 
| 4.000014
| 119.92712 
| -2473.180,  4296.283 
| 13.558,  -10.325  
| 3.611,  6.901
|-
| 16 
| 7551439.538 
| 6.000014 
| 179.91669
| -8815.254,  12.818 
| 25.636,  13.303
| -9.728,  13.301
|-
| 25 
| 11799118.905 
| 8.000014  
| 215.9122
| -11158.64,  -8081.13
| 16.580,  39.083
| -28.118,  6.602
|}


Comparison of the spiral angle with different mass: (64, 128, 250, 500, Rydberg) suggests nucleus mass and shape influences the results, however note the spiral only measures transition on a 2-D plane. For the proton:electron mass ratio; ''m'' = 1836.15267...

{| class=&quot;wikitable&quot;
|+ Spiral angle at &lt;math&gt;r_n&lt;/math&gt; = 4, 9 
! mass 
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 4 
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 9
|-
| ''m'' = 64
| 359.80318°
| 119.70323°
|-
| ''m'' = 128
| 359.903935°
| 119.854148°
|-
| ''m'' = 250
| 359.952489°
| 119.92711°
|-
| ''m'' = 500
| 359.977062°
|  
|-
| Rydberg
| 360°
| 360+120°
|}


[[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]]
==== Theory ====
=====Alpha orbital =====
The H atom has 1 proton and 1 electron orbiting the proton, the electron can be found at fixed radius (the [[w:Bohr radius |Bohr radius]]) from the proton (nucleus), these radius represent different energy levels (orbitals) at which the electron may be found orbiting the proton and so are described as quantum levels. Electron transition (to higher energy levels) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to lower energy levels.

The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron is further from the nucleus). Each ''n'' ([[w:electron shell|electron shell]]) can accommodate up to ''n''&lt;sup&gt;2&lt;/sup&gt; electrons (1, 4, 9, 16, 25...), and accounting for two states of spin, 2''n''&lt;sup&gt;2&lt;/sup&gt;. As these orbitals are fixed according to integer ''n'', the atom can be said to be quantized.


The basic (alpha) radius for each ''n'' level uses the fine structure constant alpha (α = 137.036) whereby;

&lt;math&gt;r_{orbital} = 2\alpha n^2 (\lambda_p + \lambda_e)&lt;/math&gt;


Such that at ''n'' = 1, the start radius alpha component ''r'' = 2α. We can map the electron orbit around the orbital as a series of steps. The steps are defined according to the rotation angle β;

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;


[[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]]
At specific ''n'' levels (for clarity the wavelength is not included);

:&lt;math&gt;\beta = \frac{1}{4\alpha^2 n^3}&lt;/math&gt;


This gives a length travelled per (integer) step as the inverse of the radius

:&lt;math&gt;l_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt; 

:&lt;math&gt;v_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt; 


The number of steps (orbital period) for 1 orbit of the electron then becomes 

:&lt;math&gt;t_{orbital} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha n^3&lt;/math&gt;


===== Photon orbital model =====
The electron can jump between ''n'' levels via the absorption or emission of a photon. In the [[Quantum_gravity_(Planck)#Atomic_orbitals|photon-orbital]] model&lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that &lt;math&gt;orbital \;radius + photon = zero&lt;/math&gt; (cancel).

The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius). Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role.


The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).

:&lt;math&gt;\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}&lt;/math&gt;

:&lt;math&gt;\lambda_{photon} = (+\lambda_i) - (+\lambda_f)&lt;/math&gt;


The wavelength of the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon corresponds to the wavelength of the orbital radius. The (+&lt;math&gt;\lambda_i&lt;/math&gt;) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-&lt;math&gt;\lambda_f&lt;/math&gt;), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+&lt;math&gt;\lambda_i&lt;/math&gt;) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

:&lt;math&gt;r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)&lt;/math&gt;


However, the (&lt;math&gt;\lambda_f&lt;/math&gt;) has the longer wavelength, and so after the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon has been absorbed, and for the remaining duration of this (&lt;math&gt;\lambda_f&lt;/math&gt;) photon wavelength, the orbital radius will be extended until the (&lt;math&gt;\lambda_f&lt;/math&gt;) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial &lt;math&gt;n_i = 1&lt;/math&gt; orbital to a final &lt;math&gt;n_f = 2&lt;/math&gt; orbital, first the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed (&lt;math&gt;\lambda_i + \lambda_{orbital} = zero&lt;/math&gt; which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (&lt;math&gt;\lambda_f&lt;/math&gt;) photon continues until it too is absorbed (the '''transition phase''').

:&lt;math&gt;t_{ref} \sim 2\pi 4\alpha^2 &lt;/math&gt;

:&lt;math&gt;\lambda_i = 1t_{ref}&lt;/math&gt;

:&lt;math&gt;\lambda_f = 4t_{ref}&lt;/math&gt; (''n'' = 2)


After the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed, the (&lt;math&gt;\lambda_f&lt;/math&gt;) photon still has &lt;math&gt;\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}&lt;/math&gt; steps remaining until it too is absorbed.


[[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]]
However, instead of a discrete jump between energy levels by the electron, absorption/emission takes place in discrete steps, each step corresponds to a unit of &lt;math&gt;r_{incr}&lt;/math&gt;;

:&lt;math&gt;r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}&lt;/math&gt;


As &lt;math&gt;r_{incr}&lt;/math&gt; has a minus value, the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon will shrink the orbital radius accordingly, per step

WHILE (&lt;math&gt;r_{orbital} &gt; 0&lt;/math&gt;) 

:&lt;math&gt;r_{orbital} = r_{orbital} + r_{incr}&lt;/math&gt; 


Conversely, because of its minus term, the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon will simultaneously extend the orbital radius accordingly;

&lt;math&gt;r = r_{orbital}&lt;/math&gt;

WHILE (&lt;math&gt;r &lt; 4 r_{orbital}&lt;/math&gt;) 

:&lt;math&gt;r = r - r_{incr}&lt;/math&gt; 


The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]]

:&lt;math&gt;t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}&lt;/math&gt; 

The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;

:&lt;math&gt;t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}&lt;/math&gt; 


===== Alpha spiral =====
[[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]]
A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a''').

In its simplest form, a [[w:fine structure constant|fine structure constant]] spiral (or alpha spiral) is a specific hyperbolic spiral that appears in [[w:Atomic electron transition|electron transitions]] between [[w:atomic orbital|atomic orbitals]] in a [[w:Hydrogen atom|Hydrogen atom]].


It can be represented in Cartesian coordinates by

:&lt;math&gt;x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 &lt; \varphi &lt; 4\pi&lt;/math&gt;


This spiral has only 2 revolutions approaching 720° as the radius approaches infinity. If we set start radius '''r''' = 1, then at given angles radius '''r''' will have integer values (the angle components cancel).

:&lt;math&gt;\varphi = (2)\pi, \; r = 4&lt;/math&gt; (360°)
:&lt;math&gt;\varphi = (4/3)\pi,\; r = 9&lt;/math&gt; (240°)
:&lt;math&gt;\varphi = (1)\pi, \;  r = 16&lt;/math&gt; (180°)
:&lt;math&gt;\varphi = (4/5)\pi, \;  r = 25&lt;/math&gt; (144°)
:&lt;math&gt;\varphi = (2/3)\pi, \;  r = 36&lt;/math&gt; (120°)


For a Rydberg atom, starting the simulation with &lt;math&gt;\varphi = 0, \;r = 2\alpha&lt;/math&gt; (''n''=1), such that for each step during transition;

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;

:&lt;math&gt;\varphi = \varphi + \beta&lt;/math&gt;


As &lt;math&gt;\beta&lt;/math&gt; is proportional to the radius, as the radius increases the value of &lt;math&gt;\beta&lt;/math&gt; will reduce correspondingly (likewise reducing the orbital velocity). 

{{see|Fine-structure_constant_(spiral)}}


Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and the &lt;math&gt;n_f^2&lt;/math&gt; at each step &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=4. Atomic energy levels correlate exactly to pi via a hyperbolic spiral |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;.

:&lt;math&gt;r = 2 \alpha + \frac{t}{2\pi 2\alpha}&lt;/math&gt; (number of increments ''t'' of &lt;math&gt;r_{incr}&lt;/math&gt;)

:&lt;math&gt;n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}&lt;/math&gt; (&lt;math&gt;n_f^2&lt;/math&gt; as a function of ''t'') 

:&lt;math&gt;\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}&lt;/math&gt; (&lt;math&gt;\varphi&lt;/math&gt; at any &lt;math&gt;n_f^2&lt;/math&gt;) 

[[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]]


== External links ==
* [[v:Fine-structure_constant_(spiral) | Fine structure constant hyperbolic spiral]]
* [[v:Physical_constant_(anomaly) | Physical constant anomalies]]
* [[v:Planck_units_(geometrical) | Planck units as geometrical objects]] 
* [[v:electron_(mathematical) | The mathematical electron]]
* [[v:Relativity_(Planck) | Programming relativity at the Planck scale]]
* [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck level]]
* [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]]
* [[v:God_(programmer) | The Programmer God]]
* [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] 
* [https://theprogrammergod.com/ The Programmer God, are we in a computer simulation? - eBook] 


==References==
{{Reflist}}


[[Category:Physics| ]]
[[Category:Philosophy of science| ]]</text>
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==Most general Lorentz transformations==
===General quadratic form===
The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as&lt;ref&gt;Bôcher (1907), chapter X&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align}
=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\
b=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y}=b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}'\\
\left(\mathbf{A}=\mathbf{A}^{{\rm T}}\right)\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\mathbf{g}^{-1}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q1}}}}

The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc.

===Most general Lorentz transformation===
{{CSS image crop
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|Description=The Lorentz interval is the invariant relation between axes and conjugate diameters of hyperbolas, illustrating Lorentz transformations between two inertial frames.}}
The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:&lt;ref name=ratcliffe&gt;Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1&lt;/ref&gt;&lt;ref&gt;Naimark (1964), 2 in four dimensions&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} &amp; =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\
-x_{0}y_{0}+\cdots+x_{n}y_{n} &amp; =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\downarrow\\
\begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\
 &amp; \dots\\
x_{n}^{\prime} &amp; =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn}
\end{align}
\\
\\
\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\
\downarrow\\
\begin{align}x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\
 &amp; \dots\\
x_{n} &amp; =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn}
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} &amp; =\mathbf{g}^{-1}\\
\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} &amp; =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} &amp; =\mathbf{A}\\
\\
\end{align}
\\
\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\end{matrix}
\end{matrix}&lt;/math&gt;|{{equationRef|1a}}}}

The invariance of the Lorentz interval with ''n''=1 between axes and [[w:conjugate diameters]] of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.&lt;ref&gt;Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.&lt;/ref&gt;&lt;ref&gt;Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.&lt;/ref&gt; They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881-91)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the &quot;absolute&quot; form &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=0&lt;/math&gt; as discussed by [[#Klein|Klein (1871-73)]].&lt;ref group=M&gt;Killing (1885), p. 71&lt;/ref&gt;&lt;ref&gt;Müller (1910), p. 661, in particular footnote 247.&lt;/ref&gt;&lt;ref&gt;Sommerville (1911), p. 286, section K6.&lt;/ref&gt; In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; by [[#Killing3|Killing (1888-1897)]].

===Most general Lorentz transformation of velocity===

If &lt;math&gt;x_{i},\ x_{i}^{\prime}&lt;/math&gt; in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates &lt;math&gt;u_{s},\ u_{s}^{\prime}&lt;/math&gt; follow by

:&lt;math&gt;\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)&lt;/math&gt;

defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1&lt;/math&gt; so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called &quot;the most general formula for the composition of velocities&quot; in terms of special relativity&lt;ref&gt;Synge (1955), p. 129 for ''n''=3&lt;/ref&gt; (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})):

{{NumBlk|:|&lt;math&gt;\begin{align}u_{s}^{\prime} &amp; =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\\
u_{s} &amp; =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}}
\end{align}
\left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.&lt;/math&gt;|{{equationRef|1b}}}}

Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere &lt;math&gt;-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0&lt;/math&gt; were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]].

==Historical notation==
==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Apo|label 1=History of Lorentz transformations via squeeze mappings § Apollonius}}
====Equality of difference in squares====
[[File:Apollonius-Borelli-XII.png|thumb|&lt;small&gt;Fig. 1: Apollonius' proposition illustrated by Borelli (1661) of &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Apollonius of Perga]] (c. 240–190 BC) in his 7th book on conics defined the following well known proposition (the 7th book survived in Arabian translation, and was translated into Latin in 1661 and 1710), as follows:

*The difference of the squares of the two axes of the hyperbola is equal to the difference of the squares of any two conjugate diameters. &lt;small&gt;(Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].)&lt;ref group=M name=bor1&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XII and other props. from book VII on pp. 291-292; See also the note on prop. XII on pp. 293-294, where Borelli demontrates &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt; (in later translations such as Halley (1710), the proposition was numbered as XIII.) Latin: &quot;Differentia quadratorum duorum axium hyperboles æqualis est differentiæ quadratorum quarumlibet duarum diametrorum coniugatarum.&quot;&lt;/ref&gt;&lt;/small&gt;

*In every hyperbola the difference between the squares of the axes is equal to the difference between the squares of any conjugate diameters of the section. &lt;small&gt;(Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XIII of book VII on p. 107; Latin: &quot;In omni Hyperbola differentia inter quadrata Axium aequalis est differentiae inter quadrata ex diametris quibusvis conjugatis sectionis.&quot;&lt;/ref&gt;&lt;/small&gt;

*[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).&lt;/ref&gt;&lt;/small&gt;
----
[[File:Lahire-XLII-XLIII.png|thumb|left|&lt;small&gt;Fig. 2: La Hire's (1685) illustration of &lt;math&gt;\scriptstyle \overline{AB}^{2}-\overline{DE}^{2}=\overline{NM}^{2}-\overline{LK}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[File:Lhopital Conjugate Diameters.png|thumb|&lt;small&gt;Fig. 3: l'Hôpital's (1707) illustration of &lt;math&gt;\scriptstyle \overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685) stated this proposition as follows:
{{Block indent|1=I say that the difference of the squares of any two diameters conjugated to each other, AB, DE, is equal to the difference of the squares of any two other diameters conjugated to each other, NM, LK.&lt;ref group=M name=lahire1&gt;La Hire (1685), Book IV, Proposition XLII, p. 85; Latin: &quot;Dico differentiam quadratorum duarum diametrorum quarumlibet inter se conjugatarum AB, DE esse æqualem differentiæ quadratorum duarum aliarum diametrorum quarumlibet inter se conjugatarum, NM, LK.&quot;&lt;/ref&gt;}}
and also summarized the related propositions in the 7th book of Apollonius:
{{Block indent|1=In a hyperbola, the difference of the squares of the axes is equal to the difference of the squares of any two conjugate diameters.&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of propositions XII, XIII, XXV in the 7th book of Apollonius; Latin: &quot;In hyperbola differentia quadratorum axium æqualis est differentia quadratorum duarum diametrorum conjugatarum quarumlibet.&quot;&lt;/ref&gt;}}
----
[[w:Guillaume de l'Hôpital]] (1707), using the methods of [[wikipedia:Analytic_geometry|w:analytic geometry]], demonstrated the same proposition:&lt;ref group=M name=lop&gt;l'Hôpital (1707), Third book, Prop. XII, p. 76.&lt;/ref&gt;

{{Block indent|1=The difference of the squares of any two conjugate diameters &quot;Mm, Ss&quot; is equal to the difference of the squares of the two axes &quot;Aa, Bb.&quot; We are to prove that &lt;math&gt;\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;, or &lt;math&gt;\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}&lt;/math&gt;. &lt;small&gt;(English translation 1723 by [[w:Edmund Stone]].)&lt;ref group=M&gt;l'Hôpital/Stone (1723), pp. 62-63&lt;/ref&gt;&lt;/small&gt;}}

{{Lorentzbox|Text=Apollonius' proposition can be expressed as &lt;math&gt;-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation ({{equationNote|1a}}) &quot;(n=1)&quot; can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.}}

====Equality of areas of parallelograms====

[[File:Apollonius-Borelli-XXXI.png|thumb|&lt;small&gt;Fig. 4: Apollonius' proposition illustrated by Borelli (1661) of the equality of areas of parallelogram ABCD (of the axes) and KLMN (of the conjugated diameters).&lt;/small&gt;]] 

Apollonius also gave another well known proposition in his 7th book regarding ellipses as well as conjugate sections of hyperbolas (see also Del Centina &amp; Fiocca&lt;ref&gt;Del Centina &amp; Fiocca (2020)&lt;/ref&gt; for further details on the history of this proposition): 

*In the ellipse, and in conjugate sections [the opposite branches of two conjugate hyperbolas] the parallelogram bounded by the axes is equal to the parallelogram bounded by any pair of conjugate diameters, if its angles are equal to the angles the conjugate diameters form at the centre. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del&gt;Del Centina &amp; Fiocca (2020), section 3.1&lt;/ref&gt; based on the Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].&lt;ref group=M name=bor2&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XXXI of book VII on p. 370; Note on pp. 372-374; Latin: &quot;In ellypsi, &amp; sectionibus coniugatis parallelogrammum sub axibus contentum æquale est parallelogrammo à quibuscunque duabus coniugatis diametris comprehenso, si eorum anguli æquales fuerint angulis ad centrum contentis à coniugatis diametris.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If two conjugate diameters are taken in an ellipse, or in the opposite conjugate sections; the parallelogram bounded by them is equal to the rectangle bounded by the axes, provided its angles are equal to those formed at the centre by the conjugate diameters. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del /&gt; based on the Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XXXI of book VII on p. 115–117; Latin: &quot;Si ducantur diametri quævis conjugate in Ellipsi, vel inter sectiones oppositas conjugatas; erit parallelogrammum contentam sub his diametris æquale rectangulo sub ipsis Axibus facto: modo anguli ejus æquales sint angulis ad centrum sectionis à diametris conjugatis comprehensis.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = rect. AA'·BB'. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 136, p. 235; (Apollonius, Book VII, Prop. 31).&lt;/ref&gt;&lt;/small&gt; 

{{Lorentzbox|Text=The graphical representation of Apollonius proposition in Borelli's Fig. 4 is essentially a [[w:Minkowski diagram]], being a graphical representation of the Lorentz transformation. If line AB is the x-axis of an inertial frame S1, then line FG is the x-axis of another inertial frames S2 which together with its parallel lines (such as KL and NM) represent [[w:relativity of simultaneity]]. Analogously, if line CD is the time axis of another inertial frame S2, then line HI is the time axis of S2 which together with its parallel lines (such as KN and LM) represent the [[w:worldlines]] of objects at different locations. The diagonals KE (or KM) and LE (or LN) lie on the asymptotes which form a light cone. Thus the totality of all parallelograms of equal area and conjugate diameters as constructed by Apollonius, represents the totality of all inertial frames, lines of simultaneity and worldlines within a spacetime area bounded by &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=\rm{const}&lt;/math&gt;.}}

[[File:Saint-Vincent-Hyperbola-VI-XLIX.png|thumb|175px|left|&lt;small&gt;Fig. 5: Saint-Vincent's (1647) illustration of FGHI=OPQR, as well as BADC=KNLM.&lt;/small&gt;]]
[[w:Grégoire de Saint-Vincent]] independently (1647) stated the same proposition:&lt;ref group=M name=vinc&gt;St. Vincent (1647), Book VI, Prop. XLIX, p. 560; Latin: “Si fuerint binæ hyperbolarum coniugaciones A, B, C, D: ponantur autem per E centrum duæ quoque diametrorum coniugationes per quarum vertices contingentes actæ constituant duo quadrilatera FGHI, OPQR. Dico illa esse æqualia inter se.”&lt;/ref&gt;

{{Block indent|1=The parallelograms whose opposite sides are tangent to two conjugate hyperbolas at the extremities of two conjugate diameters are equivalent among them. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref&gt;Del Centina &amp; Fiocca (2020), section 5.1&lt;/ref&gt;)&lt;/small&gt; }}

---- 

[[File:Lahire-XLII-XLIII.png|thumb|&lt;small&gt;Fig. 6 (identical to Fig. 2): La Hire's (1685) illustration of FGHI=OPQR.&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685), who was aware of both Apollonius 7th book and Saint-Vincent's book, stated this proposition as follows:&lt;ref group=M name=lahire&gt;La Hire (1685), Book IV, Proposition XLIII, pp. 85-86; Latin: &quot;In sectionibus conjugatis NA, DL, BM, KE si circumscribatur parallelogrammum FGHI à rectis parallelis duabus diametris inter se conjugatis ED, BA, &amp; per ipsorum terminos ductis, &amp; simili methodo circumscribatur aliud parallelogrammum OPQR à rectis ductis per terminos diametrorum conjugatarum, &amp; ipsis parallelis: Dico parallelogramma FGHI, OPQR esse inter se æqualia.&quot;&lt;/ref&gt; 

{{Block indent|1=If a parallelogram FGHI is circumscribed about conjugate sections NA, DL, BM, KE whose sides are parallel to two conjugate diameters ED, BA drawn through their extremities, and with similar method another parallelogram OPQR is drawn through the extremities of other two conjugate diameters, then the parallelograms FGHI, OPQR are equal. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2&gt;Del Centina &amp; Fiocca (2020), section 5.2&lt;/ref&gt;)&lt;/small&gt;}}

and also summarized the related propositions in the 7th book of Apollonius:&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of proposition XXXI in the 7th book of Apollonius; Latin: &quot;In sectionibus conjugatis &amp; Ellipsi parallelogrammum sub axibus æquale est paralelogrammo sub duabus quibuscunque diametris inter se conjugatis, in angulis ipsarum diametrorum conjugatarum.&quot;&lt;/ref&gt;

{{Block indent|1=In conjugate sections and in the ellipse, the parallelogram constructed with the axes, is equal to the parallelogram constructed with any two conjugated diameters, provided the angles are equal to those between the diameters themselves. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2 /&gt;)&lt;/small&gt;}}

{{Lorentzbox|Text=In Saint-Vincent's Fig. 5 or La Hire's Fig. 6, parallelogram FGHI contains all coordinates related to an inertial frame S3, in particular triangles EGH, EFI (Fig. 5) or CFG, CHI (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EHI, EGF (Fig. 5) or CFI, CGH (Fig. 6) contain space like intervals between events on the negative and positive x-axis. Conversely, parallelogram OPQR contains all coordinates related to another frame S4, in particular triangles EQR, EOP (Fig. 5) or CPQ, COR (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EPR, EOQ (Fig. 5) or COP, CQR (Fig. 6) contain space like intervals between events on the negative and positive x-axis.}}

===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.}}

==={{anchor|Gauss}} Gauss (1798–1818)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Gauss|label 1=History of Lorentz transformations via Möbius transformations § Gauss}}
===={{anchor|Gauss1}} Binary quadratic forms====

The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

{{Lorentzbox|Text=The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.}}

===={{anchor|Gauss2}} Ternary quadratic forms====

Gauss (1798/1801)&lt;ref group=M&gt;Gauss (1798/1801), articles 266–285&lt;/ref&gt; also discussed ternary quadratic forms with the general expression

:&lt;math&gt;\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\\
g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)\\
\hline \begin{align}x &amp; =\alpha y+\beta y'+\gamma y''\\
x' &amp; =\alpha'y+\beta'y'+\gamma'y''\\
x'' &amp; =\alpha''y+\beta''y'+\gamma''y''
\end{align}
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;'', or indefinite in the case of different signs such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:&lt;ref group=M&gt;Gauss (1798/1801), article 277&lt;/ref&gt;

:&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\\
\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval &lt;math&gt;x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt;}}

===={{anchor|Gauss4}} Homogeneous coordinates====

Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression

:&lt;math&gt;(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E&lt;/math&gt;

into

:&lt;math&gt;G+G'\cos T^{2}+G''\sin T^{2}&lt;/math&gt;

in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:&lt;ref group=M&gt;Gauss (1818), pp. 5–10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\
k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\
\hline \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =k &amp; \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =-k\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-k &amp; \beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-k\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-k &amp; \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =+k\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}\right.\\
\hline k=1\\
\begin{align}t\cos E &amp; =\alpha+\alpha'\cos T+\alpha''\sin T\\
t\sin E &amp; =\beta+\beta'\cos T+\beta''\sin T\\
t &amp; =\gamma+\gamma'\cos T+\gamma''\sin T
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}})  ''(n=2)''.

Further setting &lt;math&gt;[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}})  ''(n=2)''.}}

Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that

:&lt;math&gt;aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz&lt;/math&gt;

can be transformed into

:&lt;math&gt;Guu+G'u'u'+G''u''u''&lt;/math&gt;,

in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:&lt;ref group=M&gt;Gauss (1818), pp. 9–10&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =-\alpha x-\beta y+\gamma z\\
u' &amp; =\alpha'x+\beta'y-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}})  ''(n=2)'' satisfying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;, and can be related to Gauss' previous equations in terms of homogeneous coordinates &lt;math&gt;\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]&lt;/math&gt;.}}

==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:&lt;ref group=M&gt;Jacobi (1827), p. 235, 239–240&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\
k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\
\hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P &amp; =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\cos\vartheta &amp; =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\sin\vartheta &amp; =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\\
\cos\psi &amp; =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\cos\varphi &amp; =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\sin\varphi &amp; =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}
\end{align}
\\
\\
\hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z &amp; =m\\
\alpha'\mu+\beta'x+\gamma'y+\delta'z &amp; =m'\\
\alpha''\mu+\beta''x+\gamma''y+\delta''z &amp; =m''\\
\alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z &amp; =m'''\\
\\
Am+A'm'+A''m''+A'''m''' &amp; =\mu\\
Bm+B'm'+B''m''+B'''m''' &amp; =x\\
Cm+C'm'+C''m''+C'''m''' &amp; =y\\
Dm+D'm'+D''m''+D'''m''' &amp; =z\\
\\
\end{align}
\\
\begin{align}\alpha &amp; =-kA, &amp; \beta &amp; =-kB, &amp; \gamma &amp; =-kC, &amp; \delta &amp; =kD,\\
\alpha' &amp; =kA', &amp; \beta' &amp; =kB', &amp; \gamma' &amp; =kC', &amp; \delta' &amp; =-kD',\\
\alpha'' &amp; =kA'', &amp; \beta'' &amp; =kB'', &amp; \gamma'' &amp; =kC'', &amp; \delta'' &amp; =-kD'',\\
\alpha''' &amp; =kA''', &amp; \beta''' &amp; =kB''', &amp; \gamma''' &amp; =kC''', &amp; \delta''' &amp; =-kD''',
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta &amp; =-k\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' &amp; =k\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' &amp; =k\\
\alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' &amp; =k\\
\alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' &amp; =0\\
\alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' &amp; =0\\
\alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' &amp; =0\\
\alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' &amp; =0\\
\alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' &amp; =0\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' &amp; =0\\
\\
-\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' &amp; =k\\
-\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' &amp; =k\\
-\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' &amp; =k\\
-\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' &amp; =-k\\
-\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' &amp; =0\\
-\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' &amp; =0\\
-\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' &amp; =0\\
-\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' &amp; =0\\
-\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' &amp; =0\\
-\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' &amp; =0
\end{align}
\end{matrix}}
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] &amp; =\left[u_{1},\ u_{2},\ u_{3}\right]\\{}
[\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] &amp; =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right]
\end{align}
}&lt;/math&gt; and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing &lt;math&gt;m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}&lt;/math&gt;.}}

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):&lt;ref group=M&gt;The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\
\mathbf{(1)}\begin{align}x &amp; =\alpha s+\alpha's'+\alpha''s''\\
y &amp; =\beta s+\beta's'+\beta''s''\\
z &amp; =\gamma s+\gamma's'+\gamma''s''\\
\\
s &amp; =\alpha x+\beta y+\gamma z\\
s' &amp; =\alpha'x+\beta'y+\gamma'z\\
s'' &amp; =\alpha''x+\beta''y+\gamma''z
\end{align}
\left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma &amp; =1 &amp; \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' &amp; =1\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma' &amp; =1 &amp; \beta\beta+\beta'\beta'+\beta''\beta'' &amp; =1\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' &amp; =1 &amp; \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' &amp; =1\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma+\beta'\gamma'+\beta''\gamma'' &amp; =0\\
\alpha''\alpha+\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' &amp; =0\\
\alpha\alpha'+\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta+\alpha'\beta'+\alpha''\beta'' &amp; =0
\end{align}
\right.\\
\hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\
\left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\
\hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\
\left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\
\hline \begin{align}\cos\phi &amp; =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \cos\eta &amp; =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\
\sin\phi &amp; =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \sin\eta &amp; =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}
\end{align}
\end{matrix}\\
\hline \\
\mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\
\hline \begin{align}y &amp; =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, &amp; s' &amp; =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\
z &amp; =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, &amp; s'' &amp; =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z},
\end{align}
\end{matrix}
\end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma &amp; =1\\
\alpha'\alpha'-\beta'\beta'-\gamma'\gamma' &amp; =-1\\
\alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' &amp; =-1\\
\alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\alpha''\alpha-\beta''\beta-\gamma''\gamma &amp; =0\\
\alpha\alpha'-\beta\beta'-\gamma\gamma' &amp; =0\\
\\
\alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =1\\
\beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-1\\
\gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =-1\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.}}

Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:&lt;ref group =M&gt;Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\
\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} &amp; =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\
x_{\varkappa} &amp; =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\
\\
\frac{y_{\varkappa}}{y_{n}} &amp; =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\
\frac{x_{\varkappa}}{x_{n}} &amp; =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}
\end{align}
\left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} &amp; =0\\
\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} &amp; =1\\
\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} &amp; =0\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} &amp; =1
\end{align}
\right.\\
\hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\
\alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\
\hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} &amp; =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\
\\
\xi_{\varkappa} &amp; =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}}
\end{align}
\\
\hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.}}

He also stated the following transformation leaving invariant the Lorentz interval:&lt;ref group=M&gt;Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\
\hline {\scriptstyle \begin{align}u &amp; =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\
u_{1} &amp; =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\
 &amp; \dots\\
u_{n-1} &amp; =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\
\\
w &amp; =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\
w_{1} &amp; =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\
 &amp; \dots\\
w_{n-1} &amp; =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1}
\end{align}
\left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =-1\\
\alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =0\\
\alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} &amp; =0\\
\\
\alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} &amp; =-1\\
\alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} &amp; =0\\
\alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} &amp; =0
\end{align}
\text{ }\right.}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.}}

==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids ===

[[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:&lt;ref group=M&gt;Chasles (1829), p. 139&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(a)\\ \\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(b)\\ \\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(c)\\ \\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(d)
\end{matrix}&lt;/math&gt;

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:&lt;ref group=M&gt;Chasles (1829), p. 141&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\
\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\
\alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\
\beta''\quad\Rightarrow\quad\beta''\sqrt{-1}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:&lt;ref group=M&gt;Chasles (1829), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}x &amp; =lx'+l'y'+l''z'\\
y &amp; =mx'+m'y'+m''z'\\
z &amp; =nx'+n'y'+n''z'
\end{align}
\\
\left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} &amp; =0\\
\frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} &amp; =0\\
\frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} &amp; =0
\end{align}
\right\} \\
\hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\
\left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Chasles defined the conditional equations of ''l,m,n'' in the same way as those of &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.}}

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:&lt;ref group=M&gt;Chasles (1829), pp. 145-146&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =a^{2}\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =b^{2}\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(A)\\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(B)\\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =a^{2}\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =b^{2}\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(C)\\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(D)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates===

[[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation&lt;ref group=M&gt;Lebesgue (1837), pp. 338-341&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\
\hline {\scriptstyle \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\
\dots\\
y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} &amp; =1 &amp; (10)\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} &amp; =0 &amp; (11)\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} &amp; =1 &amp; (13)\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} &amp; =0 &amp; (14)
\end{align}
\right.}
\end{matrix}&lt;/math&gt;

In order to achieve the invariance of the Lorentz interval&lt;ref group=M&gt;Lebesgue (1837), pp. 353–354&lt;/ref&gt;

:&lt;math&gt;x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}&lt;/math&gt;

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

{{Lorentzbox|Text=These instructions give Lorentz transformation ({{equationNote|1a}}) in the form:
:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\hline \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\
\dots\\
-y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} &amp; =1\\
a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} &amp; =0\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} &amp; =1\\
a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;}}

He went on to redefine the variables of the Lorentz interval and its transformation:&lt;ref group=M&gt;Lebesgue (1837), pp. 353–355&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\downarrow\\
\begin{align}x_{1} &amp; =x_{n}\cos\theta_{1}, &amp; x_{2} &amp; =x_{n}\cos\theta_{2},\dots &amp; x_{n-1} &amp; =x_{n}\cos\theta_{n-1}\\
y_{1} &amp; =y_{n}\cos\phi_{1}, &amp; y_{2} &amp; =y_{n}\cos\phi_{2},\dots &amp; y_{n-1} &amp; =y_{n}\cos\phi_{n-1}
\end{align}
\\
\downarrow\\
\cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\
\cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\
\hline \\
\cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\
(i=1,2,3\dots n)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting &lt;math&gt;[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]&lt;/math&gt; it is equivalent to Lorentz transformation ({{equationNote|1b}}).}}

==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids===
Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&amp;alpha;), from which he derived equations (&amp;beta;) and others:&lt;ref group=M&gt;Weddle (1847), p. 274&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1, &amp; l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1, &amp; l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} &amp; =0\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1, &amp; l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\alpha)\\ \\
\left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1, &amp; l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1, &amp; l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} &amp; =0\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1, &amp; m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\beta)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;, he defined three conjugate points &lt;math&gt;(x_{1}\dots,y_{1}\dots,z_{1}\dots)&lt;/math&gt; on those two conjugate hyperboloids, related to each other in the same way as equations (&amp;alpha;, &amp;beta;) stated above:&lt;ref group=M&gt;Weddle (1847), pp. 275-276&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} &amp; =0\\
\frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} &amp; =0\\
\frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} &amp; =0
\end{align}
\quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} &amp; =1\\
\frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} &amp; =1\\
\frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} &amp; =-1
\end{align}
\\
\begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} &amp; =a^{2}\\
y_{1}^{2}+y_{2}^{2}-y_{3}^{2} &amp; =b^{2}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2} &amp; =-c^{2}
\end{align}
\quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} &amp; =0\\
x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} &amp; =0\\
y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} &amp; =0
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text= These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:&lt;ref group=M&gt;Bour (1856), pp. 61; 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\
\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\\
\hline \\
k=+1\\
t=\gamma+\gamma'\cos T+\gamma''\sin T,\\
1=u,\ \cos T=u',\ \sin T=u',\\
t=z,\ t\cos E=x,\ t\sin E=y\\
\downarrow\\
\mathbf{(2)}\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =\gamma z-\alpha x-\beta y\\
u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} &amp; =k\\
-\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} &amp; =-k\\
-\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} &amp; =-k\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =k\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;. Furthermore, setting &lt;math&gt;[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt; in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.}}

=== {{anchor|Somov}} Somov (1863) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:&lt;ref group=M&gt;Somov (1863), pp. 12–14; p. 18 for differentials.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\cos\phi &amp; =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\
\sin\phi &amp; =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''}
\end{align}
\right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\
\cos^{2}\psi+\cos^{2}\psi=1
\end{matrix}\\
\hline \mathbf{(1)}\ \begin{align}\cos\phi &amp; =x, &amp; \cos\psi &amp; =x'\\
\sin\phi &amp; =y, &amp; \sin\psi &amp; =y'
\end{align}
\ \left|\begin{align}x &amp; =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\
y &amp; =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=1\\
x^{\prime2}+y^{\prime2}=1
\end{matrix}\\
\hline \begin{align}\cos\phi &amp; =\frac{x}{z}, &amp; \cos\psi &amp; =\frac{x'}{z'}\\
\sin\phi &amp; =\frac{y}{z}, &amp; \sin\psi &amp; =\frac{y'}{z'}
\end{align}
\ \left|\begin{align}\frac{x}{z} &amp; =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\
\frac{y}{z} &amp; =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\
x^{\prime2}+y^{\prime2}=z^{\prime2}
\end{matrix}\\
\hline \mathbf{(2)}\ \left.\begin{align}x &amp; =mx'+ny'+sz'\\
y &amp; =m'x'+n'y'+s'z'\\
z &amp; =m''x'+n''y'+s''z'\\
\\
x' &amp; =mx+m'y-m''z\\
y' &amp; =nx+n'y-n''z\\
z' &amp; =-sx-s'y+s''z\\
\\
dx &amp; =mdx'+ndy'+sdz'\\
dy &amp; =m'dx'+n'dy'+s'dz'\\
dz &amp; =m''dx'+n''dy'+s''dz'
\end{align}
\right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} &amp; =1\\
n^{2}+n^{\prime2}-n^{\prime\prime2} &amp; =1\\
-s^{2}-s^{\prime2}+s^{\prime\prime2} &amp; =1\\
ns+n's'-n''s'' &amp; =0\\
sm+s'm'-s''m'' &amp; =0\\
mn+m'n'-m''n'' &amp; =0\\
\\
m^{2}+n^{2}-s^{2} &amp; =1\\
m^{\prime2}+n^{\prime2}-s^{\prime2} &amp; =1\\
-m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} &amp; =1\\
-m'm''-n'n''+s's'' &amp; =0\\
-m''m-n''n+s''s &amp; =0\\
mm'+nn'-ss' &amp; =0
\end{align}
}\\
dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Klein|label 1=History of Lorentz transformations via Möbius transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Klein|label 1=History of Lorentz transformations via squeeze mappings § Klein}}

Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an &quot;absolute&quot; ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a &quot;fundamental [[w:conic section]]&quot; in order to discuss motions such as rotation and translation in the non-Euclidean plane.&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt; This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].&lt;ref group=M&gt;Klein (1873), pp. 127-128&lt;/ref&gt;

{{Lorentzbox|Text=Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.}}

==={{anchor|Killing}} Killing (1878–1893)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Killing2|label 1=History of Lorentz transformations via hyperbolic functions § Killing}}
===={{anchor|Killing1}} Weierstrass coordinates====

[[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form

:&lt;math&gt;k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1877/78), p. 74; Killing (1880), p. 279&lt;/ref&gt; with &lt;math&gt;ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1880), eq. 25 on p. 283&lt;/ref&gt;

or&lt;ref group=M&gt;Killing (1880), p. 283&lt;/ref&gt;

:&lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt;

where ''k'' is the reciprocal measure of curvature, &lt;math&gt;k^{2}=\infty&lt;/math&gt; denotes [[w:Euclidean geometry]], &lt;math&gt;k^{2}&gt;0&lt;/math&gt; [[w:elliptic geometry]], and &lt;math&gt;k^{2}&lt;0&lt;/math&gt; hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.&lt;ref group=M&gt;Killing (1877/78), eq. 25 on p. 283&lt;/ref&gt; In (1879/80) he tried to formulate the corresponding transformations by plugging &lt;math&gt;k^{2}&lt;/math&gt; into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:&lt;ref group=M&gt;Killing (1879/80), p. 274&lt;/ref&gt;

&lt;math&gt;\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\
\hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\
-k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\
k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}
\end{matrix}\\
\left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right)
\end{matrix}&lt;/math&gt;

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:&lt;ref group=M&gt;Killing (1885), pp. 18, 28–30, 53&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\
k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\
ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\
\hline \begin{align}k^{2}p' &amp; =k^{2}wp+w'x+w''y\\
x' &amp; =ap+a'x+a''y\\
y' &amp; =bp+b'x+b''y\\
\\
k^{2}p &amp; =k^{2}wp'+ax'+by'\\
x &amp; =w'p'+a'x+b'y'\\
y &amp; =w''p'+a''x'+b''y'
\end{align}
\left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} &amp; =k^{2}\\
\frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} &amp; =1\\
\frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
aw+a'w'+a''w'' &amp; =0\\
bw+b'w'+b''w'' &amp; =0\\
\frac{ab}{k^{2}}+a'b'+a''b'' &amp; =0\\
\\
k^{2}w^{2}+a^{2}+b^{2} &amp; =k^{2}\\
\frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} &amp; =1\\
\frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} &amp; =1\\
ww'+aa'+bb' &amp; =0\\
ww''+aa''+bb'' &amp; =0\\
\frac{w'w''}{k^{2}}+a'a''+b'b'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

In (1885) he also gave the transformation for ''n'' dimensions:&lt;ref group=M&gt;Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222&lt;/ref&gt;&lt;ref&gt;Ratcliffe (1994), § 3.6&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\
\hline \left.\begin{align}k^{2}\xi_{0} &amp; =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\
\xi_{\varkappa} &amp; =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\
\\
k^{2}x_{0} &amp; =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\
x_{\varkappa} &amp; =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n}
\end{align}
\right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
\frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} &amp; =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa)
\end{align}
}
\end{matrix}&lt;/math&gt;

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.&lt;ref group=M&gt;Killing (1884/85), pp. 4–5&lt;/ref&gt; Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting &lt;math&gt;k^{2}=-1&lt;/math&gt; and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid &lt;math&gt;p^{2}-x^{2}-y^{2}=1&lt;/math&gt; (known as [[w:hyperboloid model]]),&lt;ref group=M&gt;Killing (1885), Note 9 on p. 260&lt;/ref&gt;&lt;ref name=rey /&gt; by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

{{Lorentzbox|Text=All of Killing's transformations between 1879 and 1885 don't work when &lt;math&gt;k^{2}&lt;/math&gt; is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

Finally, in (1893) he wrote:&lt;ref group=M&gt;Killing (1893), see pp. 144, 327–328&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\
\hline \begin{align}t' &amp; =at+bu+cv\\
u' &amp; =a't+b'u+c'v\\
v' &amp; =a''t+b''u+c''v
\end{align}
\left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} &amp; =k^{2}\\
k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} &amp; =1\\
k^{2}ab+a'b'+a''b'' &amp; =0\\
k^{2}ac+a'c'+a''c'' &amp; =0\\
k^{2}bc+b'c'+b''c'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

and in ''n'' dimensions&lt;ref group=M&gt;Killing (1893), pp. 314–316, 216–217&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\
ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\
\hline \begin{align}y_{0} &amp; =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\
y_{1} &amp; =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
 &amp; \,\,\,\vdots\\
y_{n} &amp; =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
\end{align}
\left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} &amp; =1\\
k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} &amp; =0\\
(\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

===={{anchor|Killing3}} Infinitesimal transformations and Lie group====

After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions, Killing (1887/88)&lt;ref group=M&gt;Killing (1887/88a), pp. 274–275&lt;/ref&gt; defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\
\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\
\text{where}\\
\left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\
\left[\iota\ne\varkappa\ne\lambda\ne\mu\right]
\end{matrix}&lt;/math&gt;

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:&lt;ref group=M&gt;Killing (1892), p. 177&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\
\text{where}\\
\left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f;
\end{matrix}&lt;/math&gt;

In (1897/98) he showed the relation between Weierstrass coordinates &lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt; and coordinates &lt;math&gt;k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0&lt;/math&gt; used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:&lt;ref group=M&gt;Killing (1897/98), pp. 255–256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} &amp; (a)\\
k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} &amp; (b)
\end{matrix}\\
\hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\
\text{where}\\
\left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\
\left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\
\left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\
\hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\
\downarrow\\
k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\
\hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota}
\end{matrix}
\end{matrix}&lt;/math&gt;

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b).

{{Lorentzbox|Text=Setting &lt;math&gt;k^{2}=-1&lt;/math&gt; denotes the group of hyperbolic motions and thus the Lorentz group.}}

=== {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Poincare3|label 1=History of Lorentz transformations via velocity § Poincaré}}

[[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).&lt;ref&gt;Gray (1997)&lt;/ref&gt; He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 220–221&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}F &amp; =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\
 &amp; =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
F &amp; =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\
 &amp; =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1
\end{align}
\\
\hline \begin{align}\xi' &amp; =\alpha\xi+\beta\eta+\gamma\zeta\\
\eta' &amp; =\alpha'\xi+\beta'\eta+\gamma'\zeta\\
\zeta' &amp; =\alpha''\xi+\beta''\eta+\gamma''\zeta
\end{align}
\left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1\\
\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

He went on to describe the properties of &quot;hyperbolic coordinates&quot;.&lt;ref group=M name=poinc&gt;Poincaré (1881b), p. 333&lt;/ref&gt;&lt;ref name=rey&gt;Reynolds (1993)&lt;/ref&gt; Poincaré mentioned the hyperboloid model also in (1887).&lt;ref group=M&gt;Poincaré (1887), p. 206&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

=== {{anchor|Cox}} Cox (1881–1891) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Cox|label 1=History of Lorentz transformations via hyperbolic functions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt; and [[w:George Salmon]] (1862)&lt;ref group=M&gt;Salmon (1862), section 212, p. 165&lt;/ref&gt; on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)&lt;ref group=M&gt;Frischauf (1876), pp. 86–87&lt;/ref&gt; in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:&lt;ref group=M&gt;Cox (1881/82), p. 186 for Weierstrass coordinates; pp. 193–194 for Lorentz transformation.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z^{2}-x^{2}-y^{2}=1\\
x^{2}-y^{2}-z^{2}=Z^{2}-Y^{2}-X^{2}\\
\hline \begin{align}x &amp; =l_{1}X+l_{2}Y+l_{3}Z\\
y &amp; =m_{1}X+m_{2}Y+m_{3}Z\\
z &amp; =n_{1}X+n_{2}Y+n_{3}Z\\
\\
X &amp; =l_{1}x+m_{1}y-n_{1}z\\
Y &amp; =l_{2}x+m_{2}y-n_{2}z\\
Z &amp; =l_{3}x+m_{3}y-n_{3}z
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} &amp; =0\\
\\
l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1\\
l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0\\
n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These equations contain several errors or misprints: &lt;math&gt;Z^{2}-Y^{2}-X^{2}&lt;/math&gt; has to be replaced by &lt;math&gt;X^{2}-Y^{2}-Z^{2}&lt;/math&gt;, and &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1
\end{align}
}&lt;/math&gt; replaced with &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1
\end{align}
}&lt;/math&gt;, and by reversing the sign of &lt;math&gt;Z&lt;/math&gt; in the inverse transformation, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

Cox (1881/82) also gave the Weierstrass coordinates and their transformation in hyperbolic space:&lt;ref group=M&gt;Cox (1881/82), pp. 199, 206–207&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\
w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\
\hline \begin{align}x &amp; =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\
y &amp; =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\
z &amp; =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\
w &amp; =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\
\\
x' &amp; =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\
y' &amp; =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\
z' &amp; =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\
w' &amp; =l_{4}x+m_{4}y+n_{4}z-r_{4}w
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} &amp; =1\\
l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} &amp; =1\\
l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} &amp; =0\\
l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} &amp; =0\\
l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} &amp; =0\\
l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By replacing &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}&lt;/math&gt; with &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}&lt;/math&gt; this represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w'&lt;/math&gt; everywhere.}}

In 1883 he formulated relations between [[w:orthogonal circles]] which he identified with the previously (1881/82) given transformations:&lt;ref group=M&gt;Cox (1883), pp. 109ff&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}-w^{2}=0\\
\hline \begin{align}x &amp; =\lambda_{1}X+\lambda_{2}Y+\lambda_{3}Z+\lambda_{4}W\\
y &amp; =\mu_{1}X+\mu_{2}Y+\mu_{3}Z+\mu_{4}W\\
z &amp; =\nu_{1}X+\nu_{2}Y+\nu_{3}Z+\nu_{4}W\\
-w &amp; =\rho_{1}X+\rho_{2}Y+\rho_{3}Z+\rho_{4}W\\
\\
X &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z+\rho_{1}w\\
Y &amp; =\lambda_{2}x+\mu_{2}y+\nu_{2}z+\rho_{2}w\\
Z &amp; =\lambda_{3}x+\mu_{3}y+\nu_{3}z+\rho_{3}w\\
-W &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z+\rho_{4}w
\end{align}
\left|{\scriptstyle \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{2}\lambda_{3}+\mu_{2}\mu_{3}+\nu_{2}\nu_{3}-\rho_{2}\rho_{3} &amp; =0\\
\lambda_{3}\lambda_{1}+\mu_{3}\mu_{1}+\nu_{3}\nu_{1}-\rho_{3}\rho_{1} &amp; =0\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0\\
\lambda_{1}\lambda_{4}+\mu_{1}\mu_{4}+\nu_{1}\nu_{4}-\rho_{1}\rho_{4} &amp; =0\\
\lambda_{2}\lambda_{4}+\mu_{2}\mu_{4}+\nu_{2}\nu_{4}-\rho_{2}\rho_{4} &amp; =0\\
\lambda_{3}\lambda_{4}+\mu_{3}\mu_{4}+\nu_{3}\nu_{4}-\rho_{3}\rho_{4} &amp; =0
\end{align}
}\right.{\scriptstyle \begin{align}\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}-\mu_{4}^{2} &amp; =1\\
\nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2}-\nu_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0\\
\lambda_{1}\nu_{1}+\lambda_{2}\nu_{2}+\lambda_{3}\nu_{3}-\lambda_{4}\nu_{4} &amp; =0\\
\lambda_{1}\rho_{1}+\lambda_{2}\rho_{2}+\lambda_{3}\rho_{3}-\lambda_{4}\rho_{4} &amp; =0\\
\mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}-\mu_{4}\nu_{4} &amp; =0\\
\mu_{1}\rho_{1}+\mu_{2}\rho_{2}+\mu_{3}\rho_{3}-\mu_{4}\rho_{4} &amp; =0\\
\nu_{1}\rho_{1}+\nu_{2}\rho_{2}+\nu_{3}\rho_{3}-\nu_{4}\rho_{4} &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The relations between &lt;math&gt;\lambda,\mu,\nu,\rho&lt;/math&gt; are correct, even though the transformation still represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w&lt;/math&gt; everywhere.}}

Finally, in a treatise on [[w:Hermann Grassmann|w:Grassmann's Ausdehnungslehre]] and circles (1891), he again provided transformations of orthogonal circle systems described by him as being &quot;identical with those for transformation of coordinates in non-Euclidean geometry&quot;:&lt;ref group=M&gt;Cox (1891), pp. 27-28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}=w^{2}\\
\hline \begin{align}x &amp; =\lambda_{1}x'+\lambda_{2}y'+\lambda_{3}z'+\lambda_{4}w' &amp; \text{(4 equations)}\\
x' &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z-\rho_{1}w\\
-w' &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z-\rho_{4}w
\end{align}
\\
\hline \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0 &amp; \text{(6 equations)}\\
\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0 &amp; \text{(6 equations)}
\end{align}
\end{matrix}\text{ }&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)''.}}

=== {{anchor|Hill}} Hill (1882) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations&lt;ref group=M&gt;Hill (1882), pp. 323–325&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\
k\left(\sin^{2}E+\cos^{2}E-1\right)\\
\hline \begin{align} &amp;  &amp; \cos E' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
 &amp; \mathbf{(1)} &amp; \sin E' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\hline \\
 &amp;  &amp; x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
 &amp;  &amp; y &amp; =\beta u+\beta'u'+\beta''u''\\
 &amp;  &amp; z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
 &amp; \mathbf{(2)}\\
 &amp;  &amp; u &amp; =-\alpha x-\beta y+\gamma z\\
 &amp;  &amp; u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
 &amp;  &amp; u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =1\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
(k=-1)\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-k\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .}}

=== {{anchor|Picard}} Picard (1882-1884) – Quadratic forms ===

[[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to &quot;hyperfuchsian&quot; functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:&lt;ref group=M&gt;Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 280-281&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\
\\
\mathbf{(1)}\ \begin{align}x &amp; =M_{1}X+P_{1}Y+R_{1}Z\\
y &amp; =M_{2}X+P_{2}Y+R_{2}Z\\
z &amp; =M_{3}X+P_{3}Y+R_{3}Z
\end{align}
\\
\\
\left[\begin{align}[][x,y,z]=\text{complex}\\
\left[x_{0},y_{0},z_{0}\right]=\text{conjugate}
\end{align}
\right]\\
\\
\hline \\
x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\
x=x'+ix'',\quad y=y'+iy''\\
\\
\mathbf{(2)}\ \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} &amp; =1\\
P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0\\
P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
\\
M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} &amp; =1\\
M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} &amp; =1\\
M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} &amp; =-1\\
\mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} &amp; =0\\
\mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} &amp; =0\\
\mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} &amp; =0\\
\\
\left[\begin{align}[][M,P,R\dots]=\text{complex}\\
\left[\mu,\pi,\rho\dots\right]=\text{conjugate}
\end{align}
\right]
\end{align}
}\right.\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;-Z&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:&lt;ref group=M&gt;Picard (1884a), p. 13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\
\hline \begin{align}U &amp; =\mathcal{A}u+\mathcal{B}v\\
V &amp; =\mathcal{C}u+\mathcal{D}v
\end{align}
\left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1\\
\mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} &amp; =0\\
\mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} &amp; =-1\\
\mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;-V&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;-v&lt;sup&gt;2&lt;/sup&gt;''.}}

Or in (1884b):&lt;ref group=M&gt;Picard (1884b), p. 416&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xx_{0}+yy_{0}-1=0\\
\hline \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884c):&lt;ref group=M&gt;Picard (1884c), pp. 123–124; 163&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\
\hline \mathbf{(1)}\ \begin{align}U &amp; =Mu+Pv+Rw\\
V &amp; =M'u+P'v+R'w\\
W &amp; =M''u+P''v+R''w\\
\\
u &amp; =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\
v &amp; =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\
w &amp; =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W
\end{align}
\left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} &amp; =1\\
PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} &amp; =1\\
RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} &amp; =0\\
MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} &amp; =0\\
PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} &amp; =0\\
\\
MM_{0}+PP_{0}-RR_{0} &amp; =1\\
M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} &amp; =1\\
M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
M_{0}M'+P_{0}P'-R_{0}R' &amp; =0\\
M_{0}M''+P_{0}P''-R_{0}R'' &amp; =0\\
M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' &amp; =0
\end{align}
}\right.\\
\hline \text{Invariance of unit hypersphere:}\\
\mathbf{(2)}\ \begin{align}\xi' &amp; =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\
\eta' &amp; =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''}
\end{align}
\left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} &amp; =1\\
BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} &amp; =1\\
CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} &amp; =-1\\
AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} &amp; =0\\
AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} &amp; =0\\
BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;+V&lt;sup&gt;2&lt;/sup&gt;-W&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;+v&lt;sup&gt;2&lt;/sup&gt;-w&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing &lt;math&gt;\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1&lt;/math&gt;.}}

=== {{anchor|Callandreau}} Callandreau (1885) – Homography ===

Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations&lt;ref group=M&gt;Callandreau (1885), pp. A.7; A.12&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\
{\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\
\hline \begin{align}\cos\varepsilon' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\sin\varepsilon' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}
\end{align}
\left|{\scriptstyle \begin{align} &amp; \left(k=1\right)\\
\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-k &amp; \alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =+k &amp; \alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =+k &amp; \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-1 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-1 &amp; \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =+1 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}})  ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.}}

==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Lie|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions.&lt;ref group=M&gt;Lie (1885/86), p. 411&lt;/ref&gt; In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):&lt;ref group=M&gt;Werner (1889), pp. 4, 28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\
\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\
\text{where}\\
\left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\
\left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\
\left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right]
\end{matrix}&lt;/math&gt;

More generally, Lie (1890)&lt;ref group=M&gt;Lie (1890a), p. 295;&lt;/ref&gt; defined non-Euclidean motions in terms of two forms &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0&lt;/math&gt; in which the imaginary form with &lt;math&gt;+1&lt;/math&gt; denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:&lt;ref group=M&gt;Lie (1890a), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\
\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n)
\end{matrix}&lt;/math&gt;

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

:&lt;math&gt;x^{2}+y^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 474&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 479&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt;.&lt;ref group=M&gt;Lie (1893), p. 481&lt;/ref&gt;

{{Lorentzbox|Text=The group of hyperbolic motions is isomorphic to the Lorentz group. The interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt; becomes the Lorentz interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0&lt;/math&gt; by setting
&lt;math&gt;(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)&lt;/math&gt;}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Gerard|label 1=History of Lorentz transformations via hyperbolic functions § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':&lt;ref group=M&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\
X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\
\hline \begin{align}X &amp; =aX'+a'Y'+a''Z'\\
Y &amp; =bX'+b'Y'+b''Z'\\
Z &amp; =cX'+c'Y'+c''Z'\\
\\
X' &amp; =aX+bY-cZ\\
Y' &amp; =a'X+b'Y-c'Z\\
Z' &amp; =-a''X-b''Y+c''Z
\end{align}
\left|\begin{align}a^{2}+b^{2}-c^{2} &amp; =1\\
a^{\prime2}+b^{\prime2}-c^{\prime2} &amp; =1\\
a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} &amp; =-1\\
aa'+bb'-cc' &amp; =0\\
a'a''+b'b''-c'c'' &amp; =0\\
a''a+b''b-c''c &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

He gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates===

[[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:&lt;ref group=M&gt;Hausdorff (1899), p. 165, pp. 181-182&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{2}-x^{2}-y^{2}=1\\
\hline \begin{align}x &amp; =a_{1}x'+a_{2}y'+x_{0}p'\\
y &amp; =b_{1}x'+b{}_{2}y'+y_{0}p'\\
p &amp; =e_{1}x'+e_{2}y'+p_{0}p'\\
\\
x' &amp; =a_{1}x+b_{1}y-e_{1}p\\
y' &amp; =a_{2}x+b_{2}y-e_{2}p\\
-p' &amp; =x_{0}x+y_{0}y-p_{0}p
\end{align}
\left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} &amp; =1\\
a_{2}^{2}+b_{2}^{2}-e_{2}^{2} &amp; =1\\
-x_{0}^{2}-y_{0}^{2}+p_{0}^{2} &amp; =1\\
a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} &amp; =0\\
a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} &amp; =0\\
a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} &amp; =0\\
\\
a_{1}^{2}+a_{2}^{2}-x_{0}^{2} &amp; =1\\
b_{1}^{2}+b_{2}^{2}-y_{0}^{2} &amp; =1\\
-e_{1}^{2}-e_{2}^{2}+p_{0}^{2} &amp; =1\\
b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} &amp; =0\\
a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} &amp; =0\\
a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Woods2|label 1=History of Lorentz transformations via hyperbolic functions § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting &lt;math&gt;k=\sqrt{-1}R&lt;/math&gt; with ''R'' as real quantity):&lt;ref group=M&gt;Woods (1901/02), p. 98, 104&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\
\hline \begin{align}u' &amp; =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
v' &amp; =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
w' &amp; =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}
\end{align}
\left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} &amp; =k^{2}\\
(i=1,2,3)\\
k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} &amp; =1\\
\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} &amp; =0\\
(i,h=1,2,3,4;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with &lt;math&gt;k=\sqrt{-1}k&lt;/math&gt; for hyperbolic space):&lt;ref group=M&gt;Woods (1903/05), pp. 45–46; p. 48)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\
ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\
\hline \begin{align}x_{1}^{\prime} &amp; =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\
x_{2}^{\prime} &amp; =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\
x_{3}^{\prime} &amp; =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\
x_{0}^{\prime} &amp; =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0}
\end{align}
\left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) &amp; =1\\
\delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) &amp; =k^{2}\\
(i=1,2,3)\\
\delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) &amp; =0\\
(i,h=0,1,2,3;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Liebmann|label 1=History of Lorentz transformations via hyperbolic functions § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''&lt;ref group=M&gt;Liebmann (1904/05), p. 168; pp. 175–176&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\
\hline \begin{align}x_{1} &amp; =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\
y_{1} &amp; =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\
x_{1} &amp; =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\
\\
x &amp; =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\
y &amp; =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\
p &amp; =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1}
\end{align}
\left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} &amp; =1\\
-\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} &amp; =1\\
-\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} &amp; =1\\
\alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} &amp; =0\\
\alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} &amp; =0\\
\alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|apo1}}
*{{#section:History of Topics in Special Relativity/mathsource|apo2}}
*{{#section:History of Topics in Special Relativity/mathsource|apo}}
*{{#section:History of Topics in Special Relativity/mathsource|bour56att}} 
*{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|chas29}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox83hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox91}} 
*{{#section:History of Topics in Special Relativity/mathsource|fris76}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau18}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|haus99}} 
*{{#section:History of Topics in Special Relativity/mathsource|hill82}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac27}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32a}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32b}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac33}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil77}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil79}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil84}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil85}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei73}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|hire1}} 
*{{#section:History of Topics in Special Relativity/mathsource|leb37}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie85}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie93}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lop}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic82}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84a}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84b}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84c}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin87}} 
*{{#section:History of Topics in Special Relativity/mathsource|sal62}} 
*{{#section:History of Topics in Special Relativity/mathsource|vinc}} 
*{{#section:History of Topics in Special Relativity/mathsource|som63}} 
*{{#section:History of Topics in Special Relativity/mathsource|wedd47}} 
*{{#section:History of Topics in Special Relativity/mathsource|wern89}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L1}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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==Most general Lorentz transformations==
===General quadratic form===
The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as&lt;ref&gt;Bôcher (1907), chapter X&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}\begin{align}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align}
=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\
b=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y}=b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}'\\
\left(\mathbf{A}=\mathbf{A}^{{\rm T}}\right)\\
\hline \left.\begin{align}\mathbf{x}' &amp; =\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\mathbf{g}^{-1}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q1}}}}

The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc.

===Most general Lorentz transformation===
{{CSS image crop
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|Description=The Lorentz interval is the invariant relation between axes and conjugate diameters of hyperbolas, illustrating Lorentz transformations between two inertial frames.}}
The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:&lt;ref name=ratcliffe&gt;Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1&lt;/ref&gt;&lt;ref&gt;Naimark (1964), 2 in four dimensions&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} &amp; =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\
-x_{0}y_{0}+\cdots+x_{n}y_{n} &amp; =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\downarrow\\
\begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\
 &amp; \dots\\
x_{n}^{\prime} &amp; =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn}
\end{align}
\\
\\
\mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\
\downarrow\\
\begin{align}x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\
 &amp; \dots\\
x_{n} &amp; =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn}
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} &amp; =\mathbf{g}^{-1}\\
\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} &amp; =\mathbf{A}\\
\mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} &amp; =\mathbf{A}\\
\\
\end{align}
\\
\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\end{matrix}
\end{matrix}&lt;/math&gt;|{{equationRef|1a}}}}

The invariance of the Lorentz interval with ''n''=1 between axes and [[w:conjugate diameters]] of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.&lt;ref&gt;Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.&lt;/ref&gt;&lt;ref&gt;Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.&lt;/ref&gt; They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881-91)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the &quot;absolute&quot; form &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=0&lt;/math&gt; as discussed by [[#Klein|Klein (1871-73)]].&lt;ref group=M&gt;Killing (1885), p. 71&lt;/ref&gt;&lt;ref&gt;Müller (1910), p. 661, in particular footnote 247.&lt;/ref&gt;&lt;ref&gt;Sommerville (1911), p. 286, section K6.&lt;/ref&gt; In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates &lt;math&gt;-x_{0}^{2}+\cdots+x_{n}^{2}=-1&lt;/math&gt; by [[#Killing3|Killing (1888-1897)]].

===Most general Lorentz transformation of velocity===

If &lt;math&gt;x_{i},\ x_{i}^{\prime}&lt;/math&gt; in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates &lt;math&gt;u_{s},\ u_{s}^{\prime}&lt;/math&gt; follow by

:&lt;math&gt;\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)&lt;/math&gt;

defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1&lt;/math&gt; so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called &quot;the most general formula for the composition of velocities&quot; in terms of special relativity&lt;ref&gt;Synge (1955), p. 129 for ''n''=3&lt;/ref&gt; (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})):

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align}u_{s}^{\prime} &amp; =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\
\\
u_{s} &amp; =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}}
\end{align}
\left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} &amp; =\left\{ \begin{align}-1\quad &amp; (j=k=0)\\
1\quad &amp; (j=k&gt;0)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} &amp; =\left\{ \begin{align}-1\quad &amp; (i=k=0)\\
1\quad &amp; (i=k&gt;0)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.&lt;/math&gt;|{{equationRef|1b}}}}

Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere &lt;math&gt;-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0&lt;/math&gt; were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]].

==Historical notation==
==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Apo|label 1=History of Lorentz transformations via squeeze mappings § Apollonius}}
====Equality of difference in squares====
[[File:Apollonius-Borelli-XII.png|thumb|&lt;small&gt;Fig. 1: Apollonius' proposition illustrated by Borelli (1661) of &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Apollonius of Perga]] (c. 240–190 BC) in his 7th book on conics defined the following well known proposition (the 7th book survived in Arabian translation, and was translated into Latin in 1661 and 1710), as follows:

*The difference of the squares of the two axes of the hyperbola is equal to the difference of the squares of any two conjugate diameters. &lt;small&gt;(Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].)&lt;ref group=M name=bor1&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XII and other props. from book VII on pp. 291-292; See also the note on prop. XII on pp. 293-294, where Borelli demontrates &lt;math&gt;\scriptstyle \overline{AC}^{2}-\overline{QR}^{2}=\overline{IL}^{2}-\overline{NO}^{2}&lt;/math&gt; (in later translations such as Halley (1710), the proposition was numbered as XIII.) Latin: &quot;Differentia quadratorum duorum axium hyperboles æqualis est differentiæ quadratorum quarumlibet duarum diametrorum coniugatarum.&quot;&lt;/ref&gt;&lt;/small&gt;

*In every hyperbola the difference between the squares of the axes is equal to the difference between the squares of any conjugate diameters of the section. &lt;small&gt;(Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XIII of book VII on p. 107; Latin: &quot;In omni Hyperbola differentia inter quadrata Axium aequalis est differentiae inter quadrata ex diametris quibusvis conjugatis sectionis.&quot;&lt;/ref&gt;&lt;/small&gt;

*[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).&lt;/ref&gt;&lt;/small&gt;
----
[[File:Lahire-XLII-XLIII.png|thumb|left|&lt;small&gt;Fig. 2: La Hire's (1685) illustration of &lt;math&gt;\scriptstyle \overline{AB}^{2}-\overline{DE}^{2}=\overline{NM}^{2}-\overline{LK}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[File:Lhopital Conjugate Diameters.png|thumb|&lt;small&gt;Fig. 3: l'Hôpital's (1707) illustration of &lt;math&gt;\scriptstyle \overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685) stated this proposition as follows:
{{Block indent|1=I say that the difference of the squares of any two diameters conjugated to each other, AB, DE, is equal to the difference of the squares of any two other diameters conjugated to each other, NM, LK.&lt;ref group=M name=lahire1&gt;La Hire (1685), Book IV, Proposition XLII, p. 85; Latin: &quot;Dico differentiam quadratorum duarum diametrorum quarumlibet inter se conjugatarum AB, DE esse æqualem differentiæ quadratorum duarum aliarum diametrorum quarumlibet inter se conjugatarum, NM, LK.&quot;&lt;/ref&gt;}}
and also summarized the related propositions in the 7th book of Apollonius:
{{Block indent|1=In a hyperbola, the difference of the squares of the axes is equal to the difference of the squares of any two conjugate diameters.&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of propositions XII, XIII, XXV in the 7th book of Apollonius; Latin: &quot;In hyperbola differentia quadratorum axium æqualis est differentia quadratorum duarum diametrorum conjugatarum quarumlibet.&quot;&lt;/ref&gt;}}
----
[[w:Guillaume de l'Hôpital]] (1707), using the methods of [[wikipedia:Analytic_geometry|w:analytic geometry]], demonstrated the same proposition:&lt;ref group=M name=lop&gt;l'Hôpital (1707), Third book, Prop. XII, p. 76.&lt;/ref&gt;

{{Block indent|1=The difference of the squares of any two conjugate diameters &quot;Mm, Ss&quot; is equal to the difference of the squares of the two axes &quot;Aa, Bb.&quot; We are to prove that &lt;math&gt;\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}&lt;/math&gt;, or &lt;math&gt;\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}&lt;/math&gt;. &lt;small&gt;(English translation 1723 by [[w:Edmund Stone]].)&lt;ref group=M&gt;l'Hôpital/Stone (1723), pp. 62-63&lt;/ref&gt;&lt;/small&gt;}}

{{Lorentzbox|Text=Apollonius' proposition can be expressed as &lt;math&gt;-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation ({{equationNote|1a}}) &quot;(n=1)&quot; can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.}}

====Equality of areas of parallelograms====

[[File:Apollonius-Borelli-XXXI.png|thumb|&lt;small&gt;Fig. 4: Apollonius' proposition illustrated by Borelli (1661) of the equality of areas of parallelogram ABCD (of the axes) and KLMN (of the conjugated diameters).&lt;/small&gt;]] 

Apollonius also gave another well known proposition in his 7th book regarding ellipses as well as conjugate sections of hyperbolas (see also Del Centina &amp; Fiocca&lt;ref&gt;Del Centina &amp; Fiocca (2020)&lt;/ref&gt; for further details on the history of this proposition): 

*In the ellipse, and in conjugate sections [the opposite branches of two conjugate hyperbolas] the parallelogram bounded by the axes is equal to the parallelogram bounded by any pair of conjugate diameters, if its angles are equal to the angles the conjugate diameters form at the centre. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del&gt;Del Centina &amp; Fiocca (2020), section 3.1&lt;/ref&gt; based on the Latin translation 1661 by [[w:Giovanni Alfonso Borelli]] and [[w:Abraham Ecchellensis]].&lt;ref group=M name=bor2&gt;Apollonius/Borelli/Ecchellensis (1661), Summary of prop. XXXI of book VII on p. 370; Note on pp. 372-374; Latin: &quot;In ellypsi, &amp; sectionibus coniugatis parallelogrammum sub axibus contentum æquale est parallelogrammo à quibuscunque duabus coniugatis diametris comprehenso, si eorum anguli æquales fuerint angulis ad centrum contentis à coniugatis diametris.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If two conjugate diameters are taken in an ellipse, or in the opposite conjugate sections; the parallelogram bounded by them is equal to the rectangle bounded by the axes, provided its angles are equal to those formed at the centre by the conjugate diameters. &lt;small&gt;(English translation by Del Centina &amp; Fiocca&lt;ref name=del /&gt; based on the Latin translation 1710 by [[w:Edmond Halley]].)&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XXXI of book VII on p. 115–117; Latin: &quot;Si ducantur diametri quævis conjugate in Ellipsi, vel inter sectiones oppositas conjugatas; erit parallelogrammum contentam sub his diametris æquale rectangulo sub ipsis Axibus facto: modo anguli ejus æquales sint angulis ad centrum sectionis à diametris conjugatis comprehensis.&quot;&lt;/ref&gt;)&lt;/small&gt;

*If PP', DD' be two conjugate diameters in an ellipse or in conjugate hyperbolas, and if tangents be drawn at the four extremities forming a parallelogram LL'MM', then the parallelogram LL'MM' = rect. AA'·BB'. &lt;small&gt;(English translation 1896 by [[w:Thomas Heath (classicist)|w:Thomas Heath]].)&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 136, p. 235; (Apollonius, Book VII, Prop. 31).&lt;/ref&gt;&lt;/small&gt; 

{{Lorentzbox|Text=The graphical representation of Apollonius proposition in Borelli's Fig. 4 is essentially a [[w:Minkowski diagram]], being a graphical representation of the Lorentz transformation. If line AB is the x-axis of an inertial frame S1, then line FG is the x-axis of another inertial frames S2 which together with its parallel lines (such as KL and NM) represent [[w:relativity of simultaneity]]. Analogously, if line CD is the time axis of another inertial frame S2, then line HI is the time axis of S2 which together with its parallel lines (such as KN and LM) represent the [[w:worldlines]] of objects at different locations. The diagonals KE (or KM) and LE (or LN) lie on the asymptotes which form a light cone. Thus the totality of all parallelograms of equal area and conjugate diameters as constructed by Apollonius, represents the totality of all inertial frames, lines of simultaneity and worldlines within a spacetime area bounded by &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=\rm{const}&lt;/math&gt;.}}

[[File:Saint-Vincent-Hyperbola-VI-XLIX.png|thumb|175px|left|&lt;small&gt;Fig. 5: Saint-Vincent's (1647) illustration of FGHI=OPQR, as well as BADC=KNLM.&lt;/small&gt;]]
[[w:Grégoire de Saint-Vincent]] independently (1647) stated the same proposition:&lt;ref group=M name=vinc&gt;St. Vincent (1647), Book VI, Prop. XLIX, p. 560; Latin: “Si fuerint binæ hyperbolarum coniugaciones A, B, C, D: ponantur autem per E centrum duæ quoque diametrorum coniugationes per quarum vertices contingentes actæ constituant duo quadrilatera FGHI, OPQR. Dico illa esse æqualia inter se.”&lt;/ref&gt;

{{Block indent|1=The parallelograms whose opposite sides are tangent to two conjugate hyperbolas at the extremities of two conjugate diameters are equivalent among them. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref&gt;Del Centina &amp; Fiocca (2020), section 5.1&lt;/ref&gt;)&lt;/small&gt; }}

---- 

[[File:Lahire-XLII-XLIII.png|thumb|&lt;small&gt;Fig. 6 (identical to Fig. 2): La Hire's (1685) illustration of FGHI=OPQR.&lt;/small&gt;]]
[[w:Philippe de La Hire]] (1685), who was aware of both Apollonius 7th book and Saint-Vincent's book, stated this proposition as follows:&lt;ref group=M name=lahire&gt;La Hire (1685), Book IV, Proposition XLIII, pp. 85-86; Latin: &quot;In sectionibus conjugatis NA, DL, BM, KE si circumscribatur parallelogrammum FGHI à rectis parallelis duabus diametris inter se conjugatis ED, BA, &amp; per ipsorum terminos ductis, &amp; simili methodo circumscribatur aliud parallelogrammum OPQR à rectis ductis per terminos diametrorum conjugatarum, &amp; ipsis parallelis: Dico parallelogramma FGHI, OPQR esse inter se æqualia.&quot;&lt;/ref&gt; 

{{Block indent|1=If a parallelogram FGHI is circumscribed about conjugate sections NA, DL, BM, KE whose sides are parallel to two conjugate diameters ED, BA drawn through their extremities, and with similar method another parallelogram OPQR is drawn through the extremities of other two conjugate diameters, then the parallelograms FGHI, OPQR are equal. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2&gt;Del Centina &amp; Fiocca (2020), section 5.2&lt;/ref&gt;)&lt;/small&gt;}}

and also summarized the related propositions in the 7th book of Apollonius:&lt;ref group=M&gt;La Hire (1685), p. 242. Summary of proposition XXXI in the 7th book of Apollonius; Latin: &quot;In sectionibus conjugatis &amp; Ellipsi parallelogrammum sub axibus æquale est paralelogrammo sub duabus quibuscunque diametris inter se conjugatis, in angulis ipsarum diametrorum conjugatarum.&quot;&lt;/ref&gt;

{{Block indent|1=In conjugate sections and in the ellipse, the parallelogram constructed with the axes, is equal to the parallelogram constructed with any two conjugated diameters, provided the angles are equal to those between the diameters themselves. &lt;small&gt;(English translation by Del Centina &amp; Fiocca.&lt;ref name=del2 /&gt;)&lt;/small&gt;}}

{{Lorentzbox|Text=In Saint-Vincent's Fig. 5 or La Hire's Fig. 6, parallelogram FGHI contains all coordinates related to an inertial frame S3, in particular triangles EGH, EFI (Fig. 5) or CFG, CHI (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EHI, EGF (Fig. 5) or CFI, CGH (Fig. 6) contain space like intervals between events on the negative and positive x-axis. Conversely, parallelogram OPQR contains all coordinates related to another frame S4, in particular triangles EQR, EOP (Fig. 5) or CPQ, COR (Fig. 6) contain time like intervals between events on the future and past light cones, while triangles EPR, EOQ (Fig. 5) or COP, CQR (Fig. 6) contain space like intervals between events on the negative and positive x-axis.}}

===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.}}

==={{anchor|Gauss}} Gauss (1798–1818)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Gauss|label 1=History of Lorentz transformations via Möbius transformations § Gauss}}
===={{anchor|Gauss1}} Binary quadratic forms====

The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

{{Lorentzbox|Text=The Lorentz interval &lt;math&gt;-x_{0}^{2}+x_{1}^{2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.}}

===={{anchor|Gauss2}} Ternary quadratic forms====

Gauss (1798/1801)&lt;ref group=M&gt;Gauss (1798/1801), articles 266–285&lt;/ref&gt; also discussed ternary quadratic forms with the general expression

:&lt;math&gt;\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\\
g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)\\
\hline \begin{align}x &amp; =\alpha y+\beta y'+\gamma y''\\
x' &amp; =\alpha'y+\beta'y'+\gamma'y''\\
x'' &amp; =\alpha''y+\beta''y'+\gamma''y''
\end{align}
\end{matrix}&lt;/math&gt;

which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;'', or indefinite in the case of different signs such as ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:&lt;ref group=M&gt;Gauss (1798/1801), article 277&lt;/ref&gt;

:&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\\
\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right),\ \left(\begin{matrix}-1, &amp; -1, &amp; 1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval &lt;math&gt;x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt; and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
&lt;math&gt;\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}m, &amp; m', &amp; m''\\
n, &amp; n', &amp; n''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt;}}

===={{anchor|Gauss4}} Homogeneous coordinates====

Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression

:&lt;math&gt;(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E&lt;/math&gt;

into

:&lt;math&gt;G+G'\cos T^{2}+G''\sin T^{2}&lt;/math&gt;

in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:&lt;ref group=M&gt;Gauss (1818), pp. 5–10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\
k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\
\hline \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =k &amp; \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =-k\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-k &amp; \beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-k\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-k &amp; \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =+k\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}\right.\\
\hline k=1\\
\begin{align}t\cos E &amp; =\alpha+\alpha'\cos T+\alpha''\sin T\\
t\sin E &amp; =\beta+\beta'\cos T+\beta''\sin T\\
t &amp; =\gamma+\gamma'\cos T+\gamma''\sin T
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}})  ''(n=2)''.

Further setting &lt;math&gt;[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}})  ''(n=2)''.}}

Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that

:&lt;math&gt;aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz&lt;/math&gt;

can be transformed into

:&lt;math&gt;Guu+G'u'u'+G''u''u''&lt;/math&gt;,

in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:&lt;ref group=M&gt;Gauss (1818), pp. 9–10&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =-\alpha x-\beta y+\gamma z\\
u' &amp; =\alpha'x+\beta'y-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma &amp; =1\\
-\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' &amp; =-1\\
-\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' &amp; =-1\\
-\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' &amp; =0\\
-\alpha''\alpha-\beta''\beta+\gamma''\gamma &amp; =0\\
-\alpha\alpha'-\beta\beta'+\gamma\gamma' &amp; =0
\end{align}
}\right.&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}})  ''(n=2)'' satisfying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;, and can be related to Gauss' previous equations in terms of homogeneous coordinates &lt;math&gt;\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]&lt;/math&gt;.}}

==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:&lt;ref group=M&gt;Jacobi (1827), p. 235, 239–240&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\
k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\
\hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P &amp; =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\cos\vartheta &amp; =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\sin P\sin\vartheta &amp; =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\
\\
\cos\psi &amp; =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\cos\varphi &amp; =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\
\sin\psi\sin\varphi &amp; =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}
\end{align}
\\
\\
\hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z &amp; =m\\
\alpha'\mu+\beta'x+\gamma'y+\delta'z &amp; =m'\\
\alpha''\mu+\beta''x+\gamma''y+\delta''z &amp; =m''\\
\alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z &amp; =m'''\\
\\
Am+A'm'+A''m''+A'''m''' &amp; =\mu\\
Bm+B'm'+B''m''+B'''m''' &amp; =x\\
Cm+C'm'+C''m''+C'''m''' &amp; =y\\
Dm+D'm'+D''m''+D'''m''' &amp; =z\\
\\
\end{align}
\\
\begin{align}\alpha &amp; =-kA, &amp; \beta &amp; =-kB, &amp; \gamma &amp; =-kC, &amp; \delta &amp; =kD,\\
\alpha' &amp; =kA', &amp; \beta' &amp; =kB', &amp; \gamma' &amp; =kC', &amp; \delta' &amp; =-kD',\\
\alpha'' &amp; =kA'', &amp; \beta'' &amp; =kB'', &amp; \gamma'' &amp; =kC'', &amp; \delta'' &amp; =-kD'',\\
\alpha''' &amp; =kA''', &amp; \beta''' &amp; =kB''', &amp; \gamma''' &amp; =kC''', &amp; \delta''' &amp; =-kD''',
\end{align}
\end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta &amp; =-k\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' &amp; =k\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' &amp; =k\\
\alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' &amp; =k\\
\alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' &amp; =0\\
\alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' &amp; =0\\
\alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' &amp; =0\\
\alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' &amp; =0\\
\alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' &amp; =0\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' &amp; =0\\
\\
-\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' &amp; =k\\
-\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' &amp; =k\\
-\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' &amp; =k\\
-\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' &amp; =-k\\
-\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' &amp; =0\\
-\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' &amp; =0\\
-\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' &amp; =0\\
-\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' &amp; =0\\
-\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' &amp; =0\\
-\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' &amp; =0
\end{align}
\end{matrix}}
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] &amp; =\left[u_{1},\ u_{2},\ u_{3}\right]\\{}
[\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] &amp; =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right]
\end{align}
}&lt;/math&gt; and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing &lt;math&gt;m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}&lt;/math&gt;.}}

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):&lt;ref group=M&gt;The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\
\mathbf{(1)}\begin{align}x &amp; =\alpha s+\alpha's'+\alpha''s''\\
y &amp; =\beta s+\beta's'+\beta''s''\\
z &amp; =\gamma s+\gamma's'+\gamma''s''\\
\\
s &amp; =\alpha x+\beta y+\gamma z\\
s' &amp; =\alpha'x+\beta'y+\gamma'z\\
s'' &amp; =\alpha''x+\beta''y+\gamma''z
\end{align}
\left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma &amp; =1 &amp; \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' &amp; =1\\
\alpha'\alpha'+\beta'\beta'+\gamma'\gamma' &amp; =1 &amp; \beta\beta+\beta'\beta'+\beta''\beta'' &amp; =1\\
\alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' &amp; =1 &amp; \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' &amp; =1\\
\alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' &amp; =0 &amp; \beta\gamma+\beta'\gamma'+\beta''\gamma'' &amp; =0\\
\alpha''\alpha+\beta''\beta+\gamma''\gamma &amp; =0 &amp; \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' &amp; =0\\
\alpha\alpha'+\beta\beta'+\gamma\gamma' &amp; =0 &amp; \alpha\beta+\alpha'\beta'+\alpha''\beta'' &amp; =0
\end{align}
\right.\\
\hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\
\left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\
\hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\
\left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\
\hline \begin{align}\cos\phi &amp; =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \cos\eta &amp; =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\
\sin\phi &amp; =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, &amp; \sin\eta &amp; =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}
\end{align}
\end{matrix}\\
\hline \\
\mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\
\hline \begin{align}y &amp; =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, &amp; s' &amp; =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\
z &amp; =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, &amp; s'' &amp; =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z},
\end{align}
\end{matrix}
\end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma &amp; =1\\
\alpha'\alpha'-\beta'\beta'-\gamma'\gamma' &amp; =-1\\
\alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' &amp; =-1\\
\alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\alpha''\alpha-\beta''\beta-\gamma''\gamma &amp; =0\\
\alpha\alpha'-\beta\beta'-\gamma\gamma' &amp; =0\\
\\
\alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' &amp; =1\\
\beta\beta-\beta'\beta'-\beta''\beta'' &amp; =-1\\
\gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' &amp; =-1\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=By setting &lt;math&gt;[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]&lt;/math&gt;, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.}}

Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:&lt;ref group =M&gt;Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\
\hline \mathbf{(1)\ }\begin{align}y_{\varkappa} &amp; =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\
x_{\varkappa} &amp; =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\
\\
\frac{y_{\varkappa}}{y_{n}} &amp; =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\
\frac{x_{\varkappa}}{x_{n}} &amp; =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}
\end{align}
\left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} &amp; =0\\
\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} &amp; =1\\
\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} &amp; =0\\
\alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} &amp; =1
\end{align}
\right.\\
\hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\
\alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\
1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\
\hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} &amp; =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\
\\
\xi_{\varkappa} &amp; =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}}
\end{align}
\\
\hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.}}

He also stated the following transformation leaving invariant the Lorentz interval:&lt;ref group=M&gt;Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\
\hline {\scriptstyle \begin{align}u &amp; =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\
u_{1} &amp; =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\
 &amp; \dots\\
u_{n-1} &amp; =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\
\\
w &amp; =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\
w_{1} &amp; =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\
 &amp; \dots\\
w_{n-1} &amp; =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1}
\end{align}
\left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =-1\\
\alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} &amp; =0\\
\alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} &amp; =0\\
\\
\alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} &amp; =+1\\
\alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} &amp; =-1\\
\alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} &amp; =0\\
\alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} &amp; =0
\end{align}
\text{ }\right.}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.}}

==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids ===

[[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:&lt;ref group=M&gt;Chasles (1829), p. 139&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(a)\\ \\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(b)\\ \\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1
\end{align}
\right\}  &amp; \dots(c)\\ \\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(d)
\end{matrix}&lt;/math&gt;

He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:&lt;ref group=M&gt;Chasles (1829), p. 141&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\
\gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\
\alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\
\beta''\quad\Rightarrow\quad\beta''\sqrt{-1}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:&lt;ref group=M&gt;Chasles (1829), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}x &amp; =lx'+l'y'+l''z'\\
y &amp; =mx'+m'y'+m''z'\\
z &amp; =nx'+n'y'+n''z'
\end{align}
\\
\left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} &amp; =0\\
\frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} &amp; =0\\
\frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} &amp; =0
\end{align}
\right\} \\
\hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\
\left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Chasles defined the conditional equations of ''l,m,n'' in the same way as those of &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.}}

He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:&lt;ref group=M&gt;Chasles (1829), pp. 145-146&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =a^{2}\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =b^{2}\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(A)\\
\left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma' &amp; =0
\end{align}
\right\}  &amp; \dots(B)\\
\left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =a^{2}\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =b^{2}\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-c^{2}
\end{align}
\right\}  &amp; \dots(C)\\
\left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right\}  &amp; \dots(D)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates===

[[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation&lt;ref group=M&gt;Lebesgue (1837), pp. 338-341&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\
\hline {\scriptstyle \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\
\dots\\
y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} &amp; =1 &amp; (10)\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} &amp; =0 &amp; (11)\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} &amp; =1 &amp; (13)\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} &amp; =0 &amp; (14)
\end{align}
\right.}
\end{matrix}&lt;/math&gt;

In order to achieve the invariance of the Lorentz interval&lt;ref group=M&gt;Lebesgue (1837), pp. 353–354&lt;/ref&gt;

:&lt;math&gt;x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}&lt;/math&gt;

he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

{{Lorentzbox|Text=These instructions give Lorentz transformation ({{equationNote|1a}}) in the form:
:&lt;math&gt;{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\hline \begin{align}x_{1} &amp; =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\
x_{2} &amp; =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\
\dots\\
x_{n} &amp; =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\
\\
y_{1} &amp; =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\
y_{2} &amp; =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\
\dots\\
-y_{n} &amp; =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n}
\end{align}
\left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} &amp; =1\\
a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} &amp; =0\\
a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} &amp; =1\\
a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} &amp; =-1\\
a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} &amp; =0
\end{align}
\right.
\end{matrix}}&lt;/math&gt;}}

He went on to redefine the variables of the Lorentz interval and its transformation:&lt;ref group=M&gt;Lebesgue (1837), pp. 353–355&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\
\downarrow\\
\begin{align}x_{1} &amp; =x_{n}\cos\theta_{1}, &amp; x_{2} &amp; =x_{n}\cos\theta_{2},\dots &amp; x_{n-1} &amp; =x_{n}\cos\theta_{n-1}\\
y_{1} &amp; =y_{n}\cos\phi_{1}, &amp; y_{2} &amp; =y_{n}\cos\phi_{2},\dots &amp; y_{n-1} &amp; =y_{n}\cos\phi_{n-1}
\end{align}
\\
\downarrow\\
\cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\
\cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\
\hline \\
\cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\
(i=1,2,3\dots n)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting &lt;math&gt;[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]&lt;/math&gt; it is equivalent to Lorentz transformation ({{equationNote|1b}}).}}

==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids===
Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&amp;alpha;), from which he derived equations (&amp;beta;) and others:&lt;ref group=M&gt;Weddle (1847), p. 274&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1, &amp; l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1, &amp; l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} &amp; =0\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1, &amp; l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\alpha)\\ \\
\left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1, &amp; l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1, &amp; l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} &amp; =0\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1, &amp; m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0
\end{align}
\right\}  &amp; \dots(\beta)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).}}

Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone &lt;math&gt;\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0&lt;/math&gt;, he defined three conjugate points &lt;math&gt;(x_{1}\dots,y_{1}\dots,z_{1}\dots)&lt;/math&gt; on those two conjugate hyperboloids, related to each other in the same way as equations (&amp;alpha;, &amp;beta;) stated above:&lt;ref group=M&gt;Weddle (1847), pp. 275-276&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\
\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\
\hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} &amp; =0\\
\frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} &amp; =0\\
\frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} &amp; =0
\end{align}
\quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} &amp; =1\\
\frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} &amp; =1\\
\frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} &amp; =-1
\end{align}
\\
\begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} &amp; =a^{2}\\
y_{1}^{2}+y_{2}^{2}-y_{3}^{2} &amp; =b^{2}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2} &amp; =-c^{2}
\end{align}
\quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} &amp; =0\\
x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} &amp; =0\\
y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} &amp; =0
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text= These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.}}

==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates===

Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:&lt;ref group=M&gt;Bour (1856), pp. 61; 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\
\hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E &amp; =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\
\sin E &amp; =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}
\end{align}
\\
\hline \\
k=+1\\
t=\gamma+\gamma'\cos T+\gamma''\sin T,\\
1=u,\ \cos T=u',\ \sin T=u',\\
t=z,\ t\cos E=x,\ t\sin E=y\\
\downarrow\\
\mathbf{(2)}\begin{align}x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
y &amp; =\beta u+\beta'u'+\beta''u''\\
z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
\\
u &amp; =\gamma z-\alpha x-\beta y\\
u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} &amp; =k\\
-\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} &amp; =-k\\
-\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} &amp; =-k\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =k\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying &lt;math&gt;x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}&lt;/math&gt;. Furthermore, setting &lt;math&gt;[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt; in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.}}

=== {{anchor|Somov}} Somov (1863) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:&lt;ref group=M&gt;Somov (1863), pp. 12–14; p. 18 for differentials.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{align}\cos\phi &amp; =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\
\sin\phi &amp; =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''}
\end{align}
\right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\
\cos^{2}\psi+\cos^{2}\psi=1
\end{matrix}\\
\hline \mathbf{(1)}\ \begin{align}\cos\phi &amp; =x, &amp; \cos\psi &amp; =x'\\
\sin\phi &amp; =y, &amp; \sin\psi &amp; =y'
\end{align}
\ \left|\begin{align}x &amp; =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\
y &amp; =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=1\\
x^{\prime2}+y^{\prime2}=1
\end{matrix}\\
\hline \begin{align}\cos\phi &amp; =\frac{x}{z}, &amp; \cos\psi &amp; =\frac{x'}{z'}\\
\sin\phi &amp; =\frac{y}{z}, &amp; \sin\psi &amp; =\frac{y'}{z'}
\end{align}
\ \left|\begin{align}\frac{x}{z} &amp; =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\
\frac{y}{z} &amp; =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}
\end{align}
\right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\
x^{\prime2}+y^{\prime2}=z^{\prime2}
\end{matrix}\\
\hline \mathbf{(2)}\ \left.\begin{align}x &amp; =mx'+ny'+sz'\\
y &amp; =m'x'+n'y'+s'z'\\
z &amp; =m''x'+n''y'+s''z'\\
\\
x' &amp; =mx+m'y-m''z\\
y' &amp; =nx+n'y-n''z\\
z' &amp; =-sx-s'y+s''z\\
\\
dx &amp; =mdx'+ndy'+sdz'\\
dy &amp; =m'dx'+n'dy'+s'dz'\\
dz &amp; =m''dx'+n''dy'+s''dz'
\end{align}
\right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} &amp; =1\\
n^{2}+n^{\prime2}-n^{\prime\prime2} &amp; =1\\
-s^{2}-s^{\prime2}+s^{\prime\prime2} &amp; =1\\
ns+n's'-n''s'' &amp; =0\\
sm+s'm'-s''m'' &amp; =0\\
mn+m'n'-m''n'' &amp; =0\\
\\
m^{2}+n^{2}-s^{2} &amp; =1\\
m^{\prime2}+n^{\prime2}-s^{\prime2} &amp; =1\\
-m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} &amp; =1\\
-m'm''-n'n''+s's'' &amp; =0\\
-m''m-n''n+s''s &amp; =0\\
mm'+nn'-ss' &amp; =0
\end{align}
}\\
dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Klein|label 1=History of Lorentz transformations via Möbius transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Klein|label 1=History of Lorentz transformations via squeeze mappings § Klein}}

Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an &quot;absolute&quot; ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a &quot;fundamental [[w:conic section]]&quot; in order to discuss motions such as rotation and translation in the non-Euclidean plane.&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt; This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].&lt;ref group=M&gt;Klein (1873), pp. 127-128&lt;/ref&gt;

{{Lorentzbox|Text=Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.}}

==={{anchor|Killing}} Killing (1878–1893)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Killing2|label 1=History of Lorentz transformations via hyperbolic functions § Killing}}
===={{anchor|Killing1}} Weierstrass coordinates====

[[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form

:&lt;math&gt;k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1877/78), p. 74; Killing (1880), p. 279&lt;/ref&gt; with &lt;math&gt;ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}&lt;/math&gt;&lt;ref group=M&gt;Killing (1880), eq. 25 on p. 283&lt;/ref&gt;

or&lt;ref group=M&gt;Killing (1880), p. 283&lt;/ref&gt;

:&lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt;

where ''k'' is the reciprocal measure of curvature, &lt;math&gt;k^{2}=\infty&lt;/math&gt; denotes [[w:Euclidean geometry]], &lt;math&gt;k^{2}&gt;0&lt;/math&gt; [[w:elliptic geometry]], and &lt;math&gt;k^{2}&lt;0&lt;/math&gt; hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.&lt;ref group=M&gt;Killing (1877/78), eq. 25 on p. 283&lt;/ref&gt; In (1879/80) he tried to formulate the corresponding transformations by plugging &lt;math&gt;k^{2}&lt;/math&gt; into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:&lt;ref group=M&gt;Killing (1879/80), p. 274&lt;/ref&gt;

&lt;math&gt;\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\
\hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\
-k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\
k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, &amp; \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}
\end{matrix}\\
\left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right)
\end{matrix}&lt;/math&gt;

In (1885) he wrote the Weierstrass coordinates and their transformation as follows:&lt;ref group=M&gt;Killing (1885), pp. 18, 28–30, 53&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\
k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\
ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\
\hline \begin{align}k^{2}p' &amp; =k^{2}wp+w'x+w''y\\
x' &amp; =ap+a'x+a''y\\
y' &amp; =bp+b'x+b''y\\
\\
k^{2}p &amp; =k^{2}wp'+ax'+by'\\
x &amp; =w'p'+a'x+b'y'\\
y &amp; =w''p'+a''x'+b''y'
\end{align}
\left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} &amp; =k^{2}\\
\frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} &amp; =1\\
\frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
aw+a'w'+a''w'' &amp; =0\\
bw+b'w'+b''w'' &amp; =0\\
\frac{ab}{k^{2}}+a'b'+a''b'' &amp; =0\\
\\
k^{2}w^{2}+a^{2}+b^{2} &amp; =k^{2}\\
\frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} &amp; =1\\
\frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} &amp; =1\\
ww'+aa'+bb' &amp; =0\\
ww''+aa''+bb'' &amp; =0\\
\frac{w'w''}{k^{2}}+a'a''+b'b'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

In (1885) he also gave the transformation for ''n'' dimensions:&lt;ref group=M&gt;Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222&lt;/ref&gt;&lt;ref&gt;Ratcliffe (1994), § 3.6&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\
\hline \left.\begin{align}k^{2}\xi_{0} &amp; =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\
\xi_{\varkappa} &amp; =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\
\\
k^{2}x_{0} &amp; =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\
x_{\varkappa} &amp; =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n}
\end{align}
\right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
\frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} &amp; =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa)
\end{align}
}
\end{matrix}&lt;/math&gt;

In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.&lt;ref group=M&gt;Killing (1884/85), pp. 4–5&lt;/ref&gt; Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting &lt;math&gt;k^{2}=-1&lt;/math&gt; and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid &lt;math&gt;p^{2}-x^{2}-y^{2}=1&lt;/math&gt; (known as [[w:hyperboloid model]]),&lt;ref group=M&gt;Killing (1885), Note 9 on p. 260&lt;/ref&gt;&lt;ref name=rey /&gt; by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space.

{{Lorentzbox|Text=All of Killing's transformations between 1879 and 1885 don't work when &lt;math&gt;k^{2}&lt;/math&gt; is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

Finally, in (1893) he wrote:&lt;ref group=M&gt;Killing (1893), see pp. 144, 327–328&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\
\hline \begin{align}t' &amp; =at+bu+cv\\
u' &amp; =a't+b'u+c'v\\
v' &amp; =a''t+b''u+c''v
\end{align}
\left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} &amp; =k^{2}\\
k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} &amp; =1\\
k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} &amp; =1\\
k^{2}ab+a'b'+a''b'' &amp; =0\\
k^{2}ac+a'c'+a''c'' &amp; =0\\
k^{2}bc+b'c'+b''c'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

and in ''n'' dimensions&lt;ref group=M&gt;Killing (1893), pp. 314–316, 216–217&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\
ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\
\hline \begin{align}y_{0} &amp; =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\
y_{1} &amp; =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
 &amp; \,\,\,\vdots\\
y_{n} &amp; =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
\end{align}
\left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} &amp; =k^{2}\\
k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} &amp; =1\\
k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} &amp; =0\\
k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} &amp; =0\\
(\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) with &lt;math&gt;k^{2}=-1&lt;/math&gt;.}}

===={{anchor|Killing3}} Infinitesimal transformations and Lie group====

After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions, Killing (1887/88)&lt;ref group=M&gt;Killing (1887/88a), pp. 274–275&lt;/ref&gt; defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere:

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\
\hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\
\text{where}\\
\left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\
\left[\iota\ne\varkappa\ne\lambda\ne\mu\right]
\end{matrix}&lt;/math&gt;

and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:&lt;ref group=M&gt;Killing (1892), p. 177&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\
\hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\
\text{where}\\
\left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f;
\end{matrix}&lt;/math&gt;

In (1897/98) he showed the relation between Weierstrass coordinates &lt;math&gt;k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}&lt;/math&gt; and coordinates &lt;math&gt;k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0&lt;/math&gt; used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:&lt;ref group=M&gt;Killing (1897/98), pp. 255–256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} &amp; (a)\\
k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} &amp; (b)
\end{matrix}\\
\hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\
\text{where}\\
\left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\
\left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\
\left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\
\hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\
\downarrow\\
k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\
\hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota}
\end{matrix}
\end{matrix}&lt;/math&gt;

He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b).

{{Lorentzbox|Text=Setting &lt;math&gt;k^{2}=-1&lt;/math&gt; denotes the group of hyperbolic motions and thus the Lorentz group.}}

=== {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Poincare3|label 1=History of Lorentz transformations via velocity § Poincaré}}

[[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).&lt;ref&gt;Gray (1997)&lt;/ref&gt; He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 220–221&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}F &amp; =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\
 &amp; =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
F &amp; =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\
 &amp; =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1
\end{align}
\\
\hline \begin{align}\xi' &amp; =\alpha\xi+\beta\eta+\gamma\zeta\\
\eta' &amp; =\alpha'\xi+\beta'\eta+\gamma'\zeta\\
\zeta' &amp; =\alpha''\xi+\beta''\eta+\gamma''\zeta
\end{align}
\left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =1\\
\beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} &amp; =1\\
\gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-1\\
\alpha\beta+\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma+\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

He went on to describe the properties of &quot;hyperbolic coordinates&quot;.&lt;ref group=M name=poinc&gt;Poincaré (1881b), p. 333&lt;/ref&gt;&lt;ref name=rey&gt;Reynolds (1993)&lt;/ref&gt; Poincaré mentioned the hyperboloid model also in (1887).&lt;ref group=M&gt;Poincaré (1887), p. 206&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

=== {{anchor|Cox}} Cox (1881–1891) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Cox|label 1=History of Lorentz transformations via hyperbolic functions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt; and [[w:George Salmon]] (1862)&lt;ref group=M&gt;Salmon (1862), section 212, p. 165&lt;/ref&gt; on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)&lt;ref group=M&gt;Frischauf (1876), pp. 86–87&lt;/ref&gt; in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:&lt;ref group=M&gt;Cox (1881/82), p. 186 for Weierstrass coordinates; pp. 193–194 for Lorentz transformation.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z^{2}-x^{2}-y^{2}=1\\
x^{2}-y^{2}-z^{2}=Z^{2}-Y^{2}-X^{2}\\
\hline \begin{align}x &amp; =l_{1}X+l_{2}Y+l_{3}Z\\
y &amp; =m_{1}X+m_{2}Y+m_{3}Z\\
z &amp; =n_{1}X+n_{2}Y+n_{3}Z\\
\\
X &amp; =l_{1}x+m_{1}y-n_{1}z\\
Y &amp; =l_{2}x+m_{2}y-n_{2}z\\
Z &amp; =l_{3}x+m_{3}y-n_{3}z
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}-n_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} &amp; =0\\
l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} &amp; =0\\
\\
l_{1}^{2}+l_{2}^{2}-l_{3}^{2} &amp; =1\\
m_{1}^{2}+m_{2}^{2}-m_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1\\
l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} &amp; =0\\
m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} &amp; =0\\
n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These equations contain several errors or misprints: &lt;math&gt;Z^{2}-Y^{2}-X^{2}&lt;/math&gt; has to be replaced by &lt;math&gt;X^{2}-Y^{2}-Z^{2}&lt;/math&gt;, and &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =1
\end{align}
}&lt;/math&gt; replaced with &lt;math&gt;{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} &amp; =-1\\
n_{1}^{2}+n_{2}^{2}-n_{3}^{2} &amp; =-1
\end{align}
}&lt;/math&gt;, and by reversing the sign of &lt;math&gt;Z&lt;/math&gt; in the inverse transformation, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

Cox (1881/82) also gave the Weierstrass coordinates and their transformation in hyperbolic space:&lt;ref group=M&gt;Cox (1881/82), pp. 199, 206–207&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\
w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\
\hline \begin{align}x &amp; =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\
y &amp; =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\
z &amp; =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\
w &amp; =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\
\\
x' &amp; =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\
y' &amp; =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\
z' &amp; =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\
w' &amp; =l_{4}x+m_{4}y+n_{4}z-r_{4}w
\end{align}
\left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} &amp; =1\\
l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} &amp; =1\\
l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} &amp; =1\\
l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} &amp; =1\\
l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} &amp; =0\\
l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} &amp; =0\\
l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} &amp; =0\\
l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} &amp; =0\\
l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By replacing &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}&lt;/math&gt; with &lt;math&gt;{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}&lt;/math&gt; this represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w'&lt;/math&gt; everywhere.}}

In 1883 he formulated relations between [[w:orthogonal circles]] which he identified with the previously (1881/82) given transformations:&lt;ref group=M&gt;Cox (1883), pp. 109ff&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}-w^{2}=0\\
\hline \begin{align}x &amp; =\lambda_{1}X+\lambda_{2}Y+\lambda_{3}Z+\lambda_{4}W\\
y &amp; =\mu_{1}X+\mu_{2}Y+\mu_{3}Z+\mu_{4}W\\
z &amp; =\nu_{1}X+\nu_{2}Y+\nu_{3}Z+\nu_{4}W\\
-w &amp; =\rho_{1}X+\rho_{2}Y+\rho_{3}Z+\rho_{4}W\\
\\
X &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z+\rho_{1}w\\
Y &amp; =\lambda_{2}x+\mu_{2}y+\nu_{2}z+\rho_{2}w\\
Z &amp; =\lambda_{3}x+\mu_{3}y+\nu_{3}z+\rho_{3}w\\
-W &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z+\rho_{4}w
\end{align}
\left|{\scriptstyle \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{2}\lambda_{3}+\mu_{2}\mu_{3}+\nu_{2}\nu_{3}-\rho_{2}\rho_{3} &amp; =0\\
\lambda_{3}\lambda_{1}+\mu_{3}\mu_{1}+\nu_{3}\nu_{1}-\rho_{3}\rho_{1} &amp; =0\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0\\
\lambda_{1}\lambda_{4}+\mu_{1}\mu_{4}+\nu_{1}\nu_{4}-\rho_{1}\rho_{4} &amp; =0\\
\lambda_{2}\lambda_{4}+\mu_{2}\mu_{4}+\nu_{2}\nu_{4}-\rho_{2}\rho_{4} &amp; =0\\
\lambda_{3}\lambda_{4}+\mu_{3}\mu_{4}+\nu_{3}\nu_{4}-\rho_{3}\rho_{4} &amp; =0
\end{align}
}\right.{\scriptstyle \begin{align}\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}-\mu_{4}^{2} &amp; =1\\
\nu_{1}^{2}+\nu_{2}^{2}+\nu_{3}^{2}-\nu_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0\\
\lambda_{1}\nu_{1}+\lambda_{2}\nu_{2}+\lambda_{3}\nu_{3}-\lambda_{4}\nu_{4} &amp; =0\\
\lambda_{1}\rho_{1}+\lambda_{2}\rho_{2}+\lambda_{3}\rho_{3}-\lambda_{4}\rho_{4} &amp; =0\\
\mu_{1}\nu_{1}+\mu_{2}\nu_{2}+\mu_{3}\nu_{3}-\mu_{4}\nu_{4} &amp; =0\\
\mu_{1}\rho_{1}+\mu_{2}\rho_{2}+\mu_{3}\rho_{3}-\mu_{4}\rho_{4} &amp; =0\\
\nu_{1}\rho_{1}+\nu_{2}\rho_{2}+\nu_{3}\rho_{3}-\nu_{4}\rho_{4} &amp; =0
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The relations between &lt;math&gt;\lambda,\mu,\nu,\rho&lt;/math&gt; are correct, even though the transformation still represents an improper antichronous Lorentz transformation, which becomes proper orthochronous Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' by reversing the sign of &lt;math&gt;w&lt;/math&gt; everywhere.}}

Finally, in a treatise on [[w:Hermann Grassmann|w:Grassmann's Ausdehnungslehre]] and circles (1891), he again provided transformations of orthogonal circle systems described by him as being &quot;identical with those for transformation of coordinates in non-Euclidean geometry&quot;:&lt;ref group=M&gt;Cox (1891), pp. 27-28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}=w^{2}\\
\hline \begin{align}x &amp; =\lambda_{1}x'+\lambda_{2}y'+\lambda_{3}z'+\lambda_{4}w' &amp; \text{(4 equations)}\\
x' &amp; =\lambda_{1}x+\mu_{1}y+\nu_{1}z-\rho_{1}w\\
-w' &amp; =\lambda_{4}x+\mu_{4}y+\nu_{4}z-\rho_{4}w
\end{align}
\\
\hline \begin{align}\lambda_{1}^{2}+\mu_{1}^{2}+\nu_{1}^{2}-\rho_{1}^{2} &amp; =1\\
\lambda_{2}^{2}+\mu_{2}^{2}+\nu_{2}^{2}-\rho_{2}^{2} &amp; =1\\
\lambda_{3}^{2}+\mu_{3}^{2}+\nu_{3}^{2}-\rho_{3}^{2} &amp; =1\\
\lambda_{4}^{2}+\mu_{4}^{2}+\nu_{4}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\lambda_{2}+\mu_{1}\mu_{2}+\nu_{1}\nu_{2}-\rho_{1}\rho_{2} &amp; =0 &amp; \text{(6 equations)}\\
\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-\lambda_{4}^{2} &amp; =1\\
\rho_{1}^{2}+\rho_{2}^{2}+\rho_{3}^{2}-\rho_{4}^{2} &amp; =-1\\
\lambda_{1}\mu_{1}+\lambda_{2}\mu_{2}+\lambda_{3}\mu_{3}-\lambda_{4}\mu_{4} &amp; =0 &amp; \text{(6 equations)}
\end{align}
\end{matrix}\text{ }&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)''.}}

=== {{anchor|Hill}} Hill (1882) – Homogeneous coordinates ===

Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations&lt;ref group=M&gt;Hill (1882), pp. 323–325&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\
k\left(\sin^{2}E+\cos^{2}E-1\right)\\
\hline \begin{align} &amp;  &amp; \cos E' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
 &amp; \mathbf{(1)} &amp; \sin E' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\hline \\
 &amp;  &amp; x &amp; =\alpha u+\alpha'u'+\alpha''u''\\
 &amp;  &amp; y &amp; =\beta u+\beta'u'+\beta''u''\\
 &amp;  &amp; z &amp; =\gamma u+\gamma'u'+\gamma''u''\\
 &amp; \mathbf{(2)}\\
 &amp;  &amp; u &amp; =-\alpha x-\beta y+\gamma z\\
 &amp;  &amp; u' &amp; =\alpha'x+\beta'y'-\gamma'z\\
 &amp;  &amp; u'' &amp; =\alpha''x+\beta''y-\gamma''z
\end{align}
\left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-1\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =1\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =1\\
\alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
(k=-1)\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =k\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =k\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =-k\\
\alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.

Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .}}

=== {{anchor|Picard}} Picard (1882-1884) – Quadratic forms ===

[[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to &quot;hyperfuchsian&quot; functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:&lt;ref group=M&gt;Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 280-281&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\
\\
\mathbf{(1)}\ \begin{align}x &amp; =M_{1}X+P_{1}Y+R_{1}Z\\
y &amp; =M_{2}X+P_{2}Y+R_{2}Z\\
z &amp; =M_{3}X+P_{3}Y+R_{3}Z
\end{align}
\\
\\
\left[\begin{align}[][x,y,z]=\text{complex}\\
\left[x_{0},y_{0},z_{0}\right]=\text{conjugate}
\end{align}
\right]\\
\\
\hline \\
x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\
x=x'+ix'',\quad y=y'+iy''\\
\\
\mathbf{(2)}\ \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} &amp; =1\\
P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0\\
P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
\\
M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} &amp; =1\\
M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} &amp; =1\\
M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} &amp; =-1\\
\mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} &amp; =0\\
\mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} &amp; =0\\
\mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} &amp; =0\\
\\
\left[\begin{align}[][M,P,R\dots]=\text{complex}\\
\left[\mu,\pi,\rho\dots\right]=\text{conjugate}
\end{align}
\right]
\end{align}
}\right.\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;-Z&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:&lt;ref group=M&gt;Picard (1884a), p. 13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\
\hline \begin{align}U &amp; =\mathcal{A}u+\mathcal{B}v\\
V &amp; =\mathcal{C}u+\mathcal{D}v
\end{align}
\left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1\\
\mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} &amp; =0\\
\mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} &amp; =-1\\
\mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} &amp; =1
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;-V&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;-v&lt;sup&gt;2&lt;/sup&gt;''.}}

Or in (1884b):&lt;ref group=M&gt;Picard (1884b), p. 416&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xx_{0}+yy_{0}-1=0\\
\hline \begin{align}X &amp; =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\
Y &amp; =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}}
\end{align}
\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} &amp; =1\\
R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} &amp; =-1\\
P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} &amp; =0\\
M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;=X&lt;sup&gt;2&lt;/sup&gt;+Y&lt;sup&gt;2&lt;/sup&gt;=1''.}}

Or in (1884c):&lt;ref group=M&gt;Picard (1884c), pp. 123–124; 163&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\
\hline \mathbf{(1)}\ \begin{align}U &amp; =Mu+Pv+Rw\\
V &amp; =M'u+P'v+R'w\\
W &amp; =M''u+P''v+R''w\\
\\
u &amp; =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\
v &amp; =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\
w &amp; =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W
\end{align}
\left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} &amp; =1\\
PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} &amp; =1\\
RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} &amp; =0\\
MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} &amp; =0\\
PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} &amp; =0\\
\\
MM_{0}+PP_{0}-RR_{0} &amp; =1\\
M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} &amp; =1\\
M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} &amp; =-1\\
M_{0}M'+P_{0}P'-R_{0}R' &amp; =0\\
M_{0}M''+P_{0}P''-R_{0}R'' &amp; =0\\
M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' &amp; =0
\end{align}
}\right.\\
\hline \text{Invariance of unit hypersphere:}\\
\mathbf{(2)}\ \begin{align}\xi' &amp; =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\
\eta' &amp; =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''}
\end{align}
\left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} &amp; =1\\
BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} &amp; =1\\
CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} &amp; =-1\\
AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} &amp; =0\\
AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} &amp; =0\\
BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U&lt;sup&gt;2&lt;/sup&gt;+V&lt;sup&gt;2&lt;/sup&gt;-W&lt;sup&gt;2&lt;/sup&gt;=u&lt;sup&gt;2&lt;/sup&gt;+v&lt;sup&gt;2&lt;/sup&gt;-w&lt;sup&gt;2&lt;/sup&gt;'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing &lt;math&gt;\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1&lt;/math&gt;.}}

=== {{anchor|Callandreau}} Callandreau (1885) – Homography ===

Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations&lt;ref group=M&gt;Callandreau (1885), pp. A.7; A.12&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\
{\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\
\hline \begin{align}\cos\varepsilon' &amp; =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\
\sin\varepsilon' &amp; =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}
\end{align}
\left|{\scriptstyle \begin{align} &amp; \left(k=1\right)\\
\alpha^{2}+\beta^{2}-\gamma^{2} &amp; =-k &amp; \alpha\alpha'+\beta\beta'-\gamma\gamma' &amp; =0\\
\alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} &amp; =+k &amp; \alpha\alpha''+\beta\beta''-\gamma\gamma'' &amp; =0\\
\alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} &amp; =+k &amp; \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' &amp; =0\\
\\
\alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} &amp; =-1 &amp; \alpha\beta-\alpha'\beta'-\alpha''\beta'' &amp; =0\\
\beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} &amp; =-1 &amp; \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' &amp; =0\\
\gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} &amp; =+1 &amp; \beta\gamma-\beta'\gamma'-\beta''\gamma'' &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}})  ''(n=2)'' with &lt;math&gt;[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]&lt;/math&gt;.}}

==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Lie|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree &lt;math&gt;\sum f_{ik}x_{i}'x_{k}'=0&lt;/math&gt; with the group of non-Euclidean motions.&lt;ref group=M&gt;Lie (1885/86), p. 411&lt;/ref&gt; In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):&lt;ref group=M&gt;Werner (1889), pp. 4, 28&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\
\hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\
\text{where}\\
\left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\
\left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\
\left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right]
\end{matrix}&lt;/math&gt;

More generally, Lie (1890)&lt;ref group=M&gt;Lie (1890a), p. 295;&lt;/ref&gt; defined non-Euclidean motions in terms of two forms &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0&lt;/math&gt; in which the imaginary form with &lt;math&gt;+1&lt;/math&gt; denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:&lt;ref group=M&gt;Lie (1890a), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\
\hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n)
\end{matrix}&lt;/math&gt;

Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form:

:&lt;math&gt;x^{2}+y^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 474&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0&lt;/math&gt;&lt;ref group=M&gt;Lie (1893), p. 479&lt;/ref&gt; or &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt;.&lt;ref group=M&gt;Lie (1893), p. 481&lt;/ref&gt;

{{Lorentzbox|Text=The group of hyperbolic motions is isomorphic to the Lorentz group. The interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-1=0&lt;/math&gt; becomes the Lorentz interval &lt;math&gt;x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0&lt;/math&gt; by setting
&lt;math&gt;(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)&lt;/math&gt;}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Gerard|label 1=History of Lorentz transformations via hyperbolic functions § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':&lt;ref group=M&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\
X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\
\hline \begin{align}X &amp; =aX'+a'Y'+a''Z'\\
Y &amp; =bX'+b'Y'+b''Z'\\
Z &amp; =cX'+c'Y'+c''Z'\\
\\
X' &amp; =aX+bY-cZ\\
Y' &amp; =a'X+b'Y-c'Z\\
Z' &amp; =-a''X-b''Y+c''Z
\end{align}
\left|\begin{align}a^{2}+b^{2}-c^{2} &amp; =1\\
a^{\prime2}+b^{\prime2}-c^{\prime2} &amp; =1\\
a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} &amp; =-1\\
aa'+bb'-cc' &amp; =0\\
a'a''+b'b''-c'c'' &amp; =0\\
a''a+b''b-c''c &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

He gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates===

[[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:&lt;ref group=M&gt;Hausdorff (1899), p. 165, pp. 181-182&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{2}-x^{2}-y^{2}=1\\
\hline \begin{align}x &amp; =a_{1}x'+a_{2}y'+x_{0}p'\\
y &amp; =b_{1}x'+b{}_{2}y'+y_{0}p'\\
p &amp; =e_{1}x'+e_{2}y'+p_{0}p'\\
\\
x' &amp; =a_{1}x+b_{1}y-e_{1}p\\
y' &amp; =a_{2}x+b_{2}y-e_{2}p\\
-p' &amp; =x_{0}x+y_{0}y-p_{0}p
\end{align}
\left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} &amp; =1\\
a_{2}^{2}+b_{2}^{2}-e_{2}^{2} &amp; =1\\
-x_{0}^{2}-y_{0}^{2}+p_{0}^{2} &amp; =1\\
a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} &amp; =0\\
a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} &amp; =0\\
a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} &amp; =0\\
\\
a_{1}^{2}+a_{2}^{2}-x_{0}^{2} &amp; =1\\
b_{1}^{2}+b_{2}^{2}-y_{0}^{2} &amp; =1\\
-e_{1}^{2}-e_{2}^{2}+p_{0}^{2} &amp; =1\\
b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} &amp; =0\\
a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} &amp; =0\\
a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} &amp; =0
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Woods2|label 1=History of Lorentz transformations via hyperbolic functions § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting &lt;math&gt;k=\sqrt{-1}R&lt;/math&gt; with ''R'' as real quantity):&lt;ref group=M&gt;Woods (1901/02), p. 98, 104&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\
\hline \begin{align}u' &amp; =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
v' &amp; =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\
w' &amp; =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}
\end{align}
\left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} &amp; =k^{2}\\
(i=1,2,3)\\
k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} &amp; =1\\
\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} &amp; =0\\
(i,h=1,2,3,4;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with &lt;math&gt;k=\sqrt{-1}k&lt;/math&gt; for hyperbolic space):&lt;ref group=M&gt;Woods (1903/05), pp. 45–46; p. 48)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\
ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\
\hline \begin{align}x_{1}^{\prime} &amp; =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\
x_{2}^{\prime} &amp; =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\
x_{3}^{\prime} &amp; =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\
x_{0}^{\prime} &amp; =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0}
\end{align}
\left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) &amp; =1\\
\delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) &amp; =k^{2}\\
(i=1,2,3)\\
\delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) &amp; =0\\
(i,h=0,1,2,3;\ i\ne h)
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Liebmann|label 1=History of Lorentz transformations via hyperbolic functions § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''&lt;ref group=M&gt;Liebmann (1904/05), p. 168; pp. 175–176&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\
\hline \begin{align}x_{1} &amp; =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\
y_{1} &amp; =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\
x_{1} &amp; =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\
\\
x &amp; =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\
y &amp; =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\
p &amp; =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1}
\end{align}
\left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} &amp; =1\\
-\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} &amp; =1\\
-\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} &amp; =1\\
\alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} &amp; =0\\
\alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} &amp; =0\\
\alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} &amp; =0
\end{align}
\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|apo1}}
*{{#section:History of Topics in Special Relativity/mathsource|apo2}}
*{{#section:History of Topics in Special Relativity/mathsource|apo}}
*{{#section:History of Topics in Special Relativity/mathsource|bour56att}} 
*{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|chas29}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox83hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox91}} 
*{{#section:History of Topics in Special Relativity/mathsource|fris76}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau18}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|haus99}} 
*{{#section:History of Topics in Special Relativity/mathsource|hill82}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac27}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32a}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac32b}} 
*{{#section:History of Topics in Special Relativity/mathsource|jac33}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil77}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil79}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil84}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil85}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei73}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|hire1}} 
*{{#section:History of Topics in Special Relativity/mathsource|leb37}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie85}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie93}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lop}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic82}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84a}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84b}} 
*{{#section:History of Topics in Special Relativity/mathsource|pic84c}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin87}} 
*{{#section:History of Topics in Special Relativity/mathsource|sal62}} 
*{{#section:History of Topics in Special Relativity/mathsource|vinc}} 
*{{#section:History of Topics in Special Relativity/mathsource|som63}} 
*{{#section:History of Topics in Special Relativity/mathsource|wedd47}} 
*{{#section:History of Topics in Special Relativity/mathsource|wern89}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L1}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>68ljgiu5cw9a86f5jkdk6hz2s6jmydo</sha1>
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    <title>History of Topics in Special Relativity/Lorentz transformation (imaginary)</title>
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      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <minor />
      <comment>/* Lorentz transformation via imaginary orthogonal transformation */</comment>
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      <text bytes="17750" sha1="9stw87euwnqm5w8fe6mop0w21yqt51w" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via imaginary orthogonal transformation==
By using the [[w:Imaginary unit|imaginary]] quantities &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime}\right]&lt;/math&gt; in '''x''' as well as &lt;math&gt;[\mathfrak{g}_{0s},\ \mathfrak{g}_{s0}]=\left[ig_{0s},\ ig_{s0}\right]&lt;/math&gt; ''(s=1,2...n)'' in '''g''', the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] assumes the form of an [[w:orthogonal transformation]] of [[w:Euclidean space]] forming the [[w:orthogonal group]] O(n) if det '''g'''=±1 or the special orthogonal group SO(n) if det '''g'''=+1, the Lorentz interval becomes the [[w:Euclidean norm]], and the Minkowski inner product becomes the [[w:dot product]]:&lt;ref&gt;Laue (1921), pp. 79–80 for n=3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{align}\mathfrak{x}_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2} &amp; =\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+\dots+x_{n}^{\prime2}\\
\mathfrak{x}_{0}\mathfrak{y}_{0}+x_{1}y_{1}+\cdots+x_{n}y_{n} &amp; =\mathfrak{x}_{0}^{\prime}\mathfrak{y}_{0}^{\prime}+x_{1}^{\prime}y_{1}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x}=\mathbf{\mathbf{g}^{-1}}\cdot\mathbf{x}'
\end{matrix}\left|\begin{align}\sum_{i=0}^{n}g_{ij}g_{ik} &amp; =\left\{ \begin{align}1\quad &amp; (j=k)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=0}^{n}g_{ij}g_{kj} &amp; =\left\{ \begin{align}1\quad &amp; (i=k)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|2a}}}}

The cases ''n=1,2,3,4'' of orthogonal transformations in terms of real coordinates were discussed by [[#Euler|Euler (1771)]] and in ''n'' dimensions by [[#Cauchy|Cauchy (1829)]]. The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by [[#Lie|Lie (1871)]] in terms of spheres with imaginary radius, while the interpretation of the imaginary coordinate as being related to the dimension of time as well as the explicit formulation of Lorentz transformations with ''n=3'' was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]].

A well known example of this orthogonal transformation is spatial [[w:rotation]] in terms of [[w:trigonometric function]]s, which become Lorentz transformations by using an imaginary angle &lt;math&gt;\phi=i\eta&lt;/math&gt;, so that trigonometric functions become equivalent to [[w:hyperbolic function]]s:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}\mathfrak{x}_{0}^{\prime} &amp; =\mathfrak{x}_{0}\cos\phi-x_{1}\sin\phi\\
x_{1}^{\prime} &amp; =\mathfrak{x}_{0}\sin\phi+x_{1}\cos\phi\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
\mathfrak{x}_{0} &amp; =\mathfrak{x}_{0}^{\prime}\cos\phi+x_{1}^{\prime}\sin\phi\\
x_{1} &amp; =-\mathfrak{x}_{0}^{\prime}\sin\phi+x_{1}^{\prime}\cos\phi\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}ix_{0}^{\prime} &amp; =ix_{0}\cos i\eta-x_{1}\sin i\eta\\
x_{1}^{\prime} &amp; =ix_{0}\sin i\eta+x_{1}\cos i\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
ix_{0} &amp; =ix_{0}^{\prime}\cos i\eta+x_{1}^{\prime}\sin i\eta\\
x_{1} &amp; =-ix_{0}^{\prime}\sin i\eta+x_{1}^{\prime}\cos i\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array}&lt;/math&gt;|{{equationRef|2b}}}}

or in exponential form using [[w:Euler's formula]] &lt;math&gt;e^{i\phi}=\cos\phi+i\sin\phi&lt;/math&gt;:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime} &amp; =e^{-i\phi}\left(x_{1}+i\mathfrak{x}_{0}\right)\\
x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime} &amp; =e^{i\phi}\left(x_{1}-i\mathfrak{x}_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\mathfrak{x}_{0} &amp; =e^{i\phi}\left(x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime}\right)\\
x_{1}-i\mathfrak{x}_{0} &amp; =e^{-i\phi}\left(x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right) &amp; =e^{-i(i\eta)}\left(x_{1}+i\left(ix_{0}\right)\right)\\
x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right) &amp; =e^{i(i\eta)}\left(x_{1}-i\left(ix_{0}\right)\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\left(ix_{0}\right) &amp; =e^{i(i\eta)}\left(x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right)\right)\\
x_{1}-i\left(ix_{0}\right) &amp; =e^{-i(i\eta)}\left(x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right)\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array}&lt;/math&gt;|{{equationRef|2c}}}}

Defining &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; as real, spatial rotation in the form ({{equationNote|2b}}-1) was introduced by [[#Euler2|Euler (1771)]] and in the form ({{equationNote|2c}}-1) by [[#Euler3|Wessel (1799)]]. The interpretation of ({{equationNote|2b}}) as Lorentz boost (i.e. Lorentz transformation ''without'' spatial rotation) in which &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; correspond to the imaginary quantities &lt;math&gt;[ix_{0},\ ix'_{0},\ i\eta]&lt;/math&gt; was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]]. As shown in the next section using hyperbolic functions, ({{equationNote|2b}}) becomes [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] while ({{equationNote|2c}}) becomes [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']].

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Orthogonal transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Euler|label 1=History of Lorentz transformations via hyperbolic functions § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Euler|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Leonhard Euler]] (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]], as well as under rotations using [[w:Euler angles]]. The case of two dimensions is given by&lt;ref group=M&gt;Euler (1771), pp. 84-85&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}=x^{2}+y^{2}\\
\hline \begin{align}X &amp; =\alpha x+\beta y\\
Y &amp; =\gamma x+\delta y
\end{align}
\left|\begin{matrix}\begin{align}1 &amp; =\alpha\alpha+\gamma\gamma\\
1 &amp; =\beta\beta+\delta\delta\\
0 &amp; =\alpha\beta+\gamma\delta
\end{align}
\end{matrix}\right.\\
\hline \begin{align}X &amp; =x\cos\zeta+y\sin\zeta\\
Y &amp; =x\sin\zeta-y\cos\zeta
\end{align}
\end{matrix}&lt;/math&gt;

or three dimensions&lt;ref group=M&gt;Euler (1771), pp. 77, 85-89&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\\
\hline \begin{align}x' &amp; =x\cos\zeta+y\sin\zeta &amp; x'' &amp; =x'\cos\eta+z'\sin\eta\\
y' &amp; =x\sin\zeta-y\cos\zeta &amp; y'' &amp; =y'\\
z' &amp; =z &amp; z'' &amp; =x'\sin\eta-z'\cos\eta\\
\\
x''' &amp; =x'' &amp; =X\\
y''' &amp; =y''\cos\theta+z''\sin\theta &amp; =Y\\
z''' &amp; =y''\sin\theta-z''\cos\theta &amp; =Z
\end{align}
\end{matrix}&lt;/math&gt;

The orthogonal transformation in four dimensions was given by him as&lt;ref group=M&gt;Euler (1771), pp. 89–91&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\
\hline \begin{align}V &amp; =Av+Bx+Cy+Dz\\
X &amp; =Ev+Fx+Gy+Hz\\
Y &amp; =Iv+Kx+Ly+Mz\\
Z &amp; =Nv+Ox+Py+Qz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+RR+II+NN &amp; 0 &amp; =AB+EF+IK+NO\\
1 &amp; =BB+FF+KK+OO &amp; 0 &amp; =AC+EG+IL+NP\\
1 &amp; =CC+GG+LL+PP &amp; 0 &amp; =AD+EH+IM+NQ\\
1 &amp; =DD+HH+MM+QQ &amp; 0 &amp; =BC+FG+KL+OP\\
0 &amp; =BD+FH+KM+OQ &amp; 0 &amp; =CD+FH+LM+PQ
\end{align}
}\right.\end{matrix}\\
\hline {\scriptstyle \begin{align}x^{I} &amp; =x\cos\alpha+y\sin\alpha &amp;  &amp;  &amp; x^{VI} &amp; =x^{V} &amp; =X\\
y^{I} &amp; =x\sin\alpha-y\cos\alpha &amp;  &amp;  &amp; y^{VI} &amp; =y^{V} &amp; =Y\\
z^{I} &amp; =z &amp; \dots &amp; \dots &amp; y^{VI} &amp; =z^{V}\cos\zeta+v^{V}\sin\zeta &amp; =Z\\
v^{I} &amp; =v &amp;  &amp;  &amp; v^{VI} &amp; =z^{V}\sin\zeta-v^{V}\cos\varepsilon\zeta &amp; =V
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=As shown by [[#Minkowski|Minkowski (1907)]], the orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) or ({{equationNote|2b}}) by making one variable as well as six of the sixteen coefficients imaginary.}}

==={{anchor|Euler3}} Wessel (1799) – Euler's formula and rotation===

The above orthogonal transformations representing Euclidean rotations can also be expressed by using [[w:Euler's formula]]. After this formula was derived by Euler in 1748&lt;ref group=M&gt;Euler (1748b), section 138.&lt;/ref&gt;

:&lt;math&gt;e^{+v\sqrt{-1}}=\cos v+\sqrt{-1}\sin v,\quad e^{-v\sqrt{-1}}=\cos v-\sqrt{-1}\sin v&lt;/math&gt;,

it was used by [[w:Caspar Wessel]] (1799) to describe Euclidean rotations in the complex plane:&lt;ref group=M&gt;Wessel (1799), § 28.&lt;/ref&gt;

:&lt;math&gt;x''+\varepsilon z''=(x'+\varepsilon z')\cdot(\cos III+\varepsilon\sin III),\ (\varepsilon=\sqrt{-1})&lt;/math&gt;

{{Lorentzbox|Text=Replacing the real quantities by imaginary ones by setting &lt;math&gt;\left[z',z'',III\right]=\left[iz',iz'',iIII\right]&lt;/math&gt;, Wessel's transformation becomes Lorentz transformation ({{equationNote|2c}}).}}

==={{anchor|Cauchy}} Cauchy (1829) – Orthogonal transformation===

[[w:Augustin-Louis Cauchy]] (1829) extended the orthogonal transformation of [[#Euler|Euler (1771)]] to arbitrary dimensions&lt;ref group=M&gt;Cauchy (1829), eq. 22, 98, 99, 101; Some misprints were corrected in Œuvres complètes, série 2, tome 9, pp. 174–195.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}+\dots=\xi^{2}+\eta^{2}+\zeta^{2}+\dots\\
\hline \begin{align}x &amp; =x_{1}\xi+x_{2}\eta+x_{3}\zeta+\dots\\
y &amp; =y_{1}\xi+y_{2}\eta+y_{3}\zeta+\dots\\
z &amp; =z_{1}\xi+z_{2}\eta+z_{3}\zeta+\dots\\
 &amp; \dots\\
\\
\xi &amp; =x_{1}x+y_{1}y+z_{1}z+\dots\\
\eta &amp; =x_{2}x+y_{2}y+z_{2}z+\dots\\
\zeta &amp; =x_{3}x+y_{3}y+z_{3}z+\dots\\
 &amp; \dots
\end{align}
\left|{\scriptstyle \begin{align}x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+\dots &amp; =1,\\
x_{2}x_{1}+y_{2}y_{1}+z_{2}z_{1}+\dots &amp; =0,\\
\dots\\
x_{n}x_{1}+y_{n}y_{1}+z_{n}z_{1}+\dots &amp; =0,\\
\\
x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+\dots &amp; =0,\\
x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+\dots &amp; =1,\\
\text{ }\dots\\
x_{n}x_{2}+y_{n}y_{2}+z_{n}z_{2}+\dots &amp; =0,\\
\\
x_{1}x_{n}+y_{1}y_{n}+z_{1}z_{n}+\dots &amp; =0,\\
x_{2}x_{n}+y_{2}y_{n}+z_{2}z_{n}+\dots &amp; =0,\\
\dots\\
x_{n}^{2}+y_{n}^{2}+z_{n}^{2}+\dots &amp; =1
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) by making one of the variables as well as certain coefficients imaginary.}}

==={{anchor|Lie}} Lie (1871) – Imaginary orthogonal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Lie3|label 1=History of Lorentz transformations in general § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

[[w:Sophus Lie]] (1871a) described a manifold whose elements can be represented by spheres, where the last coordinate ''y&lt;sub&gt;n+1&lt;/sub&gt;'' can be related to an imaginary radius by ''iy&lt;sub&gt;n+1&lt;/sub&gt;'':&lt;ref group=M&gt;Lie (1871a), pp. 199–209&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\sum_{i=1}^{i=n} (x_i-y_i)^2+y_{n+1}^2=0 \\
\downarrow\\
\sum_{i=1}^{i=n+1} (y_i^{\prime}-y_i^{\prime\prime})^2=0
\end{matrix}&lt;/math&gt;

If the second equation is satisfied, two spheres ''y′'' and ''y″'' are in contact. Lie then defined the correspondence between [[w:Contact geometry|contact transformations]] in ''R&lt;sub&gt;n&lt;/sub&gt;'' and conformal point transformations in ''R&lt;sub&gt;n+1&lt;/sub&gt;'': The sphere of space ''R&lt;sub&gt;n&lt;/sub&gt;'' consists of ''n+1'' parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space ''R&lt;sub&gt;n&lt;/sub&gt;'', it follows that ''R&lt;sub&gt;n&lt;/sub&gt;'' now corresponds to ''R&lt;sub&gt;n+1&lt;/sub&gt;''. Therefore, any transformation (to which he counted [[#Lorentz transformation via orthogonal transformation|orthogonal transformations]] and inversions) leaving invariant the condition of contact between spheres in ''R&lt;sub&gt;n&lt;/sub&gt;'', corresponds to the conformal transformation of points in ''R&lt;sub&gt;n+1&lt;/sub&gt;''.

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Minkowski|label 1=History of Lorentz transformations via velocity § Minkowski}}

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1) in agreement with ({{equationNote|2b}}):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|2b}}), and the relativistc velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|cau29sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48b}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie71a}} 
*{{#section:History of Topics in Special Relativity/mathsource|wes99}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|mink07a}}
{{#section:History of Topics in Special Relativity/relsource|mink07b}}
{{#section:History of Topics in Special Relativity/relsource|mink08}}
{{#section:History of Topics in Special Relativity/relsource|som09}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L2}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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      <text bytes="17751" sha1="33l9vkwb54nxsbqlln5b4vt7q8j8wew" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via imaginary orthogonal transformation==
By using the [[w:Imaginary unit|imaginary]] quantities &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime}\right]&lt;/math&gt; in '''x''' as well as &lt;math&gt;[\mathfrak{g}_{0s},\ \mathfrak{g}_{s0}]=\left[ig_{0s},\ ig_{s0}\right]&lt;/math&gt; ''(s=1,2...n)'' in '''g''', the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] assumes the form of an [[w:orthogonal transformation]] of [[w:Euclidean space]] forming the [[w:orthogonal group]] O(n) if det '''g'''=±1 or the special orthogonal group SO(n) if det '''g'''=+1, the Lorentz interval becomes the [[w:Euclidean norm]], and the Minkowski inner product becomes the [[w:dot product]]:&lt;ref&gt;Laue (1921), pp. 79–80 for n=3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{align}\mathfrak{x}_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2} &amp; =\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+\dots+x_{n}^{\prime2}\\
\mathfrak{x}_{0}\mathfrak{y}_{0}+x_{1}y_{1}+\cdots+x_{n}y_{n} &amp; =\mathfrak{x}_{0}^{\prime}\mathfrak{y}_{0}^{\prime}+x_{1}^{\prime}y_{1}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x}=\mathbf{\mathbf{g}^{-1}}\cdot\mathbf{x}'
\end{matrix}\left|\begin{align}\sum_{i=0}^{n}g_{ij}g_{ik} &amp; =\left\{ \begin{align}1\quad &amp; (j=k)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=0}^{n}g_{ij}g_{kj} &amp; =\left\{ \begin{align}1\quad &amp; (i=k)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|2a}}}}

The cases ''n=1,2,3,4'' of orthogonal transformations in terms of real coordinates were discussed by [[#Euler|Euler (1771)]] and in ''n'' dimensions by [[#Cauchy|Cauchy (1829)]]. The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by [[#Lie|Lie (1871)]] in terms of spheres with imaginary radius, while the interpretation of the imaginary coordinate as being related to the dimension of time as well as the explicit formulation of Lorentz transformations with ''n=3'' was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]].

A well known example of this orthogonal transformation is spatial [[w:rotation]] in terms of [[w:trigonometric function]]s, which become Lorentz transformations by using an imaginary angle &lt;math&gt;\phi=i\eta&lt;/math&gt;, so that trigonometric functions become equivalent to [[w:hyperbolic function]]s:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}\mathfrak{x}_{0}^{\prime} &amp; =\mathfrak{x}_{0}\cos\phi-x_{1}\sin\phi\\
x_{1}^{\prime} &amp; =\mathfrak{x}_{0}\sin\phi+x_{1}\cos\phi\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
\mathfrak{x}_{0} &amp; =\mathfrak{x}_{0}^{\prime}\cos\phi+x_{1}^{\prime}\sin\phi\\
x_{1} &amp; =-\mathfrak{x}_{0}^{\prime}\sin\phi+x_{1}^{\prime}\cos\phi\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}ix_{0}^{\prime} &amp; =ix_{0}\cos i\eta-x_{1}\sin i\eta\\
x_{1}^{\prime} &amp; =ix_{0}\sin i\eta+x_{1}\cos i\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
ix_{0} &amp; =ix_{0}^{\prime}\cos i\eta+x_{1}^{\prime}\sin i\eta\\
x_{1} &amp; =-ix_{0}^{\prime}\sin i\eta+x_{1}^{\prime}\cos i\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align} 
\end{array}&lt;/math&gt;|{{equationRef|2b}}}}

or in exponential form using [[w:Euler's formula]] &lt;math&gt;e^{i\phi}=\cos\phi+i\sin\phi&lt;/math&gt;:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime} &amp; =e^{-i\phi}\left(x_{1}+i\mathfrak{x}_{0}\right)\\
x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime} &amp; =e^{i\phi}\left(x_{1}-i\mathfrak{x}_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\mathfrak{x}_{0} &amp; =e^{i\phi}\left(x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime}\right)\\
x_{1}-i\mathfrak{x}_{0} &amp; =e^{-i\phi}\left(x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right) &amp; =e^{-i(i\eta)}\left(x_{1}+i\left(ix_{0}\right)\right)\\
x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right) &amp; =e^{i(i\eta)}\left(x_{1}-i\left(ix_{0}\right)\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\left(ix_{0}\right) &amp; =e^{i(i\eta)}\left(x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right)\right)\\
x_{1}-i\left(ix_{0}\right) &amp; =e^{-i(i\eta)}\left(x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right)\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array}&lt;/math&gt;|{{equationRef|2c}}}}

Defining &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; as real, spatial rotation in the form ({{equationNote|2b}}-1) was introduced by [[#Euler2|Euler (1771)]] and in the form ({{equationNote|2c}}-1) by [[#Euler3|Wessel (1799)]]. The interpretation of ({{equationNote|2b}}) as Lorentz boost (i.e. Lorentz transformation ''without'' spatial rotation) in which &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; correspond to the imaginary quantities &lt;math&gt;[ix_{0},\ ix'_{0},\ i\eta]&lt;/math&gt; was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]]. As shown in the next section using hyperbolic functions, ({{equationNote|2b}}) becomes [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] while ({{equationNote|2c}}) becomes [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']].

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Orthogonal transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Euler|label 1=History of Lorentz transformations via hyperbolic functions § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Euler|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Leonhard Euler]] (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]], as well as under rotations using [[w:Euler angles]]. The case of two dimensions is given by&lt;ref group=M&gt;Euler (1771), pp. 84-85&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}=x^{2}+y^{2}\\
\hline \begin{align}X &amp; =\alpha x+\beta y\\
Y &amp; =\gamma x+\delta y
\end{align}
\left|\begin{matrix}\begin{align}1 &amp; =\alpha\alpha+\gamma\gamma\\
1 &amp; =\beta\beta+\delta\delta\\
0 &amp; =\alpha\beta+\gamma\delta
\end{align}
\end{matrix}\right.\\
\hline \begin{align}X &amp; =x\cos\zeta+y\sin\zeta\\
Y &amp; =x\sin\zeta-y\cos\zeta
\end{align}
\end{matrix}&lt;/math&gt;

or three dimensions&lt;ref group=M&gt;Euler (1771), pp. 77, 85-89&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\\
\hline \begin{align}x' &amp; =x\cos\zeta+y\sin\zeta &amp; x'' &amp; =x'\cos\eta+z'\sin\eta\\
y' &amp; =x\sin\zeta-y\cos\zeta &amp; y'' &amp; =y'\\
z' &amp; =z &amp; z'' &amp; =x'\sin\eta-z'\cos\eta\\
\\
x''' &amp; =x'' &amp; =X\\
y''' &amp; =y''\cos\theta+z''\sin\theta &amp; =Y\\
z''' &amp; =y''\sin\theta-z''\cos\theta &amp; =Z
\end{align}
\end{matrix}&lt;/math&gt;

The orthogonal transformation in four dimensions was given by him as&lt;ref group=M&gt;Euler (1771), pp. 89–91&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\
\hline \begin{align}V &amp; =Av+Bx+Cy+Dz\\
X &amp; =Ev+Fx+Gy+Hz\\
Y &amp; =Iv+Kx+Ly+Mz\\
Z &amp; =Nv+Ox+Py+Qz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+RR+II+NN &amp; 0 &amp; =AB+EF+IK+NO\\
1 &amp; =BB+FF+KK+OO &amp; 0 &amp; =AC+EG+IL+NP\\
1 &amp; =CC+GG+LL+PP &amp; 0 &amp; =AD+EH+IM+NQ\\
1 &amp; =DD+HH+MM+QQ &amp; 0 &amp; =BC+FG+KL+OP\\
0 &amp; =BD+FH+KM+OQ &amp; 0 &amp; =CD+FH+LM+PQ
\end{align}
}\right.\end{matrix}\\
\hline {\scriptstyle \begin{align}x^{I} &amp; =x\cos\alpha+y\sin\alpha &amp;  &amp;  &amp; x^{VI} &amp; =x^{V} &amp; =X\\
y^{I} &amp; =x\sin\alpha-y\cos\alpha &amp;  &amp;  &amp; y^{VI} &amp; =y^{V} &amp; =Y\\
z^{I} &amp; =z &amp; \dots &amp; \dots &amp; y^{VI} &amp; =z^{V}\cos\zeta+v^{V}\sin\zeta &amp; =Z\\
v^{I} &amp; =v &amp;  &amp;  &amp; v^{VI} &amp; =z^{V}\sin\zeta-v^{V}\cos\varepsilon\zeta &amp; =V
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=As shown by [[#Minkowski|Minkowski (1907)]], the orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) or ({{equationNote|2b}}) by making one variable as well as six of the sixteen coefficients imaginary.}}

==={{anchor|Euler3}} Wessel (1799) – Euler's formula and rotation===

The above orthogonal transformations representing Euclidean rotations can also be expressed by using [[w:Euler's formula]]. After this formula was derived by Euler in 1748&lt;ref group=M&gt;Euler (1748b), section 138.&lt;/ref&gt;

:&lt;math&gt;e^{+v\sqrt{-1}}=\cos v+\sqrt{-1}\sin v,\quad e^{-v\sqrt{-1}}=\cos v-\sqrt{-1}\sin v&lt;/math&gt;,

it was used by [[w:Caspar Wessel]] (1799) to describe Euclidean rotations in the complex plane:&lt;ref group=M&gt;Wessel (1799), § 28.&lt;/ref&gt;

:&lt;math&gt;x''+\varepsilon z''=(x'+\varepsilon z')\cdot(\cos III+\varepsilon\sin III),\ (\varepsilon=\sqrt{-1})&lt;/math&gt;

{{Lorentzbox|Text=Replacing the real quantities by imaginary ones by setting &lt;math&gt;\left[z',z'',III\right]=\left[iz',iz'',iIII\right]&lt;/math&gt;, Wessel's transformation becomes Lorentz transformation ({{equationNote|2c}}).}}

==={{anchor|Cauchy}} Cauchy (1829) – Orthogonal transformation===

[[w:Augustin-Louis Cauchy]] (1829) extended the orthogonal transformation of [[#Euler|Euler (1771)]] to arbitrary dimensions&lt;ref group=M&gt;Cauchy (1829), eq. 22, 98, 99, 101; Some misprints were corrected in Œuvres complètes, série 2, tome 9, pp. 174–195.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}+\dots=\xi^{2}+\eta^{2}+\zeta^{2}+\dots\\
\hline \begin{align}x &amp; =x_{1}\xi+x_{2}\eta+x_{3}\zeta+\dots\\
y &amp; =y_{1}\xi+y_{2}\eta+y_{3}\zeta+\dots\\
z &amp; =z_{1}\xi+z_{2}\eta+z_{3}\zeta+\dots\\
 &amp; \dots\\
\\
\xi &amp; =x_{1}x+y_{1}y+z_{1}z+\dots\\
\eta &amp; =x_{2}x+y_{2}y+z_{2}z+\dots\\
\zeta &amp; =x_{3}x+y_{3}y+z_{3}z+\dots\\
 &amp; \dots
\end{align}
\left|{\scriptstyle \begin{align}x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+\dots &amp; =1,\\
x_{2}x_{1}+y_{2}y_{1}+z_{2}z_{1}+\dots &amp; =0,\\
\dots\\
x_{n}x_{1}+y_{n}y_{1}+z_{n}z_{1}+\dots &amp; =0,\\
\\
x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+\dots &amp; =0,\\
x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+\dots &amp; =1,\\
\text{ }\dots\\
x_{n}x_{2}+y_{n}y_{2}+z_{n}z_{2}+\dots &amp; =0,\\
\\
x_{1}x_{n}+y_{1}y_{n}+z_{1}z_{n}+\dots &amp; =0,\\
x_{2}x_{n}+y_{2}y_{n}+z_{2}z_{n}+\dots &amp; =0,\\
\dots\\
x_{n}^{2}+y_{n}^{2}+z_{n}^{2}+\dots &amp; =1
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) by making one of the variables as well as certain coefficients imaginary.}}

==={{anchor|Lie}} Lie (1871) – Imaginary orthogonal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Lie3|label 1=History of Lorentz transformations in general § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

[[w:Sophus Lie]] (1871a) described a manifold whose elements can be represented by spheres, where the last coordinate ''y&lt;sub&gt;n+1&lt;/sub&gt;'' can be related to an imaginary radius by ''iy&lt;sub&gt;n+1&lt;/sub&gt;'':&lt;ref group=M&gt;Lie (1871a), pp. 199–209&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\sum_{i=1}^{i=n} (x_i-y_i)^2+y_{n+1}^2=0 \\
\downarrow\\
\sum_{i=1}^{i=n+1} (y_i^{\prime}-y_i^{\prime\prime})^2=0
\end{matrix}&lt;/math&gt;

If the second equation is satisfied, two spheres ''y′'' and ''y″'' are in contact. Lie then defined the correspondence between [[w:Contact geometry|contact transformations]] in ''R&lt;sub&gt;n&lt;/sub&gt;'' and conformal point transformations in ''R&lt;sub&gt;n+1&lt;/sub&gt;'': The sphere of space ''R&lt;sub&gt;n&lt;/sub&gt;'' consists of ''n+1'' parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space ''R&lt;sub&gt;n&lt;/sub&gt;'', it follows that ''R&lt;sub&gt;n&lt;/sub&gt;'' now corresponds to ''R&lt;sub&gt;n+1&lt;/sub&gt;''. Therefore, any transformation (to which he counted [[#Lorentz transformation via orthogonal transformation|orthogonal transformations]] and inversions) leaving invariant the condition of contact between spheres in ''R&lt;sub&gt;n&lt;/sub&gt;'', corresponds to the conformal transformation of points in ''R&lt;sub&gt;n+1&lt;/sub&gt;''.

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Minkowski|label 1=History of Lorentz transformations via velocity § Minkowski}}

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1) in agreement with ({{equationNote|2b}}):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|2b}}), and the relativistc velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|cau29sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48b}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie71a}} 
*{{#section:History of Topics in Special Relativity/mathsource|wes99}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|mink07a}}
{{#section:History of Topics in Special Relativity/relsource|mink07b}}
{{#section:History of Topics in Special Relativity/relsource|mink08}}
{{#section:History of Topics in Special Relativity/relsource|som09}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L2}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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      <text bytes="17751" sha1="jvaur7oajspsv5e2eva3vsdfg794u9g" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via imaginary orthogonal transformation==
By using the [[w:Imaginary unit|imaginary]] quantities &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0}]=\left[ix_{0},\ ix_{0}^{\prime}\right]&lt;/math&gt; in '''x''' as well as &lt;math&gt;[\mathfrak{g}_{0s},\ \mathfrak{g}_{s0}]=\left[ig_{0s},\ ig_{s0}\right]&lt;/math&gt; ''(s=1,2...n)'' in '''g''', the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] assumes the form of an [[w:orthogonal transformation]] of [[w:Euclidean space]] forming the [[w:orthogonal group]] O(n) if det '''g'''=±1 or the special orthogonal group SO(n) if det '''g'''=+1, the Lorentz interval becomes the [[w:Euclidean norm]], and the Minkowski inner product becomes the [[w:dot product]]:&lt;ref&gt;Laue (1921), pp. 79–80 for n=3&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{align}\mathfrak{x}_{0}^{2}+x_{1}^{2}+\cdots+x_{n}^{2} &amp; =\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+\dots+x_{n}^{\prime2}\\
\mathfrak{x}_{0}\mathfrak{y}_{0}+x_{1}y_{1}+\cdots+x_{n}y_{n} &amp; =\mathfrak{x}_{0}^{\prime}\mathfrak{y}_{0}^{\prime}+x_{1}^{\prime}y_{1}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime}
\end{align}
\\
\hline \begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\mathbf{x}=\mathbf{\mathbf{g}^{-1}}\cdot\mathbf{x}'
\end{matrix}\left|\begin{align}\sum_{i=0}^{n}g_{ij}g_{ik} &amp; =\left\{ \begin{align}1\quad &amp; (j=k)\\
0\quad &amp; (j\ne k)
\end{align}
\right.\\
\sum_{j=0}^{n}g_{ij}g_{kj} &amp; =\left\{ \begin{align}1\quad &amp; (i=k)\\
0\quad &amp; (i\ne k)
\end{align}
\right.
\end{align}
\right.
\end{matrix}&lt;/math&gt;|{{equationRef|2a}}}}

The cases ''n=1,2,3,4'' of orthogonal transformations in terms of real coordinates were discussed by [[#Euler|Euler (1771)]] and in ''n'' dimensions by [[#Cauchy|Cauchy (1829)]]. The case in which one of these coordinates is imaginary and the other ones remain real was alluded to by [[#Lie|Lie (1871)]] in terms of spheres with imaginary radius, while the interpretation of the imaginary coordinate as being related to the dimension of time as well as the explicit formulation of Lorentz transformations with ''n=3'' was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]].

A well known example of this orthogonal transformation is spatial [[w:rotation]] in terms of [[w:trigonometric function]]s, which become Lorentz transformations by using an imaginary angle &lt;math&gt;\phi=i\eta&lt;/math&gt;, so that trigonometric functions become equivalent to [[w:hyperbolic function]]s:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}\mathfrak{x}_{0}^{\prime} &amp; =\mathfrak{x}_{0}\cos\phi-x_{1}\sin\phi\\
x_{1}^{\prime} &amp; =\mathfrak{x}_{0}\sin\phi+x_{1}\cos\phi\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
\mathfrak{x}_{0} &amp; =\mathfrak{x}_{0}^{\prime}\cos\phi+x_{1}^{\prime}\sin\phi\\
x_{1} &amp; =-\mathfrak{x}_{0}^{\prime}\sin\phi+x_{1}^{\prime}\cos\phi\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}ix_{0}^{\prime} &amp; =ix_{0}\cos i\eta-x_{1}\sin i\eta\\
x_{1}^{\prime} &amp; =ix_{0}\sin i\eta+x_{1}\cos i\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
ix_{0} &amp; =ix_{0}^{\prime}\cos i\eta+x_{1}^{\prime}\sin i\eta\\
x_{1} &amp; =-ix_{0}^{\prime}\sin i\eta+x_{1}^{\prime}\cos i\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array} &lt;/math&gt;|{{equationRef|2b}}}}

or in exponential form using [[w:Euler's formula]] &lt;math&gt;e^{i\phi}=\cos\phi+i\sin\phi&lt;/math&gt;:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{c|c|cc}
\mathfrak{x}_{0}^{2}+x_{1}^{2}+x_{2}^{2}=\mathfrak{x}_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp; \left(ix_{0}\right){}^{2}+x_{1}^{2}+x_{2}^{2}=\left(ix_{0}^{\prime}\right)^{2}+x_{1}^{\prime2}+x_{2}^{\prime2} &amp;  &amp; -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline (1)\begin{align}x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime} &amp; =e^{-i\phi}\left(x_{1}+i\mathfrak{x}_{0}\right)\\
x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime} &amp; =e^{i\phi}\left(x_{1}-i\mathfrak{x}_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\mathfrak{x}_{0} &amp; =e^{i\phi}\left(x_{1}^{\prime}+i\mathfrak{x}_{0}^{\prime}\right)\\
x_{1}-i\mathfrak{x}_{0} &amp; =e^{-i\phi}\left(x_{1}^{\prime}-i\mathfrak{x}_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; (2)\begin{align}x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right) &amp; =e^{-i(i\eta)}\left(x_{1}+i\left(ix_{0}\right)\right)\\
x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right) &amp; =e^{i(i\eta)}\left(x_{1}-i\left(ix_{0}\right)\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}+i\left(ix_{0}\right) &amp; =e^{i(i\eta)}\left(x_{1}^{\prime}+i\left(ix_{0}^{\prime}\right)\right)\\
x_{1}-i\left(ix_{0}\right) &amp; =e^{-i(i\eta)}\left(x_{1}^{\prime}-i\left(ix_{0}^{\prime}\right)\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
 &amp; \rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\end{array}&lt;/math&gt;|{{equationRef|2c}}}}

Defining &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; as real, spatial rotation in the form ({{equationNote|2b}}-1) was introduced by [[#Euler2|Euler (1771)]] and in the form ({{equationNote|2c}}-1) by [[#Euler3|Wessel (1799)]]. The interpretation of ({{equationNote|2b}}) as Lorentz boost (i.e. Lorentz transformation ''without'' spatial rotation) in which &lt;math&gt;[\mathfrak{x}_{0},\ \mathfrak{x}'_{0},\ \phi]&lt;/math&gt; correspond to the imaginary quantities &lt;math&gt;[ix_{0},\ ix'_{0},\ i\eta]&lt;/math&gt; was given by [[#Minkowski|Minkowski (1907)]] and [[#Sommerfeld|Sommerfeld (1909)]]. As shown in the next section using hyperbolic functions, ({{equationNote|2b}}) becomes [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] while ({{equationNote|2c}}) becomes [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']].

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Orthogonal transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Euler|label 1=History of Lorentz transformations via hyperbolic functions § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Euler|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Leonhard Euler]] (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]], as well as under rotations using [[w:Euler angles]]. The case of two dimensions is given by&lt;ref group=M&gt;Euler (1771), pp. 84-85&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}=x^{2}+y^{2}\\
\hline \begin{align}X &amp; =\alpha x+\beta y\\
Y &amp; =\gamma x+\delta y
\end{align}
\left|\begin{matrix}\begin{align}1 &amp; =\alpha\alpha+\gamma\gamma\\
1 &amp; =\beta\beta+\delta\delta\\
0 &amp; =\alpha\beta+\gamma\delta
\end{align}
\end{matrix}\right.\\
\hline \begin{align}X &amp; =x\cos\zeta+y\sin\zeta\\
Y &amp; =x\sin\zeta-y\cos\zeta
\end{align}
\end{matrix}&lt;/math&gt;

or three dimensions&lt;ref group=M&gt;Euler (1771), pp. 77, 85-89&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\\
\hline \begin{align}x' &amp; =x\cos\zeta+y\sin\zeta &amp; x'' &amp; =x'\cos\eta+z'\sin\eta\\
y' &amp; =x\sin\zeta-y\cos\zeta &amp; y'' &amp; =y'\\
z' &amp; =z &amp; z'' &amp; =x'\sin\eta-z'\cos\eta\\
\\
x''' &amp; =x'' &amp; =X\\
y''' &amp; =y''\cos\theta+z''\sin\theta &amp; =Y\\
z''' &amp; =y''\sin\theta-z''\cos\theta &amp; =Z
\end{align}
\end{matrix}&lt;/math&gt;

The orthogonal transformation in four dimensions was given by him as&lt;ref group=M&gt;Euler (1771), pp. 89–91&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}V^{2}+X^{2}+Y^{2}+Z^{2}=v^{2}+x^{2}+y^{2}+z^{2}\\
\hline \begin{align}V &amp; =Av+Bx+Cy+Dz\\
X &amp; =Ev+Fx+Gy+Hz\\
Y &amp; =Iv+Kx+Ly+Mz\\
Z &amp; =Nv+Ox+Py+Qz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+RR+II+NN &amp; 0 &amp; =AB+EF+IK+NO\\
1 &amp; =BB+FF+KK+OO &amp; 0 &amp; =AC+EG+IL+NP\\
1 &amp; =CC+GG+LL+PP &amp; 0 &amp; =AD+EH+IM+NQ\\
1 &amp; =DD+HH+MM+QQ &amp; 0 &amp; =BC+FG+KL+OP\\
0 &amp; =BD+FH+KM+OQ &amp; 0 &amp; =CD+FH+LM+PQ
\end{align}
}\right.\end{matrix}\\
\hline {\scriptstyle \begin{align}x^{I} &amp; =x\cos\alpha+y\sin\alpha &amp;  &amp;  &amp; x^{VI} &amp; =x^{V} &amp; =X\\
y^{I} &amp; =x\sin\alpha-y\cos\alpha &amp;  &amp;  &amp; y^{VI} &amp; =y^{V} &amp; =Y\\
z^{I} &amp; =z &amp; \dots &amp; \dots &amp; y^{VI} &amp; =z^{V}\cos\zeta+v^{V}\sin\zeta &amp; =Z\\
v^{I} &amp; =v &amp;  &amp;  &amp; v^{VI} &amp; =z^{V}\sin\zeta-v^{V}\cos\varepsilon\zeta &amp; =V
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=As shown by [[#Minkowski|Minkowski (1907)]], the orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) or ({{equationNote|2b}}) by making one variable as well as six of the sixteen coefficients imaginary.}}

==={{anchor|Euler3}} Wessel (1799) – Euler's formula and rotation===

The above orthogonal transformations representing Euclidean rotations can also be expressed by using [[w:Euler's formula]]. After this formula was derived by Euler in 1748&lt;ref group=M&gt;Euler (1748b), section 138.&lt;/ref&gt;

:&lt;math&gt;e^{+v\sqrt{-1}}=\cos v+\sqrt{-1}\sin v,\quad e^{-v\sqrt{-1}}=\cos v-\sqrt{-1}\sin v&lt;/math&gt;,

it was used by [[w:Caspar Wessel]] (1799) to describe Euclidean rotations in the complex plane:&lt;ref group=M&gt;Wessel (1799), § 28.&lt;/ref&gt;

:&lt;math&gt;x''+\varepsilon z''=(x'+\varepsilon z')\cdot(\cos III+\varepsilon\sin III),\ (\varepsilon=\sqrt{-1})&lt;/math&gt;

{{Lorentzbox|Text=Replacing the real quantities by imaginary ones by setting &lt;math&gt;\left[z',z'',III\right]=\left[iz',iz'',iIII\right]&lt;/math&gt;, Wessel's transformation becomes Lorentz transformation ({{equationNote|2c}}).}}

==={{anchor|Cauchy}} Cauchy (1829) – Orthogonal transformation===

[[w:Augustin-Louis Cauchy]] (1829) extended the orthogonal transformation of [[#Euler|Euler (1771)]] to arbitrary dimensions&lt;ref group=M&gt;Cauchy (1829), eq. 22, 98, 99, 101; Some misprints were corrected in Œuvres complètes, série 2, tome 9, pp. 174–195.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}+\dots=\xi^{2}+\eta^{2}+\zeta^{2}+\dots\\
\hline \begin{align}x &amp; =x_{1}\xi+x_{2}\eta+x_{3}\zeta+\dots\\
y &amp; =y_{1}\xi+y_{2}\eta+y_{3}\zeta+\dots\\
z &amp; =z_{1}\xi+z_{2}\eta+z_{3}\zeta+\dots\\
 &amp; \dots\\
\\
\xi &amp; =x_{1}x+y_{1}y+z_{1}z+\dots\\
\eta &amp; =x_{2}x+y_{2}y+z_{2}z+\dots\\
\zeta &amp; =x_{3}x+y_{3}y+z_{3}z+\dots\\
 &amp; \dots
\end{align}
\left|{\scriptstyle \begin{align}x_{1}^{2}+y_{1}^{2}+z_{1}^{2}+\dots &amp; =1,\\
x_{2}x_{1}+y_{2}y_{1}+z_{2}z_{1}+\dots &amp; =0,\\
\dots\\
x_{n}x_{1}+y_{n}y_{1}+z_{n}z_{1}+\dots &amp; =0,\\
\\
x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}+\dots &amp; =0,\\
x_{2}^{2}+y_{2}^{2}+z_{2}^{2}+\dots &amp; =1,\\
\text{ }\dots\\
x_{n}x_{2}+y_{n}y_{2}+z_{n}z_{2}+\dots &amp; =0,\\
\\
x_{1}x_{n}+y_{1}y_{n}+z_{1}z_{n}+\dots &amp; =0,\\
x_{2}x_{n}+y_{2}y_{n}+z_{2}z_{n}+\dots &amp; =0,\\
\dots\\
x_{n}^{2}+y_{n}^{2}+z_{n}^{2}+\dots &amp; =1
\end{align}
}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The orthogonal transformation can be directly used as Lorentz transformation ({{equationNote|2a}}) by making one of the variables as well as certain coefficients imaginary.}}

==={{anchor|Lie}} Lie (1871) – Imaginary orthogonal transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Lie3|label 1=History of Lorentz transformations in general § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lie2|label 1=History of Lorentz transformations via squeeze mappings § Lie}}

[[w:Sophus Lie]] (1871a) described a manifold whose elements can be represented by spheres, where the last coordinate ''y&lt;sub&gt;n+1&lt;/sub&gt;'' can be related to an imaginary radius by ''iy&lt;sub&gt;n+1&lt;/sub&gt;'':&lt;ref group=M&gt;Lie (1871a), pp. 199–209&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\sum_{i=1}^{i=n} (x_i-y_i)^2+y_{n+1}^2=0 \\
\downarrow\\
\sum_{i=1}^{i=n+1} (y_i^{\prime}-y_i^{\prime\prime})^2=0
\end{matrix}&lt;/math&gt;

If the second equation is satisfied, two spheres ''y′'' and ''y″'' are in contact. Lie then defined the correspondence between [[w:Contact geometry|contact transformations]] in ''R&lt;sub&gt;n&lt;/sub&gt;'' and conformal point transformations in ''R&lt;sub&gt;n+1&lt;/sub&gt;'': The sphere of space ''R&lt;sub&gt;n&lt;/sub&gt;'' consists of ''n+1'' parameter (coordinates plus imaginary radius), so if this sphere is taken as the element of space ''R&lt;sub&gt;n&lt;/sub&gt;'', it follows that ''R&lt;sub&gt;n&lt;/sub&gt;'' now corresponds to ''R&lt;sub&gt;n+1&lt;/sub&gt;''. Therefore, any transformation (to which he counted [[#Lorentz transformation via orthogonal transformation|orthogonal transformations]] and inversions) leaving invariant the condition of contact between spheres in ''R&lt;sub&gt;n&lt;/sub&gt;'', corresponds to the conformal transformation of points in ''R&lt;sub&gt;n+1&lt;/sub&gt;''.

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Minkowski|label 1=History of Lorentz transformations via velocity § Minkowski}}

The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation assumes a form (with ''c''=1) in agreement with ({{equationNote|2b}}):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
Using an imaginary rapidity such as Minkowski, [[w:Arnold Sommerfeld]] (1909) formulated a transformation equivalent to Lorentz boost ({{equationNote|2b}}), and the relativistc velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in terms of trigonometric functions and the [[w:spherical law of cosines]]:&lt;ref group=R&gt;Sommerfeld (1909), p. 826ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{array}{lrl}
x'= &amp; x\ \cos\varphi+l\ \sin\varphi, &amp; y'=y\\
l'= &amp; -x\ \sin\varphi+l\ \cos\varphi, &amp; z'=z
\end{array}\right\} \\
\left(\operatorname{tg}\varphi=i\beta,\ \cos\varphi=\frac{1}{\sqrt{1-\beta^{2}}},\ \sin\varphi=\frac{i\beta}{\sqrt{1-\beta^{2}}}\right)\\
\hline \beta=\frac{1}{i}\operatorname{tg}\left(\varphi_{1}+\varphi_{2}\right)=\frac{1}{i}\frac{\operatorname{tg}\varphi_{1}+\operatorname{tg}\varphi_{2}}{1-\operatorname{tg}\varphi_{1}\operatorname{tg}\varphi_{2}}=\frac{\beta_{1}+\beta_{2}}{1+\beta_{1}\beta_{2}}\\
\cos\varphi=\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1}\sin\varphi_{2}\cos\alpha\\
v^{2}=\frac{v_{1}^{2}+v_{2}^{2}+2v_{1}v_{2}\cos\alpha-\frac{1}{c^{2}}v_{1}^{2}v_{2}^{2}\sin^{2}\alpha}{\left(1+\frac{1}{c^{2}}v_{1}v_{2}\cos\alpha\right)^{2}}
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|cau29sec}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48b}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie71a}} 
*{{#section:History of Topics in Special Relativity/mathsource|wes99}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|mink07a}}
{{#section:History of Topics in Special Relativity/relsource|mink07b}}
{{#section:History of Topics in Special Relativity/relsource|mink08}}
{{#section:History of Topics in Special Relativity/relsource|som09}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L2}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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    <title>History of Topics in Special Relativity/Lorentz transformation (hyperbolic)</title>
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    <revision>
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      <comment>/* Lorentz transformation via hyperbolic functions */</comment>
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      <text bytes="57595" sha1="iov9o2macpo6xtkkcf56gw3o14wo8j2" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A ray through the unit hyperbola {{math|1=''x''&lt;sup&gt;2&lt;/sup&gt; − ''y''&lt;sup&gt;2&lt;/sup&gt; = 1}} at the point {{math|(cosh ''a'', sinh ''a'')}}.]]
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\scriptstyle\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}v}{\sqrt{1-v^{2}}}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\scriptstyle\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Goursat|Goursat (1888)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Using the idendity &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)
\end{align}
\quad\begin{align}x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)
\end{align}
\end{matrix}\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}} 

Lorentz transformations ({{equationNote|3c}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while a form similar to &lt;math&gt;k=\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; was given by [[#Lipschitz1|Lipschitz (1885/86)]], and the exponential form was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1}=\sinh\left(\eta_{2}-\eta\right)=\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1}=\cosh\left(\eta_{2}-\eta\right)=-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2}=\sinh\left(\eta_{1}+\eta\right)=\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2}=\cosh\left(\eta_{1}+\eta\right)=\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;{\scriptstyle \begin{align}\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{2}-\eta\right) &amp; \sinh\left(\eta_{2}-\eta\right)\\
\sinh\left(\eta_{2}-\eta\right) &amp; \cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{1}+\eta\right) &amp; \sinh\left(\eta_{1}+\eta\right)\\
\sinh\left(\eta_{1}+\eta\right) &amp; \cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}
\end{align}
}&lt;/math&gt;

or in exponential form as squeeze mapping analogous to ({{equationNote|3c}}): 

&lt;math&gt;\begin{align}e^{-\eta_{1}} &amp; =e^{\eta}e^{-\eta_{2}}=e^{\eta-\eta_{2}} &amp; e^{-\eta_{2}} &amp; =e^{-\eta}e^{-\eta_{1}}=e^{-\eta_{1}-\eta}\\
e^{\eta_{1}} &amp; =e^{-\eta}e^{\eta_{2}}=e^{\eta_{2}-\eta} &amp; e^{\eta_{2}} &amp; =e^{\eta}e^{\eta_{1}}=e^{\eta_{1}+\eta}
\end{align}
&lt;/math&gt;|{{equationRef|3d}}}}

Hyperbolic angle sum laws were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]] and many others, while matrix representations were given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther1|Günther (1880/81)]].

===Hyperbolic law of cosines===
By adding coordinates &lt;math&gt;x_{2}^{\prime}=x_{2}&lt;/math&gt; and &lt;math&gt;x_{3}^{\prime}=x_{3}&lt;/math&gt; in Lorentz transformation ({{equationNote|3b}}) and interpreting &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; as [[w:homogeneous coordinates]], the Lorentz transformation can be rewritten in line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] by using coordinates &lt;math&gt;[u_{1},\ u_{2},\ u_{3}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}}\right]&lt;/math&gt; defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\le1&lt;/math&gt; inside the [[w:unit sphere]] as follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
\hline \\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
u_{3}^{\prime} &amp; =\frac{u_{3}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
\hline \\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}\\
u_{3} &amp; =\frac{u_{3}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]], [[#Goursat|Goursat (1888)]], [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]], [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. This transformation becomes equivalent to the [[w:hyperbolic law of cosines]] by restriction to coordinates of the &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;-plane and &lt;math&gt;\left[u'_{1},u'_{2}\right]&lt;/math&gt;-plane and defining their scalar products in terms of trigonometric and hyperbolic identities:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{align}u_{1}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u_{1}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u_{2}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u_{2}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\frac{u_{2}}{u_{1}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \frac{u_{2}^{\prime}}{u_{1}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (B)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (A)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (A) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (B) was given by [[#Schur|Schur (1900/02)]]. By further setting &lt;math&gt;\tanh\xi=\tanh\zeta=1&lt;/math&gt; or &lt;math&gt;u'=u=1&lt;/math&gt; it follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}(A)  &amp; \ \cos\alpha=\frac{\cos\alpha'+\tanh\eta}{1+\tanh\eta\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{1+\tanh\eta\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{\cos\alpha'+\tanh\eta}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-\tanh\eta}{1+\tanh\eta}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-\tanh\eta}{1-\tanh\eta\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{1-\tanh\eta\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{\cos\alpha-\tanh\eta}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}\tan\frac{\alpha}{2}\\
\\
(B)  &amp; \ \cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}
&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3g-B) are the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity these formulas describe the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#mercator|label 1=History of Lorentz transformations via squeeze mappings § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated the following relations on a rectangular hyperbola:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations &lt;math&gt;1+a=c&lt;/math&gt;, &lt;math&gt;\sqrt{2a+a^{2}}=d&lt;/math&gt; with &lt;math&gt;c^{2}-d^{2}=1&lt;/math&gt; implicitly correspond to hyperbolic functions &lt;math&gt;c=\cosh\eta&lt;/math&gt;, &lt;math&gt;d=\sinh\eta&lt;/math&gt; with &lt;math&gt;\cosh^{2}\eta-\sinh^{2}\eta=1&lt;/math&gt; (which were explicitly introduced by [[#Riccati|Riccati (1757)]] much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|3c}}) because:&lt;br&gt;
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta}AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta}BI&lt;/math&gt;.&lt;br&gt;
Furthermore, transforming Mercator's asymptotic coordinates &lt;math&gt;AH=\tfrac{c+d}{\sqrt{2}}&lt;/math&gt;, &lt;math&gt;FH=\tfrac{c-d}{\sqrt{2}}&lt;/math&gt; into Cartesian coordinates &lt;math&gt;x_{0},x_{1}&lt;/math&gt; gives:&lt;br&gt;
:&lt;math&gt;x_{1}=\tfrac{AH+FH}{\sqrt{2}}=c=\cosh\eta,\quad x_{0}=\tfrac{AH-FH}{\sqrt{2}}=d=\sinh\eta&lt;/math&gt;&lt;br&gt;
which produces the unit hyperbola &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=1&lt;/math&gt; as in ({{equationNote|3d}}), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.}}

==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Euler|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

{{Lorentzbox|Text= These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3e}}) follows by setting &lt;math&gt;[\cos z,\sin z, \cos y,\sin y]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u, \cos x,\sin x]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Lagrange's formulas.}}

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===
[[File:Riccati-hyperbola.png|thumb|&lt;small&gt;Riccati's (1757) illustration of hyperbolic addition laws.&lt;/small&gt;]]
[[w:Vincenzo Riccati]] (1757) introduced hyperbolic functions ''cosh'' and ''sinh'', which he denoted as ''Ch.'' and ''Sh.'' related by &lt;math&gt;Ch.^{2}-Sh.^{2}=r^2&lt;/math&gt; with ''r'' being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}CA=r,\ CB=Ch.\varphi,\ BE=Sh.\varphi,\ CD=Ch.\pi,\ DF=Sh.\pi\\
CM=Ch.\overline{\varphi+\pi},\ MN=Sh.\overline{\varphi+\pi}\\
CK=\frac{r}{\sqrt{2}},\ CG=\frac{Ch.\varphi+Sh.\varphi}{\sqrt{2}},\ CH=\frac{Ch.\pi+Sh.\pi}{\sqrt{2}},\ CP=\frac{Ch.\overline{\varphi+\pi}+Sh.\overline{\varphi+\pi}}{\sqrt{2}}\\
CK:CG::CH:CP\\
\left[Ch.^{2}-Sh.^{2}=rr\right]\\
\hline Ch.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Ch.\pi+Sh.\varphi\,Sh.\pi}{r}\\
Sh.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Sh.\pi+Ch.\pi\,Sh.\varphi}{r}
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;Ch.\overline{\varphi-\pi}&lt;/math&gt; and &lt;math&gt;Sh.\overline{\varphi-\pi}&lt;/math&gt; follow by setting &lt;math&gt;Ch.\pi\Rightarrow Ch.-\pi&lt;/math&gt; and &lt;math&gt;Sh.\pi\Rightarrow Sh.-\pi&lt;/math&gt; in the above formulas.

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;\pi=\eta,\ \varphi=\eta_1,\ \overline{\varphi+\pi}=\eta_2&lt;/math&gt;.}}

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].}}

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}).}}

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

{{Lorentzbox|Text=When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].}}

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.}}

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Laisant1|label 1=History of Lorentz transformations via squeeze mappings § Laisant}}

In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

{{Lorentzbox|Text=These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).}}

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;}}

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be expressed by matrix multiplication:&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\text{by multiplication:}\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\\
\text{where}\ \left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right] \\
\Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y\end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Gunther1|label 1=History of Lorentz transformations via squeeze mappings § Günther}}

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) expressed the hyperbolic addition laws by matrix multiplication:&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}\,x, &amp; \mathfrak{Sin}\,x\\
\mathfrak{Sin}\,x, &amp; \mathfrak{Cos}\,x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y+\mathfrak{Sin}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,x\,\mathfrak{Sin}\,y+\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y\\
\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y+\mathfrak{Cos}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Sin}\,x\,\mathfrak{Sin}\,y+\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,(x+y), &amp; \mathfrak{Sin}\,(x+y)\\
\mathfrak{Sin}\,(x+y), &amp; \mathfrak{Cos}\,(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y \end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

=== {{anchor|Cox}} Cox (1881/82) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Cox|label 1=History of Lorentz transformations in general § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|w:Homersham Cox]] (1881/82) defined the case of translation in the hyperbolic plane with the ''y''-axis remaining unchanged:&lt;ref group=M name=cox&gt;Cox (1881/82), p. 194&lt;/ref&gt;
	
:&lt;math&gt;\begin{align}X &amp; =x\cosh p-z\sinh p\\	
Z &amp; =-x\sinh p+z\cosh p	\\ \\
x &amp; =X\cosh p+Z\sinh p\\
z &amp; =X\sinh p+Z\cosh p	
\end{align}	
&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) – Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lipschitz1|label 1=History of Lorentz transformations via squeeze mappings § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts ({{equationNote|3b}}-C) and ({{equationNote|3c}}) by the identity &lt;math&gt;s_{a}=\tfrac{1}{v}=\coth\eta&lt;/math&gt;. That is, by substituting &lt;math&gt;v=\tfrac{1}{s_{a}}&lt;/math&gt; in ({{equationNote|3b}}-C) or ({{equationNote|3c}}) we obtain Lipschitz's transformations.}}

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.}}

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.}}

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.}}

==={{Anchor|Goursat}} Goursat (1887/88) – Minimal surfaces===

[[w:Édouard Goursat]] defined real coordinates &lt;math&gt;x,y&lt;/math&gt; of minimal surface &lt;math&gt;S&lt;/math&gt; and imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; of the adjoint minimal surface &lt;math&gt;S_0&lt;/math&gt;, so that another real minimal surface &lt;math&gt;S_1&lt;/math&gt; follows by the (conformal) transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 144&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =\frac{1+k^{2}}{2k}x-\frac{k^{2}-1}{2k}y_{0}\\
y_{1} &amp; =\frac{1+k^{2}}{2k}y+\frac{k^{2}-1}{2k}x_{0}\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

and expressed these equations in terms of hyperbolic functions by setting &lt;math&gt;k=e^{\varphi}&lt;/math&gt;:&lt;ref group=M&gt;Goursat (1887/88), p. 145&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =x\cosh\varphi-y_{0}\sinh\varphi\\
y_{1} &amp; =y\cosh\varphi+x_{0}\sinh\varphi\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|3b}}) by replacing the imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; by real coordinates defined as &lt;math&gt;[x_{0},y_{0}]=[-x,y]&lt;/math&gt;. It can also be seen that Goursat's relation &lt;math&gt;k=e^{\varphi}&lt;/math&gt; corresponds to &lt;math&gt;k=e^{\eta}&lt;/math&gt; defined in ({{equationNote|3c}}).}}

He went on to define &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; as the direction cosines normal to surface &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;\alpha_{1},\beta_{1},\gamma_{1}&lt;/math&gt; as the ones normal to surface &lt;math&gt;S_{1}&lt;/math&gt;, connected by the transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 149f.&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha_{1} &amp; =\pm\frac{\alpha}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \alpha &amp; =\pm\frac{\alpha_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\beta_{1} &amp; =\pm\frac{\beta}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \beta &amp; =\pm\frac{\beta_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\gamma_{1} &amp; =\pm\frac{\gamma\cosh\varphi-\sinh\varphi}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \gamma &amp; =\pm\frac{\gamma_{1}\cosh\varphi+\sinh\varphi}{\cosh\varphi+\gamma_{1}\sinh\varphi}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}-A) with &lt;math&gt;\left[\alpha,\beta,\gamma\right]=\left[u_{2},u_{3},u_{1}\right]&lt;/math&gt;.}}

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lindemann|label 1=History of Lorentz transformations via squeeze mappings § Lindemann}}

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gerard|label 1=History of Lorentz transformations in general § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Killing|label 1=History of Lorentz transformations in general § Killing}}

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Elliott|label 1=History of Lorentz transformations via squeeze mappings § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3c}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.}}

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Liebmann|label 1=History of Lorentz transformations in general § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Herglotz1|label 1=History of Lorentz transformations via velocity § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}).}}

==={{anchor|Varicak}} Varićak (1910) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Varicak|label 1=History of Lorentz transformations via trigonometric functions § Varicak}}

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3f}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gour88}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>iov9o2macpo6xtkkcf56gw3o14wo8j2</sha1>
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        <id>52339</id>
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      <comment>/* Lorentz transformation via hyperbolic functions */</comment>
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      <text bytes="57596" sha1="5trtul0m2zlaaxw5a4u1v2n4gls4vxo" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A ray through the unit hyperbola {{math|1=''x''&lt;sup&gt;2&lt;/sup&gt; − ''y''&lt;sup&gt;2&lt;/sup&gt; = 1}} at the point {{math|(cosh ''a'', sinh ''a'')}}.]]
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\scriptstyle\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}v}{\sqrt{1-v^{2}}}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\scriptstyle\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Goursat|Goursat (1888)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Using the idendity &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)
\end{align}
\quad\begin{align}x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)
\end{align}
\end{matrix}\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}} 

Lorentz transformations ({{equationNote|3c}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while a form similar to &lt;math&gt;k=\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; was given by [[#Lipschitz1|Lipschitz (1885/86)]], and the exponential form was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1}=\sinh\left(\eta_{2}-\eta\right)=\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1}=\cosh\left(\eta_{2}-\eta\right)=-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2}=\sinh\left(\eta_{1}+\eta\right)=\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2}=\cosh\left(\eta_{1}+\eta\right)=\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;{\scriptstyle \begin{align}\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{2}-\eta\right) &amp; \sinh\left(\eta_{2}-\eta\right)\\
\sinh\left(\eta_{2}-\eta\right) &amp; \cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{1}+\eta\right) &amp; \sinh\left(\eta_{1}+\eta\right)\\
\sinh\left(\eta_{1}+\eta\right) &amp; \cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}
\end{align}
}&lt;/math&gt;

or in exponential form as squeeze mapping analogous to ({{equationNote|3c}}): 

&lt;math&gt;\begin{align}e^{-\eta_{1}} &amp; =e^{\eta}e^{-\eta_{2}}=e^{\eta-\eta_{2}} &amp; e^{-\eta_{2}} &amp; =e^{-\eta}e^{-\eta_{1}}=e^{-\eta_{1}-\eta}\\
e^{\eta_{1}} &amp; =e^{-\eta}e^{\eta_{2}}=e^{\eta_{2}-\eta} &amp; e^{\eta_{2}} &amp; =e^{\eta}e^{\eta_{1}}=e^{\eta_{1}+\eta}
\end{align}
&lt;/math&gt;|{{equationRef|3d}}}}

Hyperbolic angle sum laws were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]] and many others, while matrix representations were given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther1|Günther (1880/81)]].

===Hyperbolic law of cosines===
By adding coordinates &lt;math&gt;x_{2}^{\prime}=x_{2}&lt;/math&gt; and &lt;math&gt;x_{3}^{\prime}=x_{3}&lt;/math&gt; in Lorentz transformation ({{equationNote|3b}}) and interpreting &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; as [[w:homogeneous coordinates]], the Lorentz transformation can be rewritten in line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] by using coordinates &lt;math&gt;[u_{1},\ u_{2},\ u_{3}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}}\right]&lt;/math&gt; defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\le1&lt;/math&gt; inside the [[w:unit sphere]] as follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
\hline \\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
u_{3}^{\prime} &amp; =\frac{u_{3}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
\hline \\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}\\
u_{3} &amp; =\frac{u_{3}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]], [[#Goursat|Goursat (1888)]], [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]], [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. This transformation becomes equivalent to the [[w:hyperbolic law of cosines]] by restriction to coordinates of the &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;-plane and &lt;math&gt;\left[u'_{1},u'_{2}\right]&lt;/math&gt;-plane and defining their scalar products in terms of trigonometric and hyperbolic identities:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{align}u_{1}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u_{1}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u_{2}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u_{2}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\frac{u_{2}}{u_{1}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \frac{u_{2}^{\prime}}{u_{1}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (B)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (A)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (A) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (B) was given by [[#Schur|Schur (1900/02)]]. By further setting &lt;math&gt;\tanh\xi=\tanh\zeta=1&lt;/math&gt; or &lt;math&gt;u'=u=1&lt;/math&gt; it follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}(A)  &amp; \ \cos\alpha=\frac{\cos\alpha'+\tanh\eta}{1+\tanh\eta\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{1+\tanh\eta\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{\cos\alpha'+\tanh\eta}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-\tanh\eta}{1+\tanh\eta}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-\tanh\eta}{1-\tanh\eta\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{1-\tanh\eta\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{\cos\alpha-\tanh\eta}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}\tan\frac{\alpha}{2}\\
\\
(B)  &amp; \ \cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix} 
&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3g-B) are the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity these formulas describe the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#mercator|label 1=History of Lorentz transformations via squeeze mappings § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated the following relations on a rectangular hyperbola:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations &lt;math&gt;1+a=c&lt;/math&gt;, &lt;math&gt;\sqrt{2a+a^{2}}=d&lt;/math&gt; with &lt;math&gt;c^{2}-d^{2}=1&lt;/math&gt; implicitly correspond to hyperbolic functions &lt;math&gt;c=\cosh\eta&lt;/math&gt;, &lt;math&gt;d=\sinh\eta&lt;/math&gt; with &lt;math&gt;\cosh^{2}\eta-\sinh^{2}\eta=1&lt;/math&gt; (which were explicitly introduced by [[#Riccati|Riccati (1757)]] much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|3c}}) because:&lt;br&gt;
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta}AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta}BI&lt;/math&gt;.&lt;br&gt;
Furthermore, transforming Mercator's asymptotic coordinates &lt;math&gt;AH=\tfrac{c+d}{\sqrt{2}}&lt;/math&gt;, &lt;math&gt;FH=\tfrac{c-d}{\sqrt{2}}&lt;/math&gt; into Cartesian coordinates &lt;math&gt;x_{0},x_{1}&lt;/math&gt; gives:&lt;br&gt;
:&lt;math&gt;x_{1}=\tfrac{AH+FH}{\sqrt{2}}=c=\cosh\eta,\quad x_{0}=\tfrac{AH-FH}{\sqrt{2}}=d=\sinh\eta&lt;/math&gt;&lt;br&gt;
which produces the unit hyperbola &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=1&lt;/math&gt; as in ({{equationNote|3d}}), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.}}

==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Euler|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

{{Lorentzbox|Text= These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3e}}) follows by setting &lt;math&gt;[\cos z,\sin z, \cos y,\sin y]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u, \cos x,\sin x]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Lagrange's formulas.}}

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===
[[File:Riccati-hyperbola.png|thumb|&lt;small&gt;Riccati's (1757) illustration of hyperbolic addition laws.&lt;/small&gt;]]
[[w:Vincenzo Riccati]] (1757) introduced hyperbolic functions ''cosh'' and ''sinh'', which he denoted as ''Ch.'' and ''Sh.'' related by &lt;math&gt;Ch.^{2}-Sh.^{2}=r^2&lt;/math&gt; with ''r'' being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}CA=r,\ CB=Ch.\varphi,\ BE=Sh.\varphi,\ CD=Ch.\pi,\ DF=Sh.\pi\\
CM=Ch.\overline{\varphi+\pi},\ MN=Sh.\overline{\varphi+\pi}\\
CK=\frac{r}{\sqrt{2}},\ CG=\frac{Ch.\varphi+Sh.\varphi}{\sqrt{2}},\ CH=\frac{Ch.\pi+Sh.\pi}{\sqrt{2}},\ CP=\frac{Ch.\overline{\varphi+\pi}+Sh.\overline{\varphi+\pi}}{\sqrt{2}}\\
CK:CG::CH:CP\\
\left[Ch.^{2}-Sh.^{2}=rr\right]\\
\hline Ch.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Ch.\pi+Sh.\varphi\,Sh.\pi}{r}\\
Sh.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Sh.\pi+Ch.\pi\,Sh.\varphi}{r}
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;Ch.\overline{\varphi-\pi}&lt;/math&gt; and &lt;math&gt;Sh.\overline{\varphi-\pi}&lt;/math&gt; follow by setting &lt;math&gt;Ch.\pi\Rightarrow Ch.-\pi&lt;/math&gt; and &lt;math&gt;Sh.\pi\Rightarrow Sh.-\pi&lt;/math&gt; in the above formulas.

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;\pi=\eta,\ \varphi=\eta_1,\ \overline{\varphi+\pi}=\eta_2&lt;/math&gt;.}}

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].}}

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}).}}

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

{{Lorentzbox|Text=When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].}}

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.}}

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Laisant1|label 1=History of Lorentz transformations via squeeze mappings § Laisant}}

In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

{{Lorentzbox|Text=These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).}}

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;}}

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be expressed by matrix multiplication:&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\text{by multiplication:}\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\\
\text{where}\ \left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right] \\
\Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y\end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Gunther1|label 1=History of Lorentz transformations via squeeze mappings § Günther}}

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) expressed the hyperbolic addition laws by matrix multiplication:&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}\,x, &amp; \mathfrak{Sin}\,x\\
\mathfrak{Sin}\,x, &amp; \mathfrak{Cos}\,x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y+\mathfrak{Sin}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,x\,\mathfrak{Sin}\,y+\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y\\
\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y+\mathfrak{Cos}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Sin}\,x\,\mathfrak{Sin}\,y+\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,(x+y), &amp; \mathfrak{Sin}\,(x+y)\\
\mathfrak{Sin}\,(x+y), &amp; \mathfrak{Cos}\,(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y \end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

=== {{anchor|Cox}} Cox (1881/82) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Cox|label 1=History of Lorentz transformations in general § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|w:Homersham Cox]] (1881/82) defined the case of translation in the hyperbolic plane with the ''y''-axis remaining unchanged:&lt;ref group=M name=cox&gt;Cox (1881/82), p. 194&lt;/ref&gt;
	
:&lt;math&gt;\begin{align}X &amp; =x\cosh p-z\sinh p\\	
Z &amp; =-x\sinh p+z\cosh p	\\ \\
x &amp; =X\cosh p+Z\sinh p\\
z &amp; =X\sinh p+Z\cosh p	
\end{align}	
&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) – Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lipschitz1|label 1=History of Lorentz transformations via squeeze mappings § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts ({{equationNote|3b}}-C) and ({{equationNote|3c}}) by the identity &lt;math&gt;s_{a}=\tfrac{1}{v}=\coth\eta&lt;/math&gt;. That is, by substituting &lt;math&gt;v=\tfrac{1}{s_{a}}&lt;/math&gt; in ({{equationNote|3b}}-C) or ({{equationNote|3c}}) we obtain Lipschitz's transformations.}}

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.}}

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.}}

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.}}

==={{Anchor|Goursat}} Goursat (1887/88) – Minimal surfaces===

[[w:Édouard Goursat]] defined real coordinates &lt;math&gt;x,y&lt;/math&gt; of minimal surface &lt;math&gt;S&lt;/math&gt; and imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; of the adjoint minimal surface &lt;math&gt;S_0&lt;/math&gt;, so that another real minimal surface &lt;math&gt;S_1&lt;/math&gt; follows by the (conformal) transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 144&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =\frac{1+k^{2}}{2k}x-\frac{k^{2}-1}{2k}y_{0}\\
y_{1} &amp; =\frac{1+k^{2}}{2k}y+\frac{k^{2}-1}{2k}x_{0}\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

and expressed these equations in terms of hyperbolic functions by setting &lt;math&gt;k=e^{\varphi}&lt;/math&gt;:&lt;ref group=M&gt;Goursat (1887/88), p. 145&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =x\cosh\varphi-y_{0}\sinh\varphi\\
y_{1} &amp; =y\cosh\varphi+x_{0}\sinh\varphi\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|3b}}) by replacing the imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; by real coordinates defined as &lt;math&gt;[x_{0},y_{0}]=[-x,y]&lt;/math&gt;. It can also be seen that Goursat's relation &lt;math&gt;k=e^{\varphi}&lt;/math&gt; corresponds to &lt;math&gt;k=e^{\eta}&lt;/math&gt; defined in ({{equationNote|3c}}).}}

He went on to define &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; as the direction cosines normal to surface &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;\alpha_{1},\beta_{1},\gamma_{1}&lt;/math&gt; as the ones normal to surface &lt;math&gt;S_{1}&lt;/math&gt;, connected by the transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 149f.&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha_{1} &amp; =\pm\frac{\alpha}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \alpha &amp; =\pm\frac{\alpha_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\beta_{1} &amp; =\pm\frac{\beta}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \beta &amp; =\pm\frac{\beta_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\gamma_{1} &amp; =\pm\frac{\gamma\cosh\varphi-\sinh\varphi}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \gamma &amp; =\pm\frac{\gamma_{1}\cosh\varphi+\sinh\varphi}{\cosh\varphi+\gamma_{1}\sinh\varphi}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}-A) with &lt;math&gt;\left[\alpha,\beta,\gamma\right]=\left[u_{2},u_{3},u_{1}\right]&lt;/math&gt;.}}

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lindemann|label 1=History of Lorentz transformations via squeeze mappings § Lindemann}}

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gerard|label 1=History of Lorentz transformations in general § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Killing|label 1=History of Lorentz transformations in general § Killing}}

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Elliott|label 1=History of Lorentz transformations via squeeze mappings § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3c}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.}}

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Liebmann|label 1=History of Lorentz transformations in general § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Herglotz1|label 1=History of Lorentz transformations via velocity § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}).}}

==={{anchor|Varicak}} Varićak (1910) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Varicak|label 1=History of Lorentz transformations via trigonometric functions § Varicak}}

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3f}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gour88}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>5trtul0m2zlaaxw5a4u1v2n4gls4vxo</sha1>
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      <text bytes="57595" sha1="iov9o2macpo6xtkkcf56gw3o14wo8j2" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via hyperbolic functions==
===Translation in the hyperbolic plane===
[[File:Hyperbolic functions-2.svg|thumb|upright=1.4|A ray through the unit hyperbola {{math|1=''x''&lt;sup&gt;2&lt;/sup&gt; − ''y''&lt;sup&gt;2&lt;/sup&gt; = 1}} at the point {{math|(cosh ''a'', sinh ''a'')}}.]]
The case of a Lorentz transformation without spatial rotation is called a [[w:Lorentz boost]]. The simplest case can be given, for instance, by setting ''n=1'' in the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}g_{00}+x_{1}g_{01}\\
x_{1}^{\prime} &amp; =x_{0}g_{10}+x_{1}g_{11}\\
\\
x_{0} &amp; =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}\\
x_{1} &amp; =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}
\end{align}
\left|\begin{align}g_{01}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{10}^{2} &amp; =1\\
g_{01}g_{11}-g_{00}g_{10} &amp; =0\\
g_{10}^{2}-g_{00}^{2} &amp; =-1\\
g_{11}^{2}-g_{01}^{2} &amp; =1\\
g_{10}g_{11}-g_{00}g_{01} &amp; =0
\end{align}
\rightarrow\begin{align}g_{00}^{2} &amp; =g_{11}^{2}\\
g_{01}^{2} &amp; =g_{10}^{2}
\end{align}
\right.
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\scriptstyle\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}g_{00} &amp; -g_{10}\\
-g_{01} &amp; g_{11}
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}g_{00} &amp; g_{01}\\
g_{10} &amp; g_{11}
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3a}}}}

which resembles precisely the relations of [[w:hyperbolic function]]s in terms of [[w:hyperbolic angle]] &lt;math&gt;\eta&lt;/math&gt;. Thus a Lorentz boost or [[w:hyperbolic rotation]] (being the same as a rotation around an imaginary angle &lt;math&gt;i\eta=\phi&lt;/math&gt; in [[../Lorentz transformation (imaginary)#math_2b|E:'''(2b)''']] or a [[w:Translation (geometry)|translation]] in the hyperbolic plane in terms of the hyperboloid model) is given by

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline g_{00}=g_{11}=\cosh\eta,\ g_{01}=g_{10}=-\sinh\eta\\
\hline \left.\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
x_{0}^{\prime} &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta &amp;  &amp; =\frac{x_{0}-x_{1}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}-x_{1}v}{\sqrt{1-v^{2}}}\\
x_{1}^{\prime} &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta &amp;  &amp; =\frac{x_{1}-x_{0}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}-x_{0}v}{\sqrt{1-v^{2}}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}v}{\sqrt{1-v^{2}}}\\
x_{1} &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp;  &amp; =\frac{x_{1}^{\prime}+x_{0}^{\prime}v}{\sqrt{1-v^{2}}}
\end{align}
\right|{\scriptstyle \begin{align}\sinh^{2}\eta-\cosh^{2}\eta &amp; =-1 &amp; (a)\\
\cosh^{2}\eta-\sinh^{2}\eta &amp; =1 &amp; (b)\\
\frac{\sinh\eta}{\cosh\eta} &amp; =\tanh\eta=v &amp; (c)\\
\frac{1}{\sqrt{1-\tanh^{2}\eta}} &amp; =\cosh\eta &amp; (d)\\
\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}} &amp; =\sinh\eta &amp; (e)\\
\frac{\tanh q\pm\tanh\eta}{1\pm\tanh q\tanh\eta} &amp; =\tanh\left(q\pm\eta\right) &amp; (f)
\end{align}
}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;\scriptstyle\left.\begin{align}\mathbf{x}' &amp; =\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}\\
\mathbf{x} &amp; =\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}\cdot\mathbf{x}'
\end{align}
\quad\right|\quad\det\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}=1&lt;/math&gt;|{{equationRef|3b}}}}

Hyperbolic identities (a,b) on the right of ({{equationNote|3b}}) were given by [[#Riccati|Riccati (1757)]], all identities (a,b,c,d,e,f) by [[#Lambert|Lambert (1768–1770)]]. Lorentz transformations ({{equationNote|3b}}-A) were given by [[#Laisant|Laisant (1874)]], [[#Cox|Cox (1882)]], [[#Goursat|Goursat (1888)]], [[#Lindemann|Lindemann (1890/91)]], [[#Gerard|Gérard (1892)]], [[#Killing2|Killing (1893, 1897/98)]], [[#Whitehead|Whitehead (1897/98)]], [[#Woods2|Woods (1903/05)]], [[#Elliott|Elliott (1903)]] and [[#Liebmann|Liebmann (1904/05)]] in terms of Weierstrass coordinates of the [[w:hyperboloid model]], while transformations similar to ({{equationNote|3b}}-C) have been used by [[#Lipschitz1|Lipschitz (1885/86)]]. In special relativity, hyperbolic functions were used by [[#Frank|Frank (1909)]] and [[#Varicak|Varićak (1910)]].

Using the idendity &lt;math&gt;\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;, Lorentz boost ({{equationNote|3b}}) assumes a simple form by using [[w:squeeze mapping]]s in analogy to Euler's formula in [[../Lorentz transformation (imaginary)#math_2c|E:'''(2c)''']]:&lt;ref name=rind&gt;Rindler (1969), p. 45&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x_{1}^{\prime}-x_{0}^{\prime} &amp; =e^{\eta}\left(x_{1}-x_{0}\right)\\
x_{1}^{\prime}+x_{0}^{\prime} &amp; =e^{-\eta}\left(x_{1}+x_{0}\right)
\end{align}
\quad\begin{align}x_{1}-x_{0} &amp; =e^{-\eta}\left(x_{1}^{\prime}-x_{0}^{\prime}\right)\\
x_{1}+x_{0} &amp; =e^{\eta}\left(x_{1}^{\prime}+x_{0}^{\prime}\right)
\end{align}
\end{matrix}\\
\hline k=e^{\eta}=\cosh\eta+\sinh\eta=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}=\sqrt{\frac{1+v}{1-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|3c}}}} 

Lorentz transformations ({{equationNote|3c}}) for arbitrary ''k'' were given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]), while a form similar to &lt;math&gt;k=\sqrt{\tfrac{1+v}{1-v}}&lt;/math&gt; was given by [[#Lipschitz1|Lipschitz (1885/86)]], and the exponential form was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1890/91)]], [[#Elliott|Elliott (1903)]], [[#Herglotz1|Herglotz (1909)]].

Rapidity can be composed of arbitrary many rapidities &lt;math&gt;\eta_{1},\eta_{2}\dots&lt;/math&gt; as per the [[w:Hyperbolic functions#Sums of arguments|w:angle sum laws of hyperbolic sines and cosines]], so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between [[w:List of trigonometric identities#Angle sum and difference identities|w:angle sum laws of circular trigonometry]] and spatial rotations. Alternatively, the hyperbolic angle sum laws ''themselves'' can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the [[w:unit hyperbola]]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}=1\\
\hline \left[\eta=\eta_{2}-\eta_{1}\right]\\
\begin{align}x_{0}^{\prime} &amp; =\sinh\eta_{1}=\sinh\left(\eta_{2}-\eta\right)=\sinh\eta_{2}\cosh\eta-\cosh\eta_{2}\sinh\eta &amp;  &amp; =x_{0}\cosh\eta-x_{1}\sinh\eta\\
x_{1}^{\prime} &amp; =\cosh\eta_{1}=\cosh\left(\eta_{2}-\eta\right)=-\sinh\eta_{2}\sinh\eta+\cosh\eta_{2}\cosh\eta &amp;  &amp; =-x_{0}\sinh\eta+x_{1}\cosh\eta\\
\\
x_{0} &amp; =\sinh\eta_{2}=\sinh\left(\eta_{1}+\eta\right)=\sinh\eta_{1}\cosh\eta+\cosh\eta_{1}\sinh\eta &amp;  &amp; =x_{0}^{\prime}\cosh\eta+x_{1}^{\prime}\sinh\eta\\
x_{1} &amp; =\cosh\eta_{2}=\cosh\left(\eta_{1}+\eta\right)=\sinh\eta_{1}\sinh\eta+\cosh\eta_{1}\cosh\eta &amp;  &amp; =x_{0}^{\prime}\sinh\eta+x_{1}^{\prime}\cosh\eta
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation

&lt;math&gt;{\scriptstyle \begin{align}\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{2}-\eta\right) &amp; \sinh\left(\eta_{2}-\eta\right)\\
\sinh\left(\eta_{2}-\eta\right) &amp; \cosh\left(\eta_{2}-\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; -\sinh\eta\\
-\sinh\eta &amp; \cosh\eta
\end{bmatrix}\\
\begin{bmatrix}x_{1} &amp; x_{0}\\
x_{0} &amp; x_{1}
\end{bmatrix} &amp; =\begin{bmatrix}\cosh\eta_{2} &amp; \sinh\eta_{2}\\
\sinh\eta_{2} &amp; \cosh\eta_{2}
\end{bmatrix}=\begin{bmatrix}\cosh\left(\eta_{1}+\eta\right) &amp; \sinh\left(\eta_{1}+\eta\right)\\
\sinh\left(\eta_{1}+\eta\right) &amp; \cosh\left(\eta_{1}+\eta\right)
\end{bmatrix}=\begin{bmatrix}\cosh\eta_{1} &amp; \sinh\eta_{1}\\
\sinh\eta_{1} &amp; \cosh\eta_{1}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix} &amp;  &amp; =\begin{bmatrix}x_{1}^{\prime} &amp; x_{0}^{\prime}\\
x_{0}^{\prime} &amp; x_{1}^{\prime}
\end{bmatrix}\cdot\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta
\end{bmatrix}
\end{align}
}&lt;/math&gt;

or in exponential form as squeeze mapping analogous to ({{equationNote|3c}}): 

&lt;math&gt;\begin{align}e^{-\eta_{1}} &amp; =e^{\eta}e^{-\eta_{2}}=e^{\eta-\eta_{2}} &amp; e^{-\eta_{2}} &amp; =e^{-\eta}e^{-\eta_{1}}=e^{-\eta_{1}-\eta}\\
e^{\eta_{1}} &amp; =e^{-\eta}e^{\eta_{2}}=e^{\eta_{2}-\eta} &amp; e^{\eta_{2}} &amp; =e^{\eta}e^{\eta_{1}}=e^{\eta_{1}+\eta}
\end{align}
&lt;/math&gt;|{{equationRef|3d}}}}

Hyperbolic angle sum laws were given by [[#Riccati|Riccati (1757)]] and [[#Lambert|Lambert (1768–1770)]] and many others, while matrix representations were given by [[#Glaisher|Glaisher (1878)]] and [[#Gunther1|Günther (1880/81)]].

===Hyperbolic law of cosines===
By adding coordinates &lt;math&gt;x_{2}^{\prime}=x_{2}&lt;/math&gt; and &lt;math&gt;x_{3}^{\prime}=x_{3}&lt;/math&gt; in Lorentz transformation ({{equationNote|3b}}) and interpreting &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; as [[w:homogeneous coordinates]], the Lorentz transformation can be rewritten in line with equation [[../Lorentz transformation (general)#math_1b|E:'''(1b)''']] by using coordinates &lt;math&gt;[u_{1},\ u_{2},\ u_{3}]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}}\right]&lt;/math&gt; defined by &lt;math&gt;u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\le1&lt;/math&gt; inside the [[w:unit sphere]] as follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align} &amp; \quad\quad(A) &amp;  &amp; \quad\quad(B) &amp;  &amp; \quad\quad(C)\\
\hline \\
u_{1}^{\prime} &amp; =\frac{-\sinh\eta+u_{1}\cosh\eta}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{1}-\tanh\eta}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{1}-v}{1-u_{1}v}\\
u_{2}^{\prime} &amp; =\frac{u_{2}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{2}\sqrt{1-v^{2}}}{1-u_{1}v}\\
u_{3}^{\prime} &amp; =\frac{u_{3}}{\cosh\eta-u_{1}\sinh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-\tanh^{2}\eta}}{1-u_{1}\tanh\eta} &amp;  &amp; =\frac{u_{3}\sqrt{1-v^{2}}}{1-u_{1}v}\\
\\
\hline \\
u_{1} &amp; =\frac{\sinh\eta+u_{1}^{\prime}\cosh\eta}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+\tanh\eta}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v}\\
u_{2} &amp; =\frac{u_{2}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{2}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}\\
u_{3} &amp; =\frac{u_{3}^{\prime}}{\cosh\eta+u_{1}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+u_{1}^{\prime}\tanh\eta} &amp;  &amp; =\frac{u_{3}^{\prime}\sqrt{1-v^{2}}}{1+u_{1}^{\prime}v}
\end{align}
&lt;/math&gt;|{{equationRef|3e}}}}

Transformations (A) were given by [[#Escherich|Escherich (1874)]], [[#Goursat|Goursat (1888)]], [[#Killing2|Killing (1898)]], and transformations (C) by [[#Beltrami|Beltrami (1868)]], [[#Schur|Schur (1885/86, 1900/02)]] in terms of [[w:Beltrami–Klein model|Beltrami coordinates]]&lt;ref&gt;Rosenfeld (1988), p. 231&lt;/ref&gt; of hyperbolic geometry. This transformation becomes equivalent to the [[w:hyperbolic law of cosines]] by restriction to coordinates of the &lt;math&gt;\left[u_{1},u_{2}\right]&lt;/math&gt;-plane and &lt;math&gt;\left[u'_{1},u'_{2}\right]&lt;/math&gt;-plane and defining their scalar products in terms of trigonometric and hyperbolic identities:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix} &amp; \begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\
u'^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}
\end{matrix}\left|\begin{align}u_{1}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+vu'\cos\alpha'}, &amp; u_{1}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-vu\cos\alpha}\\
u_{2}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{1+vu'\cos\alpha'}, &amp; u_{2}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{1-vu\cos\alpha}\\
\frac{u_{2}}{u_{1}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-v^{2}}}{u'\cos\alpha'+v}, &amp; \frac{u_{2}^{\prime}}{u_{1}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-v^{2}}}{u\cos\alpha-v}
\end{align}
\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(vu'\sin\alpha'\right){}^{2}}}{1+vu'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(vu\sin\alpha\right){}^{2}}}{1-vu\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-u^{\prime2}}}=\frac{1}{\sqrt{1-v^{2}}}\frac{1}{\sqrt{1-u^{2}}}-\frac{v}{\sqrt{1-v^{2}}}\frac{u}{\sqrt{1-u^{2}}}\cos\alpha &amp; (B)\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha &amp; (A)
\end{matrix}&lt;/math&gt;|{{equationRef|3f}}}}

The hyperbolic law of cosines (A) was given by [[#Taurinus|Taurinus (1826) and Lobachevsky (1829/30)]] and others, while variant (B) was given by [[#Schur|Schur (1900/02)]]. By further setting &lt;math&gt;\tanh\xi=\tanh\zeta=1&lt;/math&gt; or &lt;math&gt;u'=u=1&lt;/math&gt; it follows:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}(A)  &amp; \ \cos\alpha=\frac{\cos\alpha'+\tanh\eta}{1+\tanh\eta\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{1+\tanh\eta\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\tanh^{2}\eta}}{\cos\alpha'+\tanh\eta}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-\tanh\eta}{1+\tanh\eta}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-\tanh\eta}{1-\tanh\eta\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{1-\tanh\eta\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\tanh^{2}\eta}}{\cos\alpha-\tanh\eta}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+\tanh\eta}{1-\tanh\eta}}\tan\frac{\alpha}{2}\\
\\
(B)  &amp; \ \cos\alpha=\frac{\cos\alpha'+v}{1+v\cos\alpha'}; &amp; \ \sin\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{1+v\cos\alpha'}; &amp; \ \tan\alpha=\frac{\sin\alpha'\sqrt{1-v^{2}}}{\cos\alpha'+v}; &amp; \ \tan\frac{\alpha}{2}=\sqrt{\frac{1-v}{1+v}}\tan\frac{\alpha'}{2}\\
 &amp; \ \cos\alpha'=\frac{\cos\alpha-v}{1-v\cos\alpha}; &amp; \ \sin\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{1-v\cos\alpha}; &amp; \ \tan\alpha'=\frac{\sin\alpha\sqrt{1-v^{2}}}{\cos\alpha-v}; &amp; \ \tan\frac{\alpha'}{2}=\sqrt{\frac{1+v}{1-v}}\tan\frac{\alpha}{2}
\end{matrix}
&lt;/math&gt;|{{equationRef|3g}}}}

Formulas (3g-B) are the equations of an [[w:ellipse]] of [[w:Orbital eccentricity|eccentricity]] ''v'', [[w:eccentric anomaly]] α' and [[w:true anomaly]] α, first geometrically formulated by [[#Euler|Kepler (1609)]] and explicitly written down by [[#Euler|Euler (1735, 1748), Lagrange (1770)]] and many others in relation to planetary motions. They were also used by [[../Lorentz transformation (conformal)#Darboux|E:Darboux (1873)]] as a sphere transformation. In special relativity these formulas describe the aberration of light, see [[../Lorentz transformation (velocity)#Velocity addition and aberration|E:velocity addition and aberration]].

==Historical notation==
==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#mercator|label 1=History of Lorentz transformations via squeeze mappings § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated the following relations on a rectangular hyperbola:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations &lt;math&gt;1+a=c&lt;/math&gt;, &lt;math&gt;\sqrt{2a+a^{2}}=d&lt;/math&gt; with &lt;math&gt;c^{2}-d^{2}=1&lt;/math&gt; implicitly correspond to hyperbolic functions &lt;math&gt;c=\cosh\eta&lt;/math&gt;, &lt;math&gt;d=\sinh\eta&lt;/math&gt; with &lt;math&gt;\cosh^{2}\eta-\sinh^{2}\eta=1&lt;/math&gt; (which were explicitly introduced by [[#Riccati|Riccati (1757)]] much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|3c}}) because:&lt;br&gt;
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta}AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta}BI&lt;/math&gt;.&lt;br&gt;
Furthermore, transforming Mercator's asymptotic coordinates &lt;math&gt;AH=\tfrac{c+d}{\sqrt{2}}&lt;/math&gt;, &lt;math&gt;FH=\tfrac{c-d}{\sqrt{2}}&lt;/math&gt; into Cartesian coordinates &lt;math&gt;x_{0},x_{1}&lt;/math&gt; gives:&lt;br&gt;
:&lt;math&gt;x_{1}=\tfrac{AH+FH}{\sqrt{2}}=c=\cosh\eta,\quad x_{0}=\tfrac{AH-FH}{\sqrt{2}}=d=\sinh\eta&lt;/math&gt;&lt;br&gt;
which produces the unit hyperbola &lt;math&gt;-x_{0}^{2}+x_{1}^{2}=1&lt;/math&gt; as in ({{equationNote|3d}}), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.}}

==={{anchor|Euler}} Euler (1735) – True and eccentric anomaly===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Euler|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Euler|label 1=History of Lorentz transformations via Quaternions § Euler}}

[[w:Johannes Kepler]] (1609) geometrically formulated [[w:Kepler's equation]] and the relations between the [[w:mean anomaly]], [[w:true anomaly]], and [[w:eccentric anomaly]].&lt;ref group=M&gt;Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to &lt;math&gt;{\scriptstyle \alpha=\beta+e\sin\beta}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\nu=\frac{e+\cos\beta}{1+e\cos\beta}}&lt;/math&gt; and &lt;math&gt;{\scriptstyle \cos\beta=\frac{\cos\nu-e}{1-e\cos\nu}}&lt;/math&gt;&lt;/ref&gt;&lt;ref&gt;Volk (1976), p. 366&lt;/ref&gt; The relation between the true anomaly ''z'' and the eccentric anomaly ''P'' was algebraically expressed by [[w:Leonhard Euler]] (1735/40) as follows:&lt;ref group=M&gt;Euler (1735/40), § 19&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{\cos P+v}{1+v\cos P},\ \cos P=\frac{\cos z-v}{1-v\cos z},\ \int P=\frac{\int z\sqrt{1-v^{2}}}{1-v\cos z}&lt;/math&gt;

and in 1748:&lt;ref group=M&gt;Euler (1748a), section VIII&lt;/ref&gt;

:&lt;math&gt;\cos z=\frac{n+\cos y}{1+n\cos y},\ \sin z=\frac{\sin y\sqrt{1-n^{2}}}{1+n\cos y},\ \tan z=\frac{\sin y\sqrt{1-n^{2}}}{n+\cos y}&lt;/math&gt;

while [[w:Joseph-Louis Lagrange]] (1770/71) expressed them as follows&lt;ref group=M&gt;Lagrange (1770/71), section I&lt;/ref&gt;

:&lt;math&gt;\sin u=\frac{m\sin x}{1+n\cos x},\ \cos u=\frac{n+\cos x}{1+n\cos x},\ \operatorname{tang}\frac{1}{2}u=\frac{m}{1+n}\operatorname{tang}\frac{1}{2}x,\ \left(m^{2}=1-n^{2}\right)&lt;/math&gt;

{{Lorentzbox|Text= These relations resemble formulas ({{equationNote|3g}}), while ({{equationNote|3e}}) follows by setting &lt;math&gt;[\cos z,\sin z, \cos y,\sin y]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Euler's formulas or &lt;math&gt;[\cos u,\sin u, \cos x,\sin x]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]&lt;/math&gt; in Lagrange's formulas.}}

==={{anchor|Riccati}} Riccati (1757) – hyperbolic addition===
[[File:Riccati-hyperbola.png|thumb|&lt;small&gt;Riccati's (1757) illustration of hyperbolic addition laws.&lt;/small&gt;]]
[[w:Vincenzo Riccati]] (1757) introduced hyperbolic functions ''cosh'' and ''sinh'', which he denoted as ''Ch.'' and ''Sh.'' related by &lt;math&gt;Ch.^{2}-Sh.^{2}=r^2&lt;/math&gt; with ''r'' being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:&lt;ref group=M&gt;Riccati (1757), p. 71&lt;/ref&gt;&lt;ref group=M&gt;Günther (1880/81), pp. 7–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}CA=r,\ CB=Ch.\varphi,\ BE=Sh.\varphi,\ CD=Ch.\pi,\ DF=Sh.\pi\\
CM=Ch.\overline{\varphi+\pi},\ MN=Sh.\overline{\varphi+\pi}\\
CK=\frac{r}{\sqrt{2}},\ CG=\frac{Ch.\varphi+Sh.\varphi}{\sqrt{2}},\ CH=\frac{Ch.\pi+Sh.\pi}{\sqrt{2}},\ CP=\frac{Ch.\overline{\varphi+\pi}+Sh.\overline{\varphi+\pi}}{\sqrt{2}}\\
CK:CG::CH:CP\\
\left[Ch.^{2}-Sh.^{2}=rr\right]\\
\hline Ch.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Ch.\pi+Sh.\varphi\,Sh.\pi}{r}\\
Sh.\overline{\varphi+\pi}=\frac{Ch.\varphi\,Sh.\pi+Ch.\pi\,Sh.\varphi}{r}
\end{matrix}&lt;/math&gt;

He furthermore showed that &lt;math&gt;Ch.\overline{\varphi-\pi}&lt;/math&gt; and &lt;math&gt;Sh.\overline{\varphi-\pi}&lt;/math&gt; follow by setting &lt;math&gt;Ch.\pi\Rightarrow Ch.-\pi&lt;/math&gt; and &lt;math&gt;Sh.\pi\Rightarrow Sh.-\pi&lt;/math&gt; in the above formulas.

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}) with &lt;math&gt;\pi=\eta,\ \varphi=\eta_1,\ \overline{\varphi+\pi}=\eta_2&lt;/math&gt;.}}

==={{anchor|Lambert}} Lambert (1768–1770) – hyperbolic addition===

While [[#Riccati|Riccati (1757)]] discussed the hyperbolic sine and cosine, [[w:Johann Heinrich Lambert]] (read 1767, published 1768) introduced the expression ''tang φ'' or abbreviated ''tφ'' as the [[w:tangens hyperbolicus]] &lt;math&gt;{\scriptstyle \frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}}&lt;/math&gt; of a variable ''u'', or in modern notation ''tφ=tanh(u)'':&lt;ref group=M&gt;Lambert (1761/68), pp. 309–318&lt;/ref&gt;&lt;ref&gt;Barnett (2004), pp. 22–23&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}\xi\xi-1 &amp; =\eta\eta &amp; (a)\\
1+\eta\eta &amp; =\xi\xi &amp; (b)\\
\frac{\eta}{\xi} &amp; =tang\ \phi=t\phi &amp; (c)\\
\xi &amp; =\frac{1}{\sqrt{1-t\phi^{2}}} &amp; (d)\\
\eta &amp; =\frac{t\phi}{\sqrt{1-t\phi^{2}}} &amp; (e)\\
t\phi'' &amp; =\frac{t\phi+t\phi'}{1+t\phi\cdot t\phi'} &amp; (f)\\
t\phi' &amp; =\frac{t\phi''-t\phi}{1-t\phi\cdot t\phi''} &amp; (g)
\end{align}
\right|\begin{align}2u &amp; =\log\frac{1+t\phi}{1-t\phi}\\
\xi &amp; =\frac{e^{u}+e^{-u}}{2}\\
\eta &amp; =\frac{e^{u}-e^{-u}}{2}\\
t\phi &amp; =\frac{e^{u}-e^{-u}}{e^{u}+e^{-u}}\\
e^{u} &amp; =\xi+\eta\\
e^{-u} &amp; =\xi-\eta
\end{align}&lt;/math&gt;

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:&lt;ref group=M&gt;Lambert (1770), p. 335&lt;/ref&gt;

:&lt;math&gt;\begin{align}t(y+z) &amp; =(ty+tz):(1+ty\cdot tz) &amp; (f)\\
t(y-z) &amp; =(ty-tz):(1-ty\cdot tz) &amp; (g)
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of ({{equationNote|3b}}). Relations (f,g) can also be found in ({{equationNote|3e}}). By setting ''tφ=v/c'', formula (c) becomes the relative velocity between two frames, (d) the [[w:Lorentz factor]], (e) the [[w:proper velocity]], (f) or (g) becomes the Lorentz transformation of velocity (or relativistic [[w:velocity addition formula]]) for collinear velocities in [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] and [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']].}}

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's &quot;cos&quot; and &quot;sin&quot; actually mean &quot;cosh&quot; and &quot;sinh&quot;):

:&lt;math&gt;\begin{align}\sin(y+z) &amp; =\sin y\cos z+\cos y\sin z\\
\sin(y-z) &amp; =\sin y\cos z-\cos y\sin z\\
\cos(y+z) &amp; =\cos y\cos z+\sin y\sin z\\
\cos(y-z) &amp; =\cos y\cos z-\sin y\sin z
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost ({{equationNote|3d}}).}}

==={{Anchor|Taurinus}} Taurinus (1826) – Hyperbolic law of cosines===

After the addition theorem for the tangens hyperbolicus was given by [[#Lambert|Lambert (1768)]], [[w:hyperbolic geometry]] was used by [[w:Franz Taurinus]] (1826), and later by [[w:Nikolai Lobachevsky]] (1829/30) and others, to formulate the [[w:hyperbolic law of cosines]]:&lt;ref group=M&gt;Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)&lt;/ref&gt;&lt;ref&gt;Bonola (1912), p. 79&lt;/ref&gt;&lt;ref&gt;Gray (1979), p. 242&lt;/ref&gt;

:&lt;math&gt;A=\operatorname{arccos}\frac{\cos\left(\alpha\sqrt{-1}\right)-\cos\left(\beta\sqrt{-1}\right)\cos\left(\gamma\sqrt{-1}\right)}{\sin\left(\beta\sqrt{-1}\right)\sin\left(\gamma\sqrt{-1}\right)}&lt;/math&gt;

{{Lorentzbox|Text=When solved for &lt;math&gt;\cos\left(\alpha\sqrt{-1}\right)&lt;/math&gt; it corresponds to the Lorentz transformation in Beltrami coordinates ({{equationNote|3f}}), and by defining the rapidities &lt;math&gt;{\scriptstyle \left(\left[\frac{U}{c},\ \frac{v}{c},\ \frac{u}{c}\right]=\left[\tanh\alpha,\ \tanh\beta,\ \tanh\gamma\right]\right)}&lt;/math&gt; it corresponds to the relativistic velocity addition formula [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']].}}

==={{anchor|Beltrami}} Beltrami (1868) – Beltrami coordinates===

[[w:Eugenio Beltrami]] (1868a) introduced coordinates of the [[w:Beltrami–Klein model]] of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:&lt;ref group=M&gt;Beltrami (1868a), pp. 287-288; Note I; Note II&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=R^{2}\frac{\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}\\
u^{2}+v^{2}=a^{2}\\
\hline u''=\frac{aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'},\ v''=\frac{a_{0}w_{0}v'}{a^{2}-r_{0}u'},\\
\left(r_{0}=\sqrt{u_{0}^{2}+v_{0}^{2}},\ w_{0}=\sqrt{a^{2}-r_{0}^{2}}\right)\\
\hline ds^{2}=R^{2}\frac{\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}\\
(R=R\sqrt{-1},\ a=a\sqrt{-1})
\end{matrix}&lt;/math&gt;

(where the disk radius ''a'' and the [[w:radius of curvature]] ''R'' are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)&lt;ref group=M&gt;Beltrami (1868b), pp. 232, 240–241, 253–254&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds=R\frac{\sqrt{dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}}}{x}\\
x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=a^{2}\\
\hline y_{1}=\frac{ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}\ \text{or}\ x_{1}=\frac{a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}},\ x_{r}=\pm\frac{ay_{r}\sqrt{a^{2}-a_{1}^{2}}}{ab+a_{1}y_{1}}\ (r=2,3,\dots,n)\\
\hline ds=R\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+\cdots+dx_{n}^{2}-dx^{2}}}{x}\\
x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\\
\left(R=R\sqrt{-1},\ x=x\sqrt{-1},\ a=a\sqrt{-1}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting ''a=a&lt;sub&gt;0&lt;/sub&gt;'' Beltrami's (1868a) formulas become formulas ({{equationNote|3e}}), or in his (1868b) formulas one sets ''a=b'' for arbitrary dimensions.}}

=== {{anchor|Laisant2}} Laisant (1874) – Equipollences===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Laisant1|label 1=History of Lorentz transformations via squeeze mappings § Laisant}}

In his French translation of [[w:Giusto Bellavitis]]' principal work on [[w:Equipollence (geometry)|w:equipollences]], [[w:Charles-Ange Laisant]] (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as&lt;ref group=M&gt;Laisant (1874b), pp. 134–135&lt;/ref&gt;

:(1) &lt;math&gt;\begin{matrix} &amp; \mathrm{OM}\bumpeq x\mathrm{OA}+y\mathrm{OB}\\
 &amp; \mathrm{MT}\bumpeq y\mathrm{OA}+x\mathrm{OB}\\
 &amp; \left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\
\Rightarrow &amp; \mathrm{OM}\bumpeq\cosh t\cdot\mathrm{OA}+\sinh t\cdot\mathrm{OB}
\end{matrix}&lt;/math&gt;

Here, OA and OB are [[w:Conjugate diameters|conjugate semi-diameters]] of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

:&lt;math&gt;\begin{matrix}\begin{align}\mathrm{OC} &amp; \bumpeq c\mathrm{OA}+d\mathrm{OB} &amp; \qquad &amp;  &amp; \mathrm{OA} &amp; \bumpeq c\mathrm{OC}-d\mathrm{OD}\\
\mathrm{OD} &amp; \bumpeq d\mathrm{OA}+c\mathrm{OB} &amp;  &amp;  &amp; \mathrm{OB} &amp; \bumpeq-d\mathrm{OC}+c\mathrm{OD}
\end{align}
\\
\left[c^{2}-d^{2}=1\right]
\end{matrix}&lt;/math&gt;

producing the invariant relation

:&lt;math&gt;(\mathrm{OC})^{2}-(\mathrm{OD})^{2}\bumpeq(\mathrm{OA})^{2}-(\mathrm{OB})^{2}&lt;/math&gt;.

Substituting into (1), he showed that OM retains its form

:&lt;math&gt;\begin{matrix}\mathrm{OM}\bumpeq(cx-dy)\mathrm{OC}+(cy-dx)\mathrm{OD}\\
\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]
\end{matrix}&lt;/math&gt;

He also defined velocity and acceleration by differentiation of (1).

{{Lorentzbox|Text=These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with ({{equationNote|3b}}).}}

==={{anchor|Escherich}} Escherich (1874) – Beltrami coordinates===

[[w:Gustav von Escherich]] (1874) discussed the plane of constant negative curvature&lt;ref&gt;Sommerville (1911), p. 297&lt;/ref&gt; based on the [[w:Beltrami–Klein model]] of hyperbolic geometry by [[#Beltrami|Beltrami (1868)]]. Similar to [[w:Christoph Gudermann]] (1830)&lt;ref name=guder group=M&gt;Gudermann (1830), §1–3, §18–19&lt;/ref&gt;  who introduced axial coordinates ''x''=tan(a) and ''y''=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions ''x''=tanh(a/k) and ''y''=tanh(b/k)&lt;ref group=M&gt;Escherich (1874), p. 508&lt;/ref&gt; in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:&lt;ref group=M name=escher&gt;Escherich (1874), p. 510&lt;/ref&gt;

:&lt;math&gt;x=\frac{\sinh\frac{a}{k}+x'\cosh\frac{a}{k}}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt; and &lt;math&gt;y=\frac{y'}{\cosh\frac{a}{k}+x'\sinh\frac{a}{k}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}), also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting &lt;math&gt;\tfrac{a}{k}=\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt; and multiplying ''[x,y,x′,y′]'' by 1/''c'', and equivalent to Lorentz boost ({{equationNote|3b}}) by setting &lt;math&gt;\scriptstyle (x,\ y,\ x',\ y')=\left(\frac{x_{1}}{x_{0}},\ \frac{x_{2}}{x_{0}},\ \frac{x_{1}^{\prime}}{x_{0}^{\prime}},\ \frac{x_{2}^{\prime}}{x_{0}^{\prime}}\right)&lt;/math&gt;. This is the relation between the [[w:Beltrami–Klein model|Beltrami coordinates]] in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the [[w:hyperboloid model]] introduced by [[../Lorentz transformation (general)#Killing1|E:Killing (1878–1893)]], [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]], and [[../Lorentz transformation (general)#Cox|E:Cox (1881)]]. Both coordinate systems were compared by Cox (1881).&lt;ref group=M&gt;Cox (1881), p. 186&lt;/ref&gt;}}

==={{anchor|Glaisher}} Glaisher (1878) – hyperbolic addition===

It was shown by [[w:James Whitbread Lee Glaisher]] (1878) that the hyperbolic addition laws can be expressed by matrix multiplication:&lt;ref group=M&gt;Glaisher (1878), p. 30&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\cosh x, &amp; \sinh x\\
\sinh x, &amp; \cosh x
\end{vmatrix}=1,\ \begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y
\end{vmatrix}=1\\
\text{by multiplication:}\\
\Rightarrow\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2}, &amp; s_{1}c_{2}+c_{1}s_{2}\\
c_{1}s_{2}+s_{1}c_{2}, &amp; s_{1}s_{2}+c_{1}c_{2}
\end{vmatrix}=1\\
\text{where}\ \left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right] \\
\Rightarrow\begin{vmatrix}\cosh(x+y), &amp; \sinh(x+y)\\
\sinh(x+y), &amp; \cosh(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\cosh y, &amp; \sinh y\\
\sinh y, &amp; \cosh y\end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

==={{anchor|Gunther1}} Günther (1880/81) – hyperbolic addition ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Gunther1|label 1=History of Lorentz transformations via squeeze mappings § Günther}}

Following [[#Glaisher|Glaisher (1878)]], [[w:Siegmund Günther]] (1880/81) expressed the hyperbolic addition laws by matrix multiplication:&lt;ref group=M&gt;Günther (1880/81), p. 405&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{vmatrix}\mathfrak{Cos}\,x, &amp; \mathfrak{Sin}\,x\\
\mathfrak{Sin}\,x, &amp; \mathfrak{Cos}\,x
\end{vmatrix}\cdot\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y+\mathfrak{Sin}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,x\,\mathfrak{Sin}\,y+\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y\\
\mathfrak{Sin}\,x\,\mathfrak{Cos}\,y+\mathfrak{Cos}\,x\,\mathfrak{Sin}\,y, &amp; \mathfrak{Sin}\,x\,\mathfrak{Sin}\,y+\mathfrak{Cos}\,x\,\mathfrak{Cos}\,y
\end{vmatrix}\\
=\begin{vmatrix}\mathfrak{Cos}\,(x+y), &amp; \mathfrak{Sin}\,(x+y)\\
\mathfrak{Sin}\,(x+y), &amp; \mathfrak{Cos}\,(x+y)
\end{vmatrix}=1
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix &lt;math&gt;\scriptstyle\begin{vmatrix}\mathfrak{Cos}\,y, &amp; \mathfrak{Sin}\,y\\
\mathfrak{Sin}\,y, &amp; \mathfrak{Cos}\,y \end{vmatrix}&lt;/math&gt; producing the hyperbolic addition is analogous to matrix &lt;math&gt;\scriptstyle\begin{bmatrix}\cosh\eta &amp; \sinh\eta\\
\sinh\eta &amp; \cosh\eta\end{bmatrix}&lt;/math&gt; producing Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3d}}).}}

=== {{anchor|Cox}} Cox (1881/82) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Cox|label 1=History of Lorentz transformations in general § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cox2|label 1=History of Lorentz transformations via Quaternions § Cox}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Cox|label 1=History of Lorentz transformations via sphere transformations § Cox}}

[[w:Homersham Cox (mathematician)|w:Homersham Cox]] (1881/82) defined the case of translation in the hyperbolic plane with the ''y''-axis remaining unchanged:&lt;ref group=M name=cox&gt;Cox (1881/82), p. 194&lt;/ref&gt;
	
:&lt;math&gt;\begin{align}X &amp; =x\cosh p-z\sinh p\\	
Z &amp; =-x\sinh p+z\cosh p	\\ \\
x &amp; =X\cosh p+Z\sinh p\\
z &amp; =X\sinh p+Z\cosh p	
\end{align}	
&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) – Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lipschitz1|label 1=History of Lorentz transformations via squeeze mappings § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(a)\\
\hline \begin{matrix}\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(b)\end{matrix}\\
\hline \left\{ \begin{matrix}r_{a}=\frac{\sqrt{s_{a}+1}}{\sqrt{s_{a}-1}}\\
s_{a}&gt;1
\end{matrix}\right\}\Rightarrow\begin{align}\mathfrak{x}_{a} &amp; =\frac{s_{a}x_{a}+y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}\\
\mathfrak{y}_{a} &amp; =\frac{x_{a}+s_{a}y_{a}}{\sqrt{s_{a}-1}\sqrt{s_{a}+1}}
\end{align}
\quad(c)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts ({{equationNote|3b}}-C) and ({{equationNote|3c}}) by the identity &lt;math&gt;s_{a}=\tfrac{1}{v}=\coth\eta&lt;/math&gt;. That is, by substituting &lt;math&gt;v=\tfrac{1}{s_{a}}&lt;/math&gt; in ({{equationNote|3b}}-C) or ({{equationNote|3c}}) we obtain Lipschitz's transformations.}}

==={{Anchor|Schur}} Schur (1885/86, 1900/02) – Beltrami coordinates===

[[w:Friedrich Schur]] (1885/86) discussed spaces of constant Riemann curvature, and by following [[#Beltrami|Beltrami (1868)]] he used the transformation&lt;ref group=M&gt;Schur (1885/86), p. 167&lt;/ref&gt;

:&lt;math&gt;x_{1}=R^{2}\frac{y_{1}+a_{1}}{R^{2}+a_{1}y_{1}},\ x_{2}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{2}}{R^{2}+a_{1}y_{1}},\dots,\ x_{n}=R\sqrt{R^{2}-a_{1}^{2}}\frac{y_{n}}{R^{2}+a_{1}y_{1}}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] in arbitrary dimensions by setting ''R=c'' as the speed of light and ''a&lt;sub&gt;1&lt;/sub&gt;=v'' as relative velocity.}}

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:&lt;ref group=M&gt;Schur (1900/02), p. 290; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;x'=\frac{x-a}{1-\mathfrak{k}ax},\quad y'=\frac{y\sqrt{1-\mathfrak{k}a^{2}}}{1-\mathfrak{k}ax}&lt;/math&gt;

where &lt;math&gt;\mathfrak{k}&lt;/math&gt; can have values &gt;0, &lt;0 or ∞.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}) and therefore also equivalent to the relativistic velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] by setting ''a=v'' and &lt;math&gt;\mathfrak{k}=\tfrac{1}{c^{2}}&lt;/math&gt;.}}

He also defined the triangle&lt;ref group=M&gt;Schur (1900/02), p. 291; (1909), p. 83&lt;/ref&gt;

:&lt;math&gt;\frac{1}{\sqrt{1-\mathfrak{k}c^{2}}}=\frac{1}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{1}{\sqrt{1-\mathfrak{k}b^{2}}}-\frac{a}{\sqrt{1-\mathfrak{k}a^{2}}}\cdot\frac{b}{\sqrt{1-\mathfrak{k}b^{2}}}\cos\gamma&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition ({{equationNote|3f}}, b) or [[../Lorentz transformation (velocity)#math_4e|E:'''(4e)''']] by setting &lt;math&gt;[\mathfrak{k},\ c,\ a,\ b]=\left[\tfrac{1}{c^{2}},\ \sqrt{u_{x}^{\prime2}+u_{y}^{\prime2}},\ v,\ \sqrt{u_{x}^{2}+u_{y}^{2}}\right]&lt;/math&gt;.}}

==={{Anchor|Goursat}} Goursat (1887/88) – Minimal surfaces===

[[w:Édouard Goursat]] defined real coordinates &lt;math&gt;x,y&lt;/math&gt; of minimal surface &lt;math&gt;S&lt;/math&gt; and imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; of the adjoint minimal surface &lt;math&gt;S_0&lt;/math&gt;, so that another real minimal surface &lt;math&gt;S_1&lt;/math&gt; follows by the (conformal) transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 144&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =\frac{1+k^{2}}{2k}x-\frac{k^{2}-1}{2k}y_{0}\\
y_{1} &amp; =\frac{1+k^{2}}{2k}y+\frac{k^{2}-1}{2k}x_{0}\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

and expressed these equations in terms of hyperbolic functions by setting &lt;math&gt;k=e^{\varphi}&lt;/math&gt;:&lt;ref group=M&gt;Goursat (1887/88), p. 145&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1} &amp; =x\cosh\varphi-y_{0}\sinh\varphi\\
y_{1} &amp; =y\cosh\varphi+x_{0}\sinh\varphi\\
z_{1} &amp; =z
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|3b}}) by replacing the imaginary coordinates &lt;math&gt;x_{0},y_{0}&lt;/math&gt; by real coordinates defined as &lt;math&gt;[x_{0},y_{0}]=[-x,y]&lt;/math&gt;. It can also be seen that Goursat's relation &lt;math&gt;k=e^{\varphi}&lt;/math&gt; corresponds to &lt;math&gt;k=e^{\eta}&lt;/math&gt; defined in ({{equationNote|3c}}).}}

He went on to define &lt;math&gt;\alpha,\beta,\gamma&lt;/math&gt; as the direction cosines normal to surface &lt;math&gt;S&lt;/math&gt; and &lt;math&gt;\alpha_{1},\beta_{1},\gamma_{1}&lt;/math&gt; as the ones normal to surface &lt;math&gt;S_{1}&lt;/math&gt;, connected by the transformation:&lt;ref group=M&gt;Goursat (1887/88), p. 149f.&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha_{1} &amp; =\pm\frac{\alpha}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \alpha &amp; =\pm\frac{\alpha_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\beta_{1} &amp; =\pm\frac{\beta}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \beta &amp; =\pm\frac{\beta_{1}}{\cosh\varphi+\gamma_{1}\sinh\varphi}\\
\gamma_{1} &amp; =\pm\frac{\gamma\cosh\varphi-\sinh\varphi}{\cosh\varphi-\gamma\sinh\varphi} &amp;  &amp;  &amp; \gamma &amp; =\pm\frac{\gamma_{1}\cosh\varphi+\sinh\varphi}{\cosh\varphi+\gamma_{1}\sinh\varphi}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|3e}}-A) with &lt;math&gt;\left[\alpha,\beta,\gamma\right]=\left[u_{2},u_{3},u_{1}\right]&lt;/math&gt;.}}

==={{anchor|Lindemann}} Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Lindemann|label 1=History of Lorentz transformations via squeeze mappings § Lindemann}}

[[w:Ferdinand von Lindemann]] discussed hyperbolic geometry in terms of the [[w:Cayley–Klein metric]] in his (1890/91) edition of the lectures on geometry of [[w:Alfred Clebsch]]. Citing [[../Lorentz transformation (general)#Killing|E:Killing (1885)]] and [[../Lorentz transformation (general)#Poincare|Poincaré (1887)]] in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 477–478, 524&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega_{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\
\Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ \text{and}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

and used the following transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}) with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (''a''=0) or translations (α=0):&lt;ref group=M  name=linde /&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\
\hline \begin{align}x_{2} &amp; =\xi_{2}\cos\alpha+\xi_{3}\sin\alpha, &amp; x_{1} &amp; =\xi_{1}\cos\frac{a}{i}+2ki\xi_{4}\sin\frac{a}{i},\\
x_{3} &amp; =-\xi_{2}\sin\alpha+\xi_{3}\cos\alpha, &amp; 2kx_{4} &amp; =i\xi_{1}\sin\frac{a}{i}+2k\xi_{4}\cos\frac{a}{i}.
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0 and ''2k=1''.}}

==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gerard|label 1=History of Lorentz transformations in general § Gerard}}

[[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:&lt;ref group=M name=gerard&gt;Gérard (1892), pp. 40–41&lt;/ref&gt;

:&lt;math&gt;\begin{align}X &amp; =Z_{0}X'+X_{0}Z'\\
Y &amp; =Y'\\
Z &amp; =X_{0}X'+Z_{0}Z'
\end{align}
\ \text{with}\ \begin{align}X_{0} &amp; =\operatorname{sh}OO'\\
Z_{0} &amp; =\operatorname{ch}OO'
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Killing2}} Killing (1893,97) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Killing|label 1=History of Lorentz transformations in general § Killing}}

[[w:Wilhelm Killing]] (1878–1880) gave case of translation in the form&lt;ref group=M name=killtra&gt;Killing (1893), p. 331&lt;/ref&gt;

:&lt;math&gt;y_{0}=x_{0}\operatorname{Ch}a+x_{1}\operatorname{Sh}a,\quad y_{1}=x_{0}\operatorname{Sh}a+x_{1}\operatorname{Ch}a,\quad y_{2}=x_{2}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

In 1898, Killing wrote that relation in a form similar to [[#Escherich|Escherich (1874)]], and derived the corresponding Lorentz transformation for the two cases were ''v'' is unchanged or ''u'' is unchanged:&lt;ref group=M name=kill98&gt;Killing (1898), p. 133&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi'=\frac{\xi\operatorname{Ch}\frac{\mu}{l}+l\operatorname{Sh}\frac{\mu}{l}}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}},\ \eta'=\frac{\eta}{\frac{\xi}{l}\operatorname{Sh}\frac{\mu}{l}+\operatorname{Ch}\frac{\mu}{l}}\\
\hline \frac{u}{p}=\xi,\ \frac{v}{p}=\eta\\
\hline p'=p\operatorname{Ch}\frac{\mu}{l}+\frac{u}{l}\operatorname{Sh}\frac{\mu}{l},\quad u'=pl\operatorname{Sh}\frac{\mu}{l}+u\operatorname{Ch}\frac{\mu}{l},\quad v'=v\\
\text{or}\\
p'=p\operatorname{Ch}\frac{\nu}{l}+\frac{v}{l}\operatorname{Sh}\frac{\nu}{l},\quad u'=u,\quad v'=pl\operatorname{Sh}\frac{\nu}{l}+v\operatorname{Ch}\frac{\nu}{l}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The upper transformation system is equivalent to Lorentz transformation ({{equationNote|3e}}) and the velocity addition [[../Lorentz transformation (velocity)#math_4d|E:'''(4d)''']] with ''l=c'' and &lt;math&gt;\mu=c\operatorname{atanh}\tfrac{v}{c}&lt;/math&gt;, the system below is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Whitehead}} Whitehead (1897/98) – Universal algebra===

[[w:Alfred North Whitehead]] (1898) discussed the kinematics of hyperbolic space as part of his study of [[w:universal algebra]], and obtained the following transformation:&lt;ref group=M name=white&gt;Whitehead (1898), pp. 459–460&lt;/ref&gt;

:&lt;math&gt;\begin{align}x' &amp; =\left(\eta\cosh\frac{\delta}{\gamma}+\eta_{1}\sinh\frac{\delta}{\gamma}\right)e+\left(\eta\sinh\frac{\delta}{\gamma}+\eta_{1}\cosh\frac{\delta}{\gamma}\right)e_{1}\\
 &amp; \qquad+\left(\eta_{2}\cos\alpha+\eta_{3}\sin\alpha\right)e_{2}+\left(\eta_{3}\cos\alpha-\eta_{2}\sin\alpha\right)e_{3}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with α=0.}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Elliott|label 1=History of Lorentz transformations via squeeze mappings § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) and ({{equationNote|3c}}). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.}}

==={{anchor|Woods2}} Woods (1903) – Weierstrass coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Woods|label 1=History of Lorentz transformations via Möbius transformations § Woods}}

[[w:Frederick S. Woods]] (1903, published 1905) gave the case of translation in hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 55&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x_{1}\cos kl+x_{0}\frac{\sin kl}{k},\quad x_{2}^{\prime}=x_{2},\quad x_{2}^{\prime}=x_{3},\quad x_{0}^{\prime}=-x_{1}k\sin kl+x_{0}\cos kl&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with ''k''&lt;sup&gt;2&lt;/sup&gt;=-1.}}

and the loxodromic substitution for hyperbolic space:&lt;ref group=M&gt;Woods (1903/05), p. 72&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}x_{1}^{\prime} &amp; =x_{1}\cosh\alpha-x_{0}\sinh\alpha\\
x_{2}^{\prime} &amp; =x_{2}\cos\beta-x_{3}\sin\beta\\
x_{3}^{\prime} &amp; =x_{2}\sin\beta+x_{3}\cos\beta\\
x_{0}^{\prime} &amp; =-x_{1}\sinh\alpha+x_{0}\cosh\alpha
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}) with β=0.}}

==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Liebmann|label 1=History of Lorentz transformations in general § Liebmann}}

[[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:&lt;ref group=M name=lieb&gt;Liebmann (1904/05), p. 174&lt;/ref&gt;

:&lt;math&gt;x_{1}^{\prime}=x'\operatorname{ch}a+p'\operatorname{sh}a,\quad y_{1}^{\prime}=y',\quad p_{1}^{\prime}=x'\operatorname{sh}a+p'\operatorname{ch}a&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

==={{anchor|Frank}} Frank (1909) – Special relativity===
In special relativity, hyperbolic functions were used by [[w:Philipp Frank]] (1909), who derived the Lorentz transformation using ''ψ'' as rapidity:&lt;ref group=R&gt;Frank (1909), pp. 423-425&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=x\varphi(a)\,{\rm ch}\,\psi+t\varphi(a)\,{\rm sh}\,\psi\\
t'=-x\varphi(a)\,{\rm sh}\,\psi+t\varphi(a)\,{\rm ch}\,\psi\\
\hline {\rm th}\,\psi=-a,\ {\rm sh}\,\psi=\frac{a}{\sqrt{1-a^{2}}},\ {\rm ch}\,\psi=\frac{1}{\sqrt{1-a^{2}}},\ \varphi(a)=1\\
\hline x'=\frac{x-at}{\sqrt{1-a^{2}}},\ y'=y,\ z'=z,\ t'=\frac{-ax+t}{\sqrt{1-a^{2}}}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}
 	
=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Herglotz1|label 1=History of Lorentz transformations via velocity § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

In special relativity, [[w:Gustav Herglotz]] (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}Z=Z'e^{\vartheta}\\
	
\begin{aligned}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
	
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
	
\end{aligned}
	
\end{matrix}&lt;/math&gt;	
	
{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3c}}).}}

==={{anchor|Varicak}} Varićak (1910) – Special relativity===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Varicak|label 1=History of Lorentz transformations via trigonometric functions § Varicak}}

In special relativity, hyperbolic functions were used by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, by setting ''l=ct'' and ''v/c=tanh(u)'' with ''u'' as rapidity he wrote the Lorentz transformation in agreement with ({{equationNote|4b}}):&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\begin{align}l' &amp; =-x\operatorname{sh}u+l\operatorname{ch}u,\\
x' &amp; =x\operatorname{ch}u-l\operatorname{sh}u,\\
y' &amp; =y,\quad z'=z,\\
\operatorname{ch}u &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^{2}}}
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3b}}).}}

He showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1 /&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

He also related the velocity addition to the [[w:hyperbolic law of cosines]]:&lt;ref group=R&gt;Varićak (1910), p. 94&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\operatorname{ch}{u}=\operatorname{ch}{u_{1}}\operatorname ch{u_{2}}+\operatorname{sh}{u_{1}}\operatorname{sh}{u_{2}}\cos\alpha\\
\operatorname{ch}{u_{i}}=\frac{1}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}},\ \operatorname{sh}{u_{i}}=\frac{v_{i}}{\sqrt{1-\left(\frac{v_{i}}{c}\right)^{2}}}\\
v=\sqrt{v_{1}^{2}+v_{2}^{2}-\left(\frac{v_{1}v_{2}}{c}\right)^{2}}\ \left(a=\frac{\pi}{2}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz boost ({{equationNote|3f}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bel68sag}} 
*{{#section:History of Topics in Special Relativity/mathsource|bel68fond}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|esch74}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul35}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul48a}} 
*{{#section:History of Topics in Special Relativity/mathsource|ger92}} 
*{{#section:History of Topics in Special Relativity/mathsource|glai78}} 
*{{#section:History of Topics in Special Relativity/mathsource|gour88}} 
*{{#section:History of Topics in Special Relativity/mathsource|gud30}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|kep09}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil93}} 
*{{#section:History of Topics in Special Relativity/mathsource|kil97}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam67}} 
*{{#section:History of Topics in Special Relativity/mathsource|lam70}} 
*{{#section:History of Topics in Special Relativity/mathsource|lieb04}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}}
*{{#section:History of Topics in Special Relativity/mathsource|lip86}}
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|ric57}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu85}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu00}} 
*{{#section:History of Topics in Special Relativity/mathsource|schu09}} 
*{{#section:History of Topics in Special Relativity/mathsource|tau26}} 
*{{#section:History of Topics in Special Relativity/mathsource|whit98}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo01}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo03}}

===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|frank09a}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/relsource|var10}}
*{{#section:History of Topics in Special Relativity/relsource|var12}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L3}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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    <title>History of Topics in Special Relativity/Lorentz transformation (velocity)</title>
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      <contributor>
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        <id>52339</id>
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      <comment>/* Lorentz transformation via velocity */</comment>
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      <text bytes="57414" sha1="1xbdz669gyrm02wu98xvgij7l4xneu6" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Written in four dimensions by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x&lt;/math&gt; and adding  &lt;math&gt;y,z&lt;/math&gt; the familiar form follows

{{NumBlk|:|&lt;math&gt;\scriptstyle(A)\quad\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation:

&lt;math&gt;\scriptstyle(B)\quad\begin{matrix}\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\beta\gamma &amp; 0 &amp; 0\\
-\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x};\quad\mathbf{x}=\begin{bmatrix}\gamma &amp; \beta\gamma &amp; 0 &amp; 0\\
\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x}'\\
\det\begin{bmatrix}\gamma &amp; -\beta\gamma\\
-\beta\gamma &amp; \gamma
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;

or in terms of &lt;math&gt;ct,x&lt;/math&gt; as squeeze mapping in line with [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']]:

&lt;math&gt;\scriptstyle(C)\quad\begin{matrix}uw=-x_{0}^{2}+x_{1}^{2}=u'w'=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x'-ct' &amp; =\sqrt{\frac{c+v}{c-v}}\left(x-ct\right)\\
x'+ct' &amp; =\sqrt{\frac{c-v}{c+v}}\left(x+ct\right)
\end{align}
\quad\begin{align}x-ct &amp; =\sqrt{\frac{c-v}{c+v}}\left(x'-ct'\right)\\
x+ct &amp; =\sqrt{\frac{c+v}{c-v}}\left(x'+ct'\right)
\end{align}
\end{matrix}\\
\hline k=\sqrt{\frac{c+v}{c-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Transformations analogous to (A) have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompressible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. 

The matrix form (B) is a special case of the general boost matrix given by [[#Hahn|Hahn (1912)]] in terms of imaginary time, while variant (C) for arbitrary ''k'' was given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]) with the choice equivalent to &lt;math&gt;k=\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; given by [[#Born|Born (1921)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ \tfrac{u_{z}}{c}\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{z}{ct}\right]&lt;/math&gt; in ({{equationNote|4b}}-A), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]):

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{z}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
\hline \\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{z} &amp; =\frac{cz'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}&lt;/math&gt;|{{equationRef|4c}}}}

By restriction to velocities in the &lt;math&gt;\left[x,y\right]&lt;/math&gt; plane and using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|\begin{align}u_{x}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{x}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u_{y}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{y}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\frac{u_{y}}{u_{x}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \frac{u_{y}^{\prime}}{u_{x}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the well known [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix} &lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Frank|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Lorentz transformation in arbitrary directions===
Lorentz boosts for arbitrary directions&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt; in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] are in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

and the vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] follows by:

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

The special case of parallel and perpendicular directions in ({{equationNote|4f}}) was given by [[#Minkowski2|Minkowski (1907/8)]] while the complete transformation was formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911), Tamaki (1911)]]. General velocity addition ({{equationNote|4g}}) was given in equivalent form by [[#Herglotz2|Ignatowski (1910)]].

Rewritten in matrix notation, the general Lorentz boost has the form:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\hline \begin{align}\mathbf{g} &amp; =\begin{pmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{pmatrix}\end{align}
\\
\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]
\end{matrix}&lt;/math&gt;|{{equationRef|4h}}}}

While [[#Minkowski3|Minkowski (1907/8)]] formulated the matrix form of Lorentz transformations in general terms, he didn't explicitly express the velocity related components of the general boost matrix. A complete representation of ({{equationNote|4h}}) was given by [[#Hahn|Hahn (1912)]].

===Other formulations===
Important contributions to the mathematical understanding of the Lorentz transformation of space and time also include: [[#Minkowski|Minkowski (1907–1908)]] as well as [[#Frank|Frank (1909) and Varićak (1910)]] showed the relation to imaginary and hyperbolic functions, [[#Herglotz1|Herglotz (1909/10)]] used exponential squeeze mappings and Möbius transformations, [[#Ignatowski|Ignatowski (1910)]] didn't use the light speed postulate, [[#klein|Klein and Noether (1908-11) as well as Conway and Silberstein (1911)]] used Biquaternions, [[#Plummer|Plummer (1910) and Gruner (1921)]] used trigonometric Lorentz boosts, [[#Borel|Borel (1913–14)]] used Cayley-Hermite parameter.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Poincare|label 1=History of Lorentz transformations in general § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
====Imaginary Lorentz transformation====
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)|label 1=History of Lorentz transformations via imaginary orthogonal transformation}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)|label 1=History of Lorentz transformations via hyperbolic functions}}
The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation ({{equationNote|4b}}-A) assumes the form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

===={{anchor|Minkowski2}} Vector representation==== 
Minkowski wrote the Lorentz transformation ({{equationNote|4f}}) in vectorial form for the special case of directions being only parallel (&lt;math&gt;\mathfrak{r_{v}}&lt;/math&gt;) or perpendicular (&lt;math&gt;\mathfrak{r_{\bar{v}}}&lt;/math&gt;) to the velocity:&lt;ref group=R&gt;Minkowski (1907/08), pp. 62-63&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\mathfrak{r'_{v}}=\frac{\mathfrak{r_{v}}-qt}{\sqrt{1-q^{2}}},\quad\mathfrak{r'_{\bar{v}}}=\mathfrak{r_{\bar{v}}},\quad t'=\frac{-q\mathfrak{r_{v}}+t}{\sqrt{1-q^{2}}}\\
\mathfrak{r_{v}}=\frac{\mathfrak{r'_{v}}+qt'}{\sqrt{1-q^{2}}},\quad\mathfrak{r_{\bar{v}}}=\mathfrak{r'_{\bar{v}}},\quad t=\frac{q\mathfrak{r'_{v}}+t'}{\sqrt{1-q^{2}}}\\
\left[\mathfrak{r}=\left(x,y,z\right)=\left(\mathfrak{r_{v}},\mathfrak{r_{\bar{v}}}\right),\ |\mathfrak{v}|=q\right]
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski3}} Matrix representation====
Minkowski used matrices in order to write the [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']], of which boost matrix ({{equationNote|4h}}) is a special case:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski4}} Minkowski diagram====
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Apo|label 1=History of Lorentz transformations in general - Apollonius}}
Minkowski (1908/09) introduced the [[w:Minkowski diagram]] as a graphical representation of the Lorentz transformation, which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank, Varicak (1909-10) – Hyperbolic functions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Frank|label 1=History of Lorentz transformations via hyperbolic functions § Frank}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Varicak|label 1=History of Lorentz transformations via hyperbolic functions § Varicak}}

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
{{Main|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Sommerfeld|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Sommerfeld}}

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation and squeeze mappings===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Herglotz1|label 1=History of Lorentz transformations via Möbius transformations § Herglotz}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

==={{anchor|Plummer}} Plummer, Gruner (1910-21) – Trigonometric Lorentz boosts===
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Plummer|label 1=History of Lorentz transformations via trigonometric functions § Plummer}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Gruner|label 1=History of Lorentz transformations via trigonometric functions § Gruner}}

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|klein}} Klein, Noether, Conway, Silberstein (1908-11) – Biquaternions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein and Noether}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Conway|label 1=History of Lorentz transformations via Quaternions § Conway and Silberstein}}

==={{anchor|Herglotz2}} Ignatowski, Herglotz, Tamaki (1910-11) – Vector transformation===
[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
&lt;/math&gt;

Kajuro Tamaki (1911) represented ({{equationNote|4g}}) as follows (as his paper was based on a 4-vector calculus, Tamaki's schematic is not representing a matrix despite looking very similar to the boost matrix in ({{equationNote|4h}})):&lt;ref group=R&gt;Tamaki (1911), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{array}{c|c|c|c|c}
 &amp; x'_{1} &amp; x'_{2} &amp; x'_{3} &amp; x'_{4}\\
\hline x_{1} &amp; 1+l^{2}\left(\cos\psi-1\right) &amp; lm\left(\cos\psi-1\right) &amp; ln\left(\cos\psi-1\right) &amp; l\sin\psi\\
\hline x_{2} &amp; lm\left(\cos\psi-1\right) &amp; 1+m^{2}\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; m\sin\psi\\
\hline x_{3} &amp; ln\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; 1+n^{2}\left(\cos\psi-1\right) &amp; n\sin\psi\\
\hline x_{4} &amp; -l\sin\psi &amp; -m\sin\psi &amp; -n\sin\psi &amp; \cos\psi
\end{array}\\
\hline \psi=i\varphi,\ -i\tan i\varphi=-\frac{v}{c},\ \cos i\varphi=\frac{1}{\sqrt{1-(v/c)^{2}}}=\beta,\ -\sin i\varphi=\frac{i(v/c)}{\sqrt{1-(v/c)^{2}}}=i\beta(v/c)\\
\hline \mathbf{r}=\mathbf{r}'+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r}')+\beta\mathbf{v}t'\\
\mathbf{r}'=\mathbf{r}+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r})-\beta\mathbf{v}t
\end{matrix}&lt;/math&gt;

==={{anchor|Hahn}} Hahn (1912) – Matrix transformation ===
{{See|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Hahn|label 1=History of Lorentz transformations via Quaternions § Hahn}}
Elaborating on [[#Minkowski3|Minkowski's (1907/8)]] matrix representation of the Lorentz transformations, Emil Hahn (1912) used matrix calculus in order to define the Lorentz boost for arbitrary directions (including the exponential form of the boost matrix) in line with ({{equationNote|4h}}), using imaginary rapidity &lt;math&gt;i\psi&lt;/math&gt; and imaginary time &lt;math&gt;x_4 =i\omega t&lt;/math&gt;:&lt;ref group=R&gt;Hahn (1912), pp. 30-32 and p. 36.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}'_{0}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\\
\hline \begin{align}\mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{J}\mathbb{G}_{\mathbf{c}}(u)\mathbb{J}^{-1} &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(r-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \frac{iur(u)}{\omega}\mathbf{c}\\
-\frac{iur(u)}{\omega}\overset{\perp}{\mathbf{c}}; &amp; r(u)
\end{matrix}\right) &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(\cos i\psi-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \sin i\psi\mathbf{c}\\
-\sin i\psi\mathbf{c}; &amp; \cos i\psi
\end{matrix}\right) &amp; (8,p.30)\\
 &amp; =\left(\begin{matrix}1+(\cos i\psi-1)c_{1}c_{1} &amp; (\cos i\psi-1)c_{1}c_{2} &amp; (\cos i\psi-1)c_{1}c_{3} &amp; \sin i\psi\,c_{1}\\
(\cos i\psi-1)c_{2}c_{1} &amp; 1+(\cos i\psi-1)c_{2}c_{2} &amp; (\cos i\psi-1)c_{2}c_{3} &amp; \sin i\psi\,c_{2}\\
(\cos i\psi-1)c_{3}c_{1} &amp; (\cos i\psi-1)c_{3}c_{2} &amp; 1+(\cos i\psi-1)c_{3}c_{3} &amp; \sin i\psi\,c_{3}\\
-\sin i\psi\,c_{1} &amp; -\sin i\psi\,c_{2} &amp; -\sin i\psi\,c_{3} &amp; 1+(\cos i\psi-1)
\end{matrix}\right) &amp; (1,p.36)\\
\hline \mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{E}-\sin i\psi\left\langle \mathbf{0},\mathbf{c}\right\rangle +(1-\cos i\psi)\left\langle \mathbf{0},\mathbf{c}\right\rangle ^{2} &amp; (10,p.30)\\
\mathbb{I}_{\mathbf{c}}(u) &amp; =e^{\left\langle \mathbf{0},\mathbf{c}\right\rangle i\psi} &amp; (p.31)
\end{align}
\\
\hline \boldsymbol{x}=\left(\begin{matrix}x_{1} &amp; 0 &amp; 0 &amp; 0\\
x_{2} &amp; 0 &amp; 0 &amp; 0\\
x_{3} &amp; 0 &amp; 0 &amp; 0\\
x_{4} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \boldsymbol{x}'=\left(\begin{matrix}x_{1}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{2}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{3}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{4}^{\prime} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \mathbb{J}=\left(\begin{matrix}1 &amp; 0 &amp; 0 &amp; 0\\
0 &amp; 1 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; i\omega
\end{matrix}\right),\ \mathbb{G}=\left(\begin{matrix}r_{11} &amp; r_{12} &amp; r_{13} &amp; p_{1}\\
r_{21} &amp; r_{22} &amp; r_{23} &amp; p_{2}\\
r_{31} &amp; r_{32} &amp; r_{33} &amp; p_{3}\\
q_{1} &amp; q_{2} &amp; q_{3} &amp; r
\end{matrix}\right)\\
r(u)=\frac{1}{\sqrt{1-\frac{u^{2}}{\omega^{2}}}}=\cos i\psi,\ |\mathbb{I}|=1,\ \left\langle \mathbf{0},\mathbf{c}\right\rangle =\left(\begin{matrix}0 &amp; 0 &amp; 0 &amp; c_{1}\\
0 &amp; 0 &amp; 0 &amp; c_{2}\\
0 &amp; 0 &amp; 0 &amp; c_{3}\\
c_{1} &amp; c_{2} &amp; c_{3} &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Borel|label 1=History of Lorentz transformations via Cayley–Hermite transformation § Borel}}

==={{anchor|Born}} Born (1921) – Squeeze mapping===
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Born|label 1=History of Lorentz transformations via squeeze mappings § Born}}

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|hahn}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|tamaki11b}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>1xbdz669gyrm02wu98xvgij7l4xneu6</sha1>
    </revision>
    <revision>
      <id>2692688</id>
      <parentid>2692668</parentid>
      <timestamp>2024-12-19T20:41:31Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <minor />
      <comment>/* Lorentz transformation via velocity */</comment>
      <origin>2692688</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57416" sha1="7q96ntbkj8d10zc0d1epcnxy60t0eh1" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via velocity==
===Boosts===
In the [[w:theory of relativity]], Lorentz transformations exhibit the symmetry of [[w:Minkowski spacetime]] by using a constant ''c'' as the [[w:speed of light]], and a parameter ''v'' as the relative [[w:velocity]] between two [[w:inertial reference frames]]. The corresponding formulas are identical to [[../Lorentz transformation (hyperbolic)|E:Lorentz transformations via hyperbolic functions]] introduced long before relativity was developed. In particular, the hyperbolic angle &lt;math&gt;\eta&lt;/math&gt; can be interpreted as the velocity related [[w:rapidity]] &lt;math&gt;\tanh\eta=\beta=v/c&lt;/math&gt;, so that &lt;math&gt;\gamma=\cosh\eta&lt;/math&gt; is the [[w:Lorentz factor]], &lt;math&gt;\beta\gamma=\sinh\eta&lt;/math&gt; the [[w:proper velocity]], &lt;math&gt;u'=c\tanh q&lt;/math&gt; the velocity of another object, &lt;math&gt;u=c\tanh(q+\eta)&lt;/math&gt; the [[w:velocity-addition formula]], thus transformation [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']] becomes:

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\gamma-x_{1}\beta\gamma\\
x_{1}^{\prime} &amp; =-x_{0}\beta\gamma+x_{1}\gamma\\
\\
x_{0} &amp; =x_{0}^{\prime}\gamma+x_{1}^{\prime}\beta\gamma\\
x_{1} &amp; =x_{0}^{\prime}\beta\gamma+x_{1}^{\prime}\gamma
\end{align}
\left|{\scriptstyle \begin{align}\beta^{2}\gamma^{2}-\gamma^{2} &amp; =-1 &amp; (a)\\
\gamma^{2}-\beta^{2}\gamma^{2} &amp; =1 &amp; (b)\\
\frac{\beta\gamma}{\gamma} &amp; =\beta &amp; (c)\\
\frac{1}{\sqrt{1-\beta^{2}}} &amp; =\gamma &amp; (d)\\
\frac{\beta}{\sqrt{1-\beta^{2}}} &amp; =\beta\gamma &amp; (e)\\
\frac{u'+v}{1+\frac{u'v}{c^{2}}} &amp; =u &amp; (f)
\end{align}
}\right.
\end{matrix}&lt;/math&gt;|{{equationRef|4a}}}}

Written in four dimensions by setting &lt;math&gt;x_{0}=ct,\ x_{1}=x&lt;/math&gt; and adding  &lt;math&gt;y,z&lt;/math&gt; the familiar form follows

{{NumBlk|:|&lt;math&gt;\scriptstyle(A)\quad\begin{matrix}-c^{2}t^{2}+x^{2}+y^{2}+z^{2}=-c^{2}t^{\prime2}+x^{\prime2}+y^{\prime2}+z^{\prime2}\\
\hline \left.\begin{align}t' &amp; =\gamma\left(t-x\frac{v}{c^{2}}\right)\\
x' &amp; =\gamma(x-vt)\\
y' &amp; =y\\
z' &amp; =z
\end{align}
\right|\begin{align}t &amp; =\gamma\left(t'+x\frac{v}{c^{2}}\right)\\
x &amp; =\gamma(x'+vt')\\
y &amp; =y'\\
z &amp; =z'
\end{align}
\end{matrix}&lt;/math&gt;

or in matrix notation:

&lt;math&gt;\scriptstyle(B)\quad\begin{matrix}\mathbf{x}'=\begin{bmatrix}\gamma &amp; -\beta\gamma &amp; 0 &amp; 0\\
-\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x};\quad\mathbf{x}=\begin{bmatrix}\gamma &amp; \beta\gamma &amp; 0 &amp; 0\\
\beta\gamma &amp; \gamma &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; 1
\end{bmatrix}\cdot\mathbf{x}'\\
\det\begin{bmatrix}\gamma &amp; -\beta\gamma\\
-\beta\gamma &amp; \gamma
\end{bmatrix}=1
\end{matrix}&lt;/math&gt;

or in terms of &lt;math&gt;ct,x&lt;/math&gt; as squeeze mapping in line with [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']]:

&lt;math&gt;\scriptstyle(C)\quad\begin{matrix}uw=-x_{0}^{2}+x_{1}^{2}=u'w'=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{matrix}\begin{align}u' &amp; =ku\\
w' &amp; =\frac{1}{k}w
\end{align}
 &amp; \Rightarrow &amp; \begin{align}x'-ct' &amp; =\sqrt{\frac{c+v}{c-v}}\left(x-ct\right)\\
x'+ct' &amp; =\sqrt{\frac{c-v}{c+v}}\left(x+ct\right)
\end{align}
\quad\begin{align}x-ct &amp; =\sqrt{\frac{c-v}{c+v}}\left(x'-ct'\right)\\
x+ct &amp; =\sqrt{\frac{c+v}{c-v}}\left(x'+ct'\right)
\end{align}
\end{matrix}\\
\hline k=\sqrt{\frac{c+v}{c-v}}
\end{matrix}&lt;/math&gt;|{{equationRef|4b}}}}

Transformations analogous to (A) have been introduced by [[#Voigt|Voigt (1887)]] in terms of an incompressible medium, and by [[#Lorentz1|Lorentz (1892, 1895)]] who analyzed [[w:Maxwell's equations]], they were completed by [[#Larmor|Larmor (1897, 1900)]] and [[#Lorentz2|Lorentz (1899, 1904)]], and brought into their modern form by [[#Poincare3|Poincaré (1905)]] who gave the transformation the name of Lorentz.&lt;ref&gt;Miller (1981), chapter 1&lt;/ref&gt; Eventually, [[#Einstein|Einstein (1905)]] showed in his development of [[w:special relativity]] that the transformations follow from the [[w:principle of relativity]] and constant light speed alone by modifying the traditional concepts of space and time, without requiring a [[w:Lorentz ether theory|mechanical aether]] in contradistinction to Lorentz and Poincaré.&lt;ref&gt;Miller (1981), chapter 4–7&lt;/ref&gt; [[#Minkowski|Minkowski (1907–1908)]] used them to argue that space and time are inseparably connected as [[w:spacetime]]. 

The matrix form (B) is a special case of the general boost matrix given by [[#Hahn|Hahn (1912)]] in terms of imaginary time, while variant (C) for arbitrary ''k'' was given by many authors (see [[../Lorentz transformation (squeeze)|E:Lorentz transformations via squeeze mappings]]) with the choice equivalent to &lt;math&gt;k=\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; given by [[#Born|Born (1921)]].

===Velocity addition and aberration===
In exact analogy to Beltrami coordinates in equation [[../Lorentz transformation (hyperbolic)#math_3e|E:'''(3e)''']], one can substitute &lt;math&gt;\left[\tfrac{u_{x}}{c},\ \tfrac{u_{y}}{c},\ \tfrac{u_{z}}{c}\right]=\left[\tfrac{x}{ct},\ \tfrac{y}{ct},\ \tfrac{z}{ct}\right]&lt;/math&gt; in ({{equationNote|4b}}-A), producing the Lorentz transformation of velocities (or [[w:velocity addition formula]]):

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{align}u_{x}^{\prime} &amp; =\frac{-c^{2}\sinh\eta+u_{x}c\cosh\eta}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{x}-c\tanh\eta}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}-v}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{y}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
u_{z}^{\prime} &amp; =\frac{cu_{y}}{c\cosh\eta-u_{x}\sinh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\tanh^{2}\eta}}{1-\frac{u_{x}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u{}_{x}}\\
\\
\hline \\
u_{x} &amp; =\frac{c^{2}\sinh\eta+u_{x}^{\prime}c\cosh\eta}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+c\tanh\eta}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{x}^{\prime}+v}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{y} &amp; =\frac{cy'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}\\
u_{z} &amp; =\frac{cz'}{c\cosh\eta+u_{x}^{\prime}\sinh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\tanh^{2}\eta}}{1+\frac{u_{x}^{\prime}}{c}\tanh\eta} &amp;  &amp; =\frac{u_{z}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}}
\end{align}&lt;/math&gt;|{{equationRef|4c}}}}

By restriction to velocities in the &lt;math&gt;\left[x,y\right]&lt;/math&gt; plane and using trigonometric and hyperbolic identities as in equation [[../Lorentz transformation (hyperbolic)#math_3f|E:'''(3f)''']], it becomes the hyperbolic law of cosines:&lt;ref name=pau&gt;Pauli (1921), p. 561&lt;/ref&gt;&lt;ref group=R name=var&gt;Varićak (1912), p. 108&lt;/ref&gt;&lt;ref name=barr&gt;Barrett (2006), chapter 4, section 2&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix} &amp; \begin{matrix}u^{2}=u_{x}^{2}+u_{y}^{2}\\
u'^{2}=u_{x}^{\prime2}+u_{y}^{\prime2}
\end{matrix}\left|\begin{align}u_{x}=u\cos\alpha &amp; =\frac{u'\cos\alpha'+v}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{x}^{\prime}=u'\cos\alpha' &amp; =\frac{u\cos\alpha-v}{1-\frac{v}{c^{2}}u\cos\alpha}\\
u_{y}=u\sin\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u'\cos\alpha'}, &amp; u_{y}^{\prime}=u'\sin\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\frac{u_{y}}{u_{x}}=\tan\alpha &amp; =\frac{u'\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{u'\cos\alpha'+v}, &amp; \frac{u_{y}^{\prime}}{u_{x}^{\prime}}=\tan\alpha' &amp; =\frac{u\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{u\cos\alpha-v}
\end{align}
\right.\\
\\
\Rightarrow &amp; u=\frac{\sqrt{v^{2}+u^{\prime2}+2vu'\cos\alpha'-\left(\frac{vu'\sin\alpha'}{c}\right){}^{2}}}{1+\frac{v}{c^{2}}u'\cos\alpha'},\quad u'=\frac{\sqrt{-v^{2}-u^{2}+2vu\cos\alpha+\left(\frac{vu\sin\alpha}{c}\right){}^{2}}}{1-\frac{v}{c^{2}}u\cos\alpha}\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\frac{u^{\prime2}}{c^{2}}}}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}-\frac{v/c}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\frac{u/c}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\cos\alpha\\
\Rightarrow &amp; \frac{1}{\sqrt{1-\tanh^{2}\xi}}=\frac{1}{\sqrt{1-\tanh^{2}\eta}}\frac{1}{\sqrt{1-\tanh^{2}\zeta}}-\frac{\tanh\eta}{\sqrt{1-\tanh^{2}\eta}}\frac{\tanh\zeta}{\sqrt{1-\tanh^{2}\zeta}}\cos\alpha\\
\Rightarrow &amp; \cosh\xi=\cosh\eta\cosh\zeta-\sinh\eta\sinh\zeta\cos\alpha
\end{matrix}&lt;/math&gt;|{{equationRef|4d}}}}

and by further setting ''u=u′=c'' one gets the well known [[../Lorentz transformation (hyperbolic)#math_3g|E:Kepler formulas '''(3g)''']], which  express the relativistic [[w:aberration of light]]:&lt;ref&gt;Pauli (1921), pp. 562; 565–566&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\cos\alpha=\frac{\cos\alpha'+\frac{v}{c}}{1+\frac{v}{c}\cos\alpha'},\ \sin\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c}\cos\alpha'},\ \tan\alpha=\frac{\sin\alpha'\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha'+\frac{v}{c}},\ \tan\frac{\alpha}{2}=\sqrt{\frac{c-v}{c+v}}\tan\frac{\alpha'}{2}\\
\cos\alpha'=\frac{\cos\alpha-\frac{v}{c}}{1-\frac{v}{c}\cos\alpha},\ \sin\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c}\cos\alpha},\ \tan\alpha'=\frac{\sin\alpha\sqrt{1-\frac{v^{2}}{c^{2}}}}{\cos\alpha-\frac{v}{c}},\ \tan\frac{\alpha'}{2}=\sqrt{\frac{c+v}{c-v}}\tan\frac{\alpha}{2}
\end{matrix} &lt;/math&gt;|{{equationRef|4e}}}}

Formulas ({{equationNote|4c}}, {{equationNote|4d}}) were given by [[#Einstein|Einstein (1905)]] and [[#Poincare3|Poincaré (1905/06)]], while the relations to the spherical and hyperbolic law of cosines were given by [[#Sommerfeld|Sommerfeld (1909)]] and [[#Frank|Varićak (1910)]]. The aberration formula for cos(α) was given by [[#Einstein|Einstein (1905)]].&lt;ref group=R name=plum&gt;Plummer (1910), pp. 258-259: After deriving the relativistic expressions for the aberration angles φ' and φ, Plummer remarked on p. 259: ''Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β.''&lt;/ref&gt;&lt;ref name=robin&gt;Robinson (1990), chapter 3-4, analyzed the relation between &quot;Kepler's formula&quot; and the &quot;physical velocity addition formula&quot; in special relativity.&lt;/ref&gt;

===Lorentz transformation in arbitrary directions===
Lorentz boosts for arbitrary directions&lt;ref&gt;Møller (1952/55), Chapter II, § 18&lt;/ref&gt; in line with [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']] are in vector notation

{{NumBlk|:|&lt;math&gt;\begin{align}t' &amp; =\gamma\left(t-\frac{v\mathbf{n}\cdot\mathbf{r}}{c^{2}}\right)\\
\mathbf{r}' &amp; =\mathbf{r}+(\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n}-\gamma tv\mathbf{n}
\end{align}
&lt;/math&gt;|{{equationRef|4f}}}}

and the vectorial velocity addition formula in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation '''(1b)''']] follows by:

{{NumBlk|:|&lt;math&gt;\mathbf{u}'=\frac{1}{1+\frac{\mathbf{v}\cdot\mathbf{u}}{c^{2}}}\left[\frac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}(\mathbf{u}\cdot\mathbf{v})\mathbf{v}\right]&lt;/math&gt;|{{equationRef|4g}}}}

The special case of parallel and perpendicular directions in ({{equationNote|4f}}) was given by [[#Minkowski2|Minkowski (1907/8)]] while the complete transformation was formulated by [[#Herglotz2|Ignatowski (1910), Herglotz (1911), Tamaki (1911)]]. General velocity addition ({{equationNote|4g}}) was given in equivalent form by [[#Herglotz2|Ignatowski (1910)]].

Rewritten in matrix notation, the general Lorentz boost has the form:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\
\hline \begin{align}\mathbf{g} &amp; =\begin{pmatrix}\gamma &amp; -\gamma\beta n_{x} &amp; -\gamma\beta n_{y} &amp; -\gamma\beta n_{z}\\
-\gamma\beta n_{x} &amp; 1+(\gamma-1)n_{x}^{2} &amp; (\gamma-1)n_{x}n_{y} &amp; (\gamma-1)n_{x}n_{z}\\
-\gamma\beta n_{y} &amp; (\gamma-1)n_{y}n_{x} &amp; 1+(\gamma-1)n_{y}^{2} &amp; (\gamma-1)n_{y}n_{z}\\
-\gamma\beta n_{z} &amp; (\gamma-1)n_{z}n_{x} &amp; (\gamma-1)n_{z}n_{y} &amp; 1+(\gamma-1)n_{z}^{2}
\end{pmatrix}\end{align}
\\
\left[\mathbf{n}=\frac{\mathbf{v}}{v}\right]
\end{matrix}\,&lt;/math&gt;|{{equationRef|4h}}}}

While [[#Minkowski3|Minkowski (1907/8)]] formulated the matrix form of Lorentz transformations in general terms, he didn't explicitly express the velocity related components of the general boost matrix. A complete representation of ({{equationNote|4h}}) was given by [[#Hahn|Hahn (1912)]].

===Other formulations===
Important contributions to the mathematical understanding of the Lorentz transformation of space and time also include: [[#Minkowski|Minkowski (1907–1908)]] as well as [[#Frank|Frank (1909) and Varićak (1910)]] showed the relation to imaginary and hyperbolic functions, [[#Herglotz1|Herglotz (1909/10)]] used exponential squeeze mappings and Möbius transformations, [[#Ignatowski|Ignatowski (1910)]] didn't use the light speed postulate, [[#klein|Klein and Noether (1908-11) as well as Conway and Silberstein (1911)]] used Biquaternions, [[#Plummer|Plummer (1910) and Gruner (1921)]] used trigonometric Lorentz boosts, [[#Borel|Borel (1913–14)]] used Cayley-Hermite parameter.

==Historical notation==
==={{anchor|Voigt}} Voigt (1887) ===

[[w:Woldemar Voigt]] (1887)&lt;ref group=R&gt;Voigt (1887), p. 45&lt;/ref&gt; developed a transformation in connection with the [[w:Doppler effect]] and an incompressible medium, being in modern notation:&lt;ref&gt;Miller (1981), 114–115&lt;/ref&gt;&lt;ref name=pais&gt;Pais (1982), Kap. 6b&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\xi_{1} &amp; =x_{1}-\varkappa t\\
\eta_{1} &amp; =y_{1}q\\
\zeta_{1} &amp; =z_{1}q\\
\tau &amp; =t-\frac{\varkappa x_{1}}{\omega^{2}}\\
q &amp; =\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =\frac{y}{\gamma}\\
z^{\prime} &amp; =\frac{z}{\gamma}\\
t^{\prime} &amp; =t-\frac{vx}{c^{2}}\\
\frac{1}{\gamma} &amp; =\sqrt{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

If the right-hand sides of his equations are multiplied by γ they are the modern Lorentz transformation ({{equationNote|4b}}). In Voigt's theory the speed of light is invariant, but his transformations mix up a relativistic boost together with a rescaling of space-time. Optical phenomena in free space are [[w:Scale invariance|scale]], [[w:Conformal map|conformal]] (using the factor λ discussed [[#Lorsph|above]]), and [[w:Lorentz covariance|Lorentz invariant]], so the combination is invariant too.&lt;ref name=pais /&gt; For instance, Lorentz transformations can be extended by using &lt;math&gt;l=\sqrt{\lambda}&lt;/math&gt;:&lt;ref group=R&gt;Lorentz (1915/16), p. 197&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz,\quad t^{\prime}=\gamma l\left(t-x\frac{v}{c^{2}}\right)&lt;/math&gt;.

''l''=1/γ gives the Voigt transformation, ''l''=1 the Lorentz transformation. But scale transformations are not a symmetry of all the laws of nature, only of electromagnetism, so these transformations cannot be used to formulate a [[w:principle of relativity]] in general. It was demonstrated by Poincaré and Einstein that one has to set ''l''=1 in order to make the above transformation symmetric and to form a group as required by the relativity principle, therefore the Lorentz transformation is the only viable choice.

Voigt sent his 1887 paper to Lorentz in 1908,&lt;ref&gt;Voigt's transformations and the beginning of the relativistic revolution, Ricardo Heras, arXiv:1411.2559 [https://arxiv.org/abs/1411.2559]&lt;/ref&gt; and that was acknowledged in 1909: {{Quote|In a paper &quot;Über das Doppler'sche Princip&quot;, published in 1887 (Gött. Nachrichten, p. 41) and which to my regret has escaped my notice all these years, Voigt has applied to equations of the form (7) (§ 3 of this book) [namely &lt;math&gt;\Delta\Psi-\tfrac{1}{c^{2}}\tfrac{\partial^{2}\Psi}{\partial t^{2}}=0&lt;/math&gt;] a transformation equivalent to the formulae (287) and (288) [namely &lt;math&gt;x^{\prime}=\gamma l\left(x-vt\right),\ y^{\prime}=ly,\ z^{\prime}=lz,\ t^{\prime}=\gamma l\left(t-\tfrac{v}{c^{2}}x\right)&lt;/math&gt;]. The idea of the transformations used above (and in § 44) might therefore have been borrowed from Voigt and the proof that it does not alter the form of the equations for the ''free'' ether is contained in his paper.&lt;ref group=R&gt;Lorentz (1915/16), p. 198&lt;/ref&gt;}}

Also [[w:Hermann Minkowski]] said in 1908 that the transformations which play the main role in the principle of relativity were first examined by Voigt in 1887. Voigt responded in the same paper by saying that his theory was based on an elastic theory of light, not an electromagnetic one. However, he concluded that some results were actually the same.&lt;ref group=R&gt;Bucherer (1908), p. 762&lt;/ref&gt;

==={{anchor|Heaviside}} Heaviside (1888), Thomson (1889), Searle (1896)===

In 1888, [[w:Oliver Heaviside]]&lt;ref group=R&gt;Heaviside (1888), p. 324&lt;/ref&gt; investigated the properties of [[w:Relativistic electromagnetism|charges in motion]] according to Maxwell's electrodynamics. He calculated, among other things, anisotropies in the electric field of moving bodies represented by this formula:&lt;ref&gt;Brown (2003)&lt;/ref&gt;

:&lt;math&gt;\mathrm{E}=\left(\frac{q\mathrm{r}}{r^{2}}\right)\left(1-\frac{v^{2}\sin^{2}\theta}{c^{2}}\right)^{-3/2}&lt;/math&gt;.

Consequently, [[w:Joseph John Thomson]] (1889)&lt;ref group=R&gt;Thomson (1889), p. 12&lt;/ref&gt; found a way to substantially simplify calculations concerning moving charges by using the following mathematical transformation (like other authors such as Lorentz or Larmor, also Thomson implicitly used the [[w:Galilean transformation]] ''z-vt'' in his equation&lt;ref name=mil /&gt;):

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}z &amp; =\left\{ 1-\frac{\omega^{2}}{v^{2}}\right\} ^{\frac{1}{2}}z'\end{align}
\right| &amp; \begin{align}z^{\ast}=z-vt &amp; =\frac{z'}{\gamma}\end{align}
\end{matrix}&lt;/math&gt;

Thereby, [[w:inhomogeneous electromagnetic wave equation]]s are transformed into a [[w:Poisson equation]].&lt;ref name=mil&gt;Miller (1981), 98–99&lt;/ref&gt; Eventually, [[w:George Frederick Charles Searle]]&lt;ref group=R&gt;Searle (1886), p. 333&lt;/ref&gt; noted in (1896) that Heaviside's expression leads to a deformation of electric fields which he called &quot;Heaviside-Ellipsoid&quot; of [[w:axial ratio]]

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align} &amp; \sqrt{\alpha}:1:1\\
\alpha= &amp; 1-\frac{u^{2}}{v^{2}}
\end{align}
\right| &amp; \begin{align} &amp; \frac{1}{\gamma}:1:1\\
\frac{1}{\gamma^{2}} &amp; =1-\frac{v^{2}}{c^{2}}
\end{align}
\end{matrix}&lt;/math&gt;&lt;ref name=mil /&gt;

=== {{anchor|Lorentz1}} Lorentz (1892, 1895) ===

In order to explain the [[w:aberration of light]] and the result of the [[w:Fizeau experiment]] in accordance with [[w:Maxwell's equations]], Lorentz in 1892 developed a model (&quot;[[w:Lorentz ether theory]]&quot;) in which the aether is completely motionless, and the speed of light in the aether is constant in all directions. In order to calculate the optics of moving bodies, Lorentz introduced the following quantities to transform from the aether system into a moving system (it's unknown whether he was influenced by Voigt, Heaviside, and Thomson)&lt;ref group=R&gt;Lorentz (1892a), p. 141&lt;/ref&gt;&lt;ref name=milf&gt;Miller (1982), 1.4 &amp; 1.5&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}\mathfrak{x} &amp; =\frac{V}{\sqrt{V^{2}-p^{2}}}x\\
t' &amp; =t-\frac{\varepsilon}{V}\mathfrak{x}\\
\varepsilon &amp; =\frac{p}{\sqrt{V^{2}-p^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)\\
\gamma\frac{v}{c} &amp; =\frac{v}{\sqrt{c^{2}-v^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

where ''x&lt;sup&gt;*&lt;/sup&gt;'' is the [[w:Galilean transformation]] ''x-vt''. Except the additional γ in the time transformation, this is the complete Lorentz transformation ({{equationNote|4b}}).&lt;ref name=milf /&gt; While ''t'' is the &quot;true&quot; time for observers resting in the aether, ''t′'' is an auxiliary variable only for calculating processes for moving systems. It is also important that Lorentz and later also Larmor formulated this transformation in two steps. At first an implicit Galilean transformation, and later the expansion into the &quot;fictitious&quot; electromagnetic system with the aid of the Lorentz transformation. In order to explain the negative result of the [[w:Michelson–Morley experiment]], he (1892b)&lt;ref group=R&gt;Lorentz (1892b), p. 141&lt;/ref&gt; introduced the additional hypothesis that also intermolecular forces are affected in a similar way and introduced [[w:length contraction]] in his theory (without proof as he admitted). The same hypothesis was already made by [[w:George FitzGerald]] in 1889 based on Heaviside's work. While length contraction was a real physical effect for Lorentz, he considered the time transformation only as a heuristic working hypothesis and a mathematical stipulation.

In 1895, Lorentz further elaborated on his theory and introduced the &quot;theorem of corresponding states&quot;. This theorem states that a moving observer (relative to the ether) in his &quot;fictitious&quot; field makes the same observations as a resting observers in his &quot;real&quot; field for velocities to first order in ''v/c''. Lorentz showed that the dimensions of electrostatic systems in the ether and a moving frame are connected by this transformation:&lt;ref group=R&gt;Lorentz (1895), p. 37&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =x^{\prime}\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{x^{\prime}}{\gamma}\\
y &amp; =y^{\prime}\\
z &amp; =z^{\prime}\\
t &amp; =t^{\prime}
\end{align}
\end{matrix}&lt;/math&gt;

For solving optical problems Lorentz used the following transformation, in which the modified time variable was called &quot;local time&quot; ({{lang-de|Ortszeit}}) by him:&lt;ref group=R&gt;Lorentz (1895), p. 49 for local time and p. 56 for spatial coordinates.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\mathrm{x}-\mathfrak{p}_{x}t\\
y &amp; =\mathrm{y}-\mathfrak{p}_{y}t\\
z &amp; =\mathrm{z}-\mathfrak{p}_{z}t\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}}x-\frac{\mathfrak{p}_{y}}{V^{2}}y-\frac{\mathfrak{p}_{z}}{V^{2}}z
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-v_{x}t\\
y^{\prime} &amp; =y-v_{y}t\\
z^{\prime} &amp; =z-v_{z}t\\
t^{\prime} &amp; =t-\frac{v_{x}}{c^{2}}x'-\frac{v_{y}}{c^{2}}y'-\frac{v_{z}}{c^{2}}z'
\end{align}
\end{matrix}&lt;/math&gt;

With this concept Lorentz could explain the [[w:Doppler effect]], the [[w:aberration of light]], and the [[w:Fizeau experiment]].&lt;ref&gt;Janssen (1995), 3.1&lt;/ref&gt;

=== {{anchor|Larmor}} Larmor (1897, 1900) ===

In 1897, Larmor extended the work of Lorentz and derived the following transformation&lt;ref group=R&gt;Larmor (1897), p. 229&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =x\varepsilon^{\frac{1}{2}}\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-vx/c^{2}\\
dt_{1} &amp; =dt^{\prime}\varepsilon^{-\frac{1}{2}}\\
\varepsilon &amp; =\left(1-v^{2}/c^{2}\right)^{-1}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y_{1} &amp; =y\\
z_{1} &amp; =z\\
t^{\prime} &amp; =t-\frac{vx^{\ast}}{c^{2}}=t-\frac{v(x-vt)}{c^{2}}\\
dt_{1} &amp; =\frac{dt^{\prime}}{\gamma}\\
\gamma^{2} &amp; =\frac{1}{1-\frac{v^{2}}{c^{2}}}
\end{align}
\end{matrix}&lt;/math&gt;

Larmor noted that if it is assumed that the constitution of molecules is electrical then the FitzGerald–Lorentz contraction is a consequence of this transformation, explaining the [[w:Michelson–Morley experiment]]. It's notable that Larmor was the first who recognized that some sort of [[w:time dilation]] is a consequence of this transformation as well, because &quot;individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio 1/γ&quot;.&lt;ref&gt;Darrigol (2000), Chap. 8.5&lt;/ref&gt;&lt;ref&gt;Macrossan (1986)&lt;/ref&gt; Larmor wrote his electrodynamical equations and transformations neglecting terms of higher order than ''(v/c)''&lt;sup&gt;2&lt;/sup&gt; – when his 1897 paper was reprinted in 1929, Larmor added the following comment in which he described how they can be made valid to all orders of ''v/c'':&lt;ref group=R&gt;Larmor (1897/1929), p. 39&lt;/ref&gt;

{{Quote|Nothing need be neglected: the transformation is ''exact'' if ''v/c''&lt;sup&gt;2&lt;/sup&gt; is replaced by ''εv/c''&lt;sup&gt;2&lt;/sup&gt; in the equations and also in the change following from ''t'' to ''t′'', as is worked out in ''Aether and Matter'' (1900), p. 168, and as Lorentz found it to be in 1904, thereby stimulating the modern schemes of intrinsic relational relativity.}}

In line with that comment, in his book Aether and Matter published in 1900, Larmor used a modified local time ''t″=t′-εvx′/c&lt;sup&gt;2&lt;/sup&gt;'' instead of the 1897 expression ''t′=t-vx/c&lt;sup&gt;2&lt;/sup&gt;'' by replacing ''v/c''&lt;sup&gt;2&lt;/sup&gt; with ''εv/c''&lt;sup&gt;2&lt;/sup&gt;, so that ''t″'' is now identical to the one given by Lorentz in 1892, which he combined with a Galilean transformation for the ''x′, y′, z′, t′'' coordinates:&lt;ref group=R&gt;Larmor (1900), p. 168&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime} &amp; =t^{\prime}-\varepsilon vx^{\prime}/c^{2}
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =x-vt\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t\\
t^{\prime\prime}=t^{\prime}-\frac{\gamma^{2}vx^{\prime}}{c^{2}} &amp; =\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Larmor knew that the Michelson–Morley experiment was accurate enough to detect an effect of motion depending on the factor ''(v/c)''&lt;sup&gt;2&lt;/sup&gt;, and so he sought the transformations which were &quot;accurate to second order&quot; (as he put it). Thus he wrote the final transformations (where ''x′=x-vt'' and ''t″'' as given above) as:&lt;ref group=R&gt;Larmor (1900), p. 174&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x_{1} &amp; =\varepsilon^{\frac{1}{2}}x^{\prime}\\
y_{1} &amp; =y^{\prime}\\
z_{1} &amp; =z^{\prime}\\
dt_{1} &amp; =\varepsilon^{-\frac{1}{2}}dt^{\prime\prime}=\varepsilon^{-\frac{1}{2}}\left(dt^{\prime}-\frac{v}{c^{2}}\varepsilon dx^{\prime}\right)\\
t_{1} &amp; =\varepsilon^{-\frac{1}{2}}t^{\prime}-\frac{v}{c^{2}}\varepsilon^{\frac{1}{2}}x^{\prime}
\end{align}
\right| &amp; \begin{align}x_{1} &amp; =\gamma x^{\prime}=\gamma(x-vt)\\
y_{1} &amp; =y'=y\\
z_{1} &amp; =z'=z\\
dt_{1} &amp; =\frac{dt^{\prime\prime}}{\gamma}=\frac{1}{\gamma}\left(dt^{\prime}-\frac{\gamma^{2}vdx^{\prime}}{c^{2}}\right)=\gamma\left(dt-\frac{vdx}{c^{2}}\right)\\
t_{1} &amp; =\frac{t^{\prime}}{\gamma}-\frac{\gamma vx^{\prime}}{c^{2}}=\gamma\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

by which he arrived at the complete Lorentz transformation ({{equationNote|4b}}). Larmor showed that Maxwell's equations were invariant under this two-step transformation, &quot;to second order in ''v/c''&quot; – it was later shown by Lorentz (1904) and Poincaré (1905) that they are indeed invariant under this transformation to all orders in ''v/c''.

Larmor gave credit to Lorentz in two papers published in 1904, in which he used the term &quot;Lorentz transformation&quot; for Lorentz's first order transformations of coordinates and field configurations:

{{Quote|p. 583: [..] Lorentz's transformation for passing from the field of activity of a stationary electrodynamic material system to that of one moving with uniform velocity of translation through the aether.&lt;br /&gt; p. 585: [..] the Lorentz transformation has shown us what is not so immediately obvious [..]&lt;ref group=R&gt;Larmor (1904a), p. 583, 585&lt;/ref&gt; &lt;br /&gt; p. 622: [..] the transformation first developed by Lorentz: namely, each point in space is to have its own origin from which time is measured, its &quot;local time&quot; in Lorentz's phraseology, and then the values of the electric and magnetic vectors [..] at all points in the aether between the molecules in the system at rest, are the same as those of the vectors [..] at the corresponding points in the convected system at the same local times.&lt;ref group=R&gt;Larmor (1904b), p. 622&lt;/ref&gt;}}

=== {{anchor|Lorentz2}} Lorentz (1899, 1904) ===

Also Lorentz extended his theorem of corresponding states in 1899. First he wrote a transformation equivalent to the one from 1892 (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1899), p. 429&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}x\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\mathfrak{p}_{x}}{V^{2}-\mathfrak{p}_{x}^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma x^{\ast}=\gamma(x-vt)\\
y^{\prime} &amp; =y\\
z^{\prime} &amp; =z\\
t^{\prime} &amp; =t-\frac{\gamma^{2}vx^{\ast}}{c^{2}}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Then he introduced a factor ε of which he said he has no means of determining it, and modified his transformation as follows (where the above value of ''t′'' has to be inserted):&lt;ref group=R&gt;Lorentz (1899), p. 439&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x &amp; =\frac{\varepsilon}{k}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon x^{\prime\prime}\\
t^{\prime} &amp; =k\varepsilon t^{\prime\prime}\\
k &amp; =\frac{V}{\sqrt{V^{2}-\mathfrak{p}_{x}^{2}}}
\end{align}
\right| &amp; \begin{align}x^{\ast}=x-vt &amp; =\frac{\varepsilon}{\gamma}x^{\prime\prime}\\
y &amp; =\varepsilon y^{\prime\prime}\\
z &amp; =\varepsilon z^{\prime\prime}\\
t^{\prime}=\gamma^{2}\left(t-\frac{vx}{c^{2}}\right) &amp; =\gamma\varepsilon t^{\prime\prime}\\
\gamma &amp; =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
\end{align}
\end{matrix}&lt;/math&gt;

This is equivalent to the complete Lorentz transformation ({{equationNote|4b}}) when solved for ''x″'' and ''t″'' and with ε=1. Like Larmor, Lorentz noticed in 1899&lt;ref group=R&gt;Lorentz (1899), p. 442&lt;/ref&gt; also some sort of time dilation effect in relation to the frequency of oscillating electrons ''&quot;that in ''S'' the time of vibrations be ''kε'' times as great as in ''S&lt;sub&gt;0&lt;/sub&gt;''&quot;'', where ''S&lt;sub&gt;0&lt;/sub&gt;'' is the aether frame.&lt;ref&gt;Janssen (1995), Kap. 3.3&lt;/ref&gt;

In 1904 he rewrote the equations in the following form by setting ''l''=1/ε (again, ''x''* must be replaced by ''x-vt''):&lt;ref group=R&gt;Lorentz (1904), p. 812&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\text{original} &amp; \text{modern}\\
\hline \left.\begin{align}x^{\prime} &amp; =klx\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =\frac{l}{k}t-kl\frac{w}{c^{2}}x
\end{align}
\right| &amp; \begin{align}x^{\prime} &amp; =\gamma lx^{\ast}=\gamma l(x-vt)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t^{\prime} &amp; =\frac{lt}{\gamma}-\frac{\gamma lvx^{\ast}}{c^{2}}=\gamma l\left(t-\frac{vx}{c^{2}}\right)
\end{align}
\end{matrix}&lt;/math&gt;

Under the assumption that ''l=1'' when ''v''=0, he demonstrated that ''l=1'' must be the case at all velocities, therefore length contraction can only arise in the line of motion. So by setting the factor ''l'' to unity, Lorentz's transformations now assumed the same form as Larmor's and are now completed. Unlike Larmor, who restricted himself to show the covariance of Maxwell's equations to second order, Lorentz tried to widen its covariance to all orders in ''v/c''. He also derived the correct formulas for the velocity dependence of [[w:electromagnetic mass]], and concluded that the transformation formulas must apply to all forces of nature, not only electrical ones.&lt;ref group=R&gt;Lorentz (1904), p. 826&lt;/ref&gt; However, he didn't achieve full covariance of the transformation equations for charge density and velocity.&lt;ref&gt;Miller (1981), Chap. 1.12.2&lt;/ref&gt; When the 1904 paper was reprinted in 1913, Lorentz therefore added the following remark:&lt;ref&gt;Janssen (1995), Chap. 3.5.6&lt;/ref&gt;

{{Quote|One will notice that in this work the transformation equations of Einstein’s Relativity Theory have not quite been attained. [..] On this circumstance depends the clumsiness of many of the further considerations in this work.}}

Lorentz's 1904 transformation was cited and used by [[w:Alfred Bucherer]] in July 1904:&lt;ref group=R&gt;Bucherer, p. 129; Definition of s on p. 32&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}&lt;/math&gt;

or by [[w:Wilhelm Wien]] in July 1904:&lt;ref group=R&gt;Wien (1904), p. 394&lt;/ref&gt;

:&lt;math&gt;x=kx',\quad y=y',\quad z=z',\quad t'=kt-\frac{v}{kc^{2}}x&lt;/math&gt;

or by [[w:Emil Cohn]] in November 1904 (setting the speed of light to unity):&lt;ref group=R&gt;Cohn (1904a), pp. 1296-1297&lt;/ref&gt;

:&lt;math&gt;x=\frac{x_{0}}{k},\quad y=y_{0},\quad z=z_{0},\quad t=kt_{0},\quad t_{1}=t_{0}-w\cdot r_{0},\quad k^{2}=\frac{1}{1-w^{2}}&lt;/math&gt;

or by [[w:Richard Gans]] in February 1905:&lt;ref group=R&gt;Gans (1905), p. 169&lt;/ref&gt;

:&lt;math&gt;x^{\prime}=kx,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{k}-\frac{kwx}{c^{2}},\quad k^{2}=\frac{c^{2}}{c^{2}-w^{2}}&lt;/math&gt;

=== {{anchor|Poincare3}} Poincaré (1900, 1905) ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Poincare|label 1=History of Lorentz transformations in general § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Poincare2|label 1=History of Lorentz transformations via Möbius transformations § Poincaré}}

==== Local time ====

Neither Lorentz or Larmor gave a clear physical interpretation of the origin of local time. However, [[w:Henri Poincaré]] in 1900 commented on the origin of Lorentz's &quot;wonderful invention&quot; of local time.&lt;ref&gt;Darrigol (2005), Kap. 4&lt;/ref&gt; He remarked that it arose when clocks in a moving reference frame are synchronised by exchanging signals which are assumed to travel with the same speed &lt;math&gt;c&lt;/math&gt; in both directions, which lead to what is nowadays called [[w:relativity of simultaneity]], although Poincaré's calculation does not involve length contraction or time dilation.&lt;ref group=R&gt;Poincaré (1900), pp. 272–273&lt;/ref&gt; In order to synchronise the clocks here on Earth (the ''x*, t''* frame) a light signal from one clock (at the origin) is sent to another (at ''x''*), and is sent back. It's supposed that the Earth is moving with speed ''v'' in the ''x''-direction (= ''x''*-direction) in some rest system (''x, t'') (''i.e.'' the [[w:luminiferous aether]] system for Lorentz and Larmor). The time of flight outwards is

:&lt;math&gt;\delta t_{a}=\frac{x^{\ast}}{\left(c-v\right)}&lt;/math&gt;

and the time of flight back is

:&lt;math&gt;\delta t_{b}=\frac{x^{\ast}}{\left(c+v\right)}&lt;/math&gt;.

The elapsed time on the clock when the signal is returned is ''δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;'' and the time ''t*=(δt&lt;sub&gt;a&lt;/sub&gt;+δt&lt;sub&gt;b&lt;/sub&gt;)/2'' is ascribed to the moment when the light signal reached the distant clock. In the rest frame the time ''t=δt&lt;sub&gt;a&lt;/sub&gt;'' is ascribed to that same instant. Some algebra gives the relation between the different time coordinates ascribed to the moment of reflection. Thus

:&lt;math&gt;t^{\ast}=t-\frac{\gamma^{2}vx^{*}}{c^{2}}&lt;/math&gt;

identical to Lorentz (1892). By dropping the factor γ&lt;sup&gt;2&lt;/sup&gt; under the assumption that &lt;math&gt;\tfrac{v^{2}}{c^{2}}\ll1&lt;/math&gt;, Poincaré gave the result ''t*=t-vx*/c&lt;sup&gt;2&lt;/sup&gt;'', which is the form used by Lorentz in 1895.

Similar physical interpretations of local time were later given by [[w:Emil Cohn]] (1904)&lt;ref group=R&gt;Cohn (1904b), p. 1408&lt;/ref&gt; and [[w:Max Abraham]] (1905).&lt;ref group=R&gt;Abraham (1905), § 42&lt;/ref&gt;

==== Lorentz transformation ====

On June 5, 1905 (published June 9) Poincaré formulated transformation equations which are algebraically equivalent to those of Larmor and Lorentz and gave them the modern form ({{equationNote|4b}}):&lt;ref group=R&gt;Poincaré (1905), p. 1505&lt;/ref&gt;

:&lt;math&gt;\begin{align}x^{\prime} &amp; =kl(x+\varepsilon t)\\
y^{\prime} &amp; =ly\\
z^{\prime} &amp; =lz\\
t' &amp; =kl(t+\varepsilon x)\\
k &amp; =\frac{1}{\sqrt{1-\varepsilon^{2}}}
\end{align}
&lt;/math&gt;.

Apparently Poincaré was unaware of Larmor's contributions, because he only mentioned Lorentz and therefore used for the first time the name &quot;Lorentz transformation&quot;.&lt;ref&gt;Pais (1982), Chap. 6c&lt;/ref&gt;&lt;ref&gt;Katzir (2005), 280–288&lt;/ref&gt; Poincaré set the speed of light to unity, pointed out the group characteristics of the transformation by setting ''l''=1, and modified/corrected Lorentz's derivation of the equations of electrodynamics in some details in order to fully satisfy the principle of relativity, ''i.e.'' making them fully Lorentz covariant.&lt;ref&gt;Miller (1981), Chap. 1.14&lt;/ref&gt;

In July 1905 (published in January 1906)&lt;ref group=R&gt;Poincaré (1905/06), pp. 129ff&lt;/ref&gt; Poincaré showed in detail how the transformations and electrodynamic equations are a consequence of the [[w:principle of least action]]; he demonstrated in more detail the group characteristics of the transformation, which he called [[w:Lorentz group]], and he showed that the combination ''x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;+z&lt;sup&gt;2&lt;/sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;'' is invariant. He noticed that the Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing &lt;math&gt;ct\sqrt{-1}&lt;/math&gt; as a fourth imaginary coordinate, and he used an early form of [[w:four-vector]]s. He also formulated the velocity addition formula ({{equationNote|4c}}), which he had already derived in unpublished letters to Lorentz from May 1905:&lt;ref group=R&gt;Poincaré (1905/06), p. 144&lt;/ref&gt;

:&lt;math&gt;\xi'=\frac{\xi+\varepsilon}{1+\xi\varepsilon},\ \eta'=\frac{\eta}{k(1+\xi\varepsilon)}&lt;/math&gt;.

==={{anchor|Einstein}} Einstein (1905) – Special relativity===

On June 30, 1905 (published September 1905) Einstein published what is now called [[w:special relativity]] and gave a new derivation of the transformation, which was based only on the principle on relativity and the principle of the constancy of the speed of light. While Lorentz considered &quot;local time&quot; to be a mathematical stipulation device for explaining the Michelson-Morley experiment, Einstein showed that the coordinates given by the Lorentz transformation were in fact the inertial coordinates of relatively moving frames of reference. For quantities of first order in ''v/c'' this was also done by Poincaré in 1900, while Einstein derived the complete transformation by this method. Unlike Lorentz and Poincaré who still distinguished between real time in the aether and apparent time for moving observers, Einstein showed that the transformations concern the nature of space and time.&lt;ref&gt;Miller (1981), Chap. 6&lt;/ref&gt;&lt;ref&gt;Pais (1982), Kap. 7&lt;/ref&gt;&lt;ref&gt;Darrigol (2005), Chap. 6&lt;/ref&gt;

The notation for this transformation is equivalent to Poincaré's of 1905 and ({{equationNote|4b}}), except that Einstein didn't set the speed of light to unity:&lt;ref group=R&gt;Einstein (1905), p. 902&lt;/ref&gt;

:&lt;math&gt;\begin{align}\tau &amp; =\beta\left(t-\frac{v}{V^{2}}x\right)\\
\xi &amp; =\beta(x-vt)\\
\eta &amp; =y\\
\zeta &amp; =z\\
\beta &amp; =\frac{1}{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}
\end{align}
&lt;/math&gt;

Einstein also defined the velocity addition formula ({{equationNote|4c}}, {{equationNote|4d}}):&lt;ref group=R&gt;Einstein (1905), § 5 and § 9&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=\frac{w_{\xi}+v}{1+\frac{vw_{\xi}}{V^{2}}}t,\ y=\frac{\sqrt{1-\left(\frac{v}{V}\right)^{2}}}{1+\frac{vw_{\xi}}{V^{2}}}w_{\eta}t\\
U^{2}=\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2},\ w^{2}=w_{\xi}^{2}+w_{\eta}^{2},\ \alpha=\operatorname{arctg}\frac{w_{y}}{w_{x}}\\
U=\frac{\sqrt{\left(v^{2}+w^{2}+2vw\cos\alpha\right)-\left(\frac{vw\sin\alpha}{V}\right)^{2}}}{1+\frac{vw\cos\alpha}{V^{2}}}
\end{matrix}\left|\begin{matrix}\frac{u_{x}-v}{1-\frac{u_{x}v}{V^{2}}}=u_{\xi}\\
\frac{u_{y}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\eta}\\
\frac{u_{z}}{\beta\left(1-\frac{u_{x}v}{V^{2}}\right)}=u_{\zeta}
\end{matrix}\right.&lt;/math&gt;

and the light aberration formula ({{equationNote|4e}}):&lt;ref group=R&gt;Einstein (1905), § 7&lt;/ref&gt;

:&lt;math&gt;\cos\varphi'=\frac{\cos\varphi-\frac{v}{V}}{1-\frac{v}{V}\cos\varphi}&lt;/math&gt;

=== {{anchor|Minkowski}} Minkowski (1907–1908) – Spacetime ===
====Imaginary Lorentz transformation====
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)|label 1=History of Lorentz transformations via imaginary orthogonal transformation}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)|label 1=History of Lorentz transformations via hyperbolic functions}}
The work on the principle of relativity by Lorentz, Einstein, [[w:Max Planck|Planck]], together with Poincaré's four-dimensional approach, were further elaborated and combined with the [[w:hyperboloid model]] by [[w:Hermann Minkowski]] in 1907 and 1908.&lt;ref group=R&gt;Minkowski (1907/15), pp. 927ff&lt;/ref&gt;&lt;ref group=R&gt;Minkowski (1907/08), pp. 53ff&lt;/ref&gt; Minkowski particularly reformulated electrodynamics in a four-dimensional way ([[w:Minkowski spacetime]]).&lt;ref&gt;Walter (1999a), (1999b), (2018)&lt;/ref&gt; For instance, he wrote ''x, y, z, it'' in the form ''x&lt;sub&gt;1&lt;/sub&gt;, x&lt;sub&gt;2&lt;/sub&gt;, x&lt;sub&gt;3&lt;/sub&gt;, x&lt;sub&gt;4&lt;/sub&gt;''. By defining ψ as the angle of rotation around the ''z''-axis, the Lorentz transformation ({{equationNote|4b}}-A) assumes the form (with ''c''=1):&lt;ref group=R name=mink1&gt;Minkowski (1907/08), p. 59&lt;/ref&gt;

:&lt;math&gt;\begin{align}x'_{1} &amp; =x_{1}\\
x'_{2} &amp; =x_{2}\\
x'_{3} &amp; =x_{3}\cos i\psi+x_{4}\sin i\psi\\
x'_{4} &amp; =-x_{3}\sin i\psi+x_{4}\cos i\psi\\
\cos i\psi &amp; =\frac{1}{\sqrt{1-q^{2}}}
\end{align}
&lt;/math&gt;

Even though Minkowski used the imaginary number iψ, he for once&lt;ref group=R name=mink1 /&gt; directly used the [[w:tangens hyperbolicus]] in the equation for velocity

:&lt;math&gt;-i\tan i\psi=\frac{e^{\psi}-e^{-\psi}}{e^{\psi}+e^{-\psi}}=q&lt;/math&gt; with &lt;math&gt;\psi=\frac{1}{2}\ln\frac{1+q}{1-q}&lt;/math&gt;.

Minkowski's expression can also by written as ψ=atanh(q) and was later called [[w:rapidity]].

===={{anchor|Minkowski2}} Vector representation==== 
Minkowski wrote the Lorentz transformation ({{equationNote|4f}}) in vectorial form for the special case of directions being only parallel (&lt;math&gt;\mathfrak{r_{v}}&lt;/math&gt;) or perpendicular (&lt;math&gt;\mathfrak{r_{\bar{v}}}&lt;/math&gt;) to the velocity:&lt;ref group=R&gt;Minkowski (1907/08), pp. 62-63&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\mathfrak{r'_{v}}=\frac{\mathfrak{r_{v}}-qt}{\sqrt{1-q^{2}}},\quad\mathfrak{r'_{\bar{v}}}=\mathfrak{r_{\bar{v}}},\quad t'=\frac{-q\mathfrak{r_{v}}+t}{\sqrt{1-q^{2}}}\\
\mathfrak{r_{v}}=\frac{\mathfrak{r'_{v}}+qt'}{\sqrt{1-q^{2}}},\quad\mathfrak{r_{\bar{v}}}=\mathfrak{r'_{\bar{v}}},\quad t=\frac{q\mathfrak{r'_{v}}+t'}{\sqrt{1-q^{2}}}\\
\left[\mathfrak{r}=\left(x,y,z\right)=\left(\mathfrak{r_{v}},\mathfrak{r_{\bar{v}}}\right),\ |\mathfrak{v}|=q\right]
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski3}} Matrix representation====
Minkowski used matrices in order to write the [[../Lorentz transformation (general)#math_1a|E:general Lorentz transformation '''(1a)''']], of which boost matrix ({{equationNote|4h}}) is a special case:&lt;ref group=R&gt;Minkowski (1907/08), pp. 65–66, 81–82&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}+x_{4}^{\prime2}\\
\left(x_{1}^{\prime}=x',\ x_{2}^{\prime}=y',\ x_{3}^{\prime}=z',\ x_{4}^{\prime}=it'\right)\\
-x^{2}-y^{2}-z^{2}+t^{2}=-x^{\prime2}-y^{\prime2}-z^{\prime2}+t^{\prime2}\\
\hline x_{h}=\alpha_{h1}x_{1}^{\prime}+\alpha_{h2}x_{2}^{\prime}+\alpha_{h3}x_{3}^{\prime}+\alpha_{h4}x_{4}^{\prime}\\
\mathrm{A}=\mathrm{\left|\begin{matrix}\alpha_{11}, &amp; \alpha_{12}, &amp; \alpha_{13}, &amp; \alpha_{14}\\
\alpha_{21}, &amp; \alpha_{22}, &amp; \alpha_{23}, &amp; \alpha_{24}\\
\alpha_{31}, &amp; \alpha_{32}, &amp; \alpha_{33}, &amp; \alpha_{34}\\
\alpha_{41}, &amp; \alpha_{42}, &amp; \alpha_{43}, &amp; \alpha_{44}
\end{matrix}\right|,\ \begin{align}\bar{\mathrm{A}}\mathrm{A} &amp; =1\\
\left(\det \mathrm{A}\right)^{2} &amp; =1\\
\det \mathrm{A} &amp; =1\\
\alpha_{44} &amp; &gt;0
\end{align}
}
\end{matrix}&lt;/math&gt;

===={{anchor|Minkowski4}} Minkowski diagram====
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Apo|label 1=History of Lorentz transformations in general - Apollonius}}
Minkowski (1908/09) introduced the [[w:Minkowski diagram]] as a graphical representation of the Lorentz transformation, which became a standard tool in textbooks and research articles on relativity:&lt;ref group=R&gt;Minkowski (1908/09), p. 77&lt;/ref&gt;

[[File:Minkowski1.png|center|thumb|400px|Original spacetime diagram by Minkowski in 1908.]]

==={{anchor|Frank}} Frank, Varicak (1909-10) – Hyperbolic functions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Frank|label 1=History of Lorentz transformations via hyperbolic functions § Frank}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Varicak|label 1=History of Lorentz transformations via hyperbolic functions § Varicak}}

==={{Anchor|Sommerfeld}} Sommerfeld (1909) – Spherical trigonometry===
{{Main|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Sommerfeld|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Sommerfeld}}

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Möbius transformation and squeeze mappings===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Herglotz1|label 1=History of Lorentz transformations via Möbius transformations § Herglotz}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

==={{anchor|Plummer}} Plummer, Gruner (1910-21) – Trigonometric Lorentz boosts===
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Plummer|label 1=History of Lorentz transformations via trigonometric functions § Plummer}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Gruner|label 1=History of Lorentz transformations via trigonometric functions § Gruner}}

=== {{anchor|Ignatowski}} Ignatowski (1910) ===

While earlier derivations and formulations of the Lorentz transformation relied from the outset on optics, electrodynamics, or the invariance of the speed of light, [[w:Vladimir Ignatowski]] (1910) showed that it is possible to use the principle of relativity (and related [[w:Group theory|group theoretical]] principles) alone, in order to derive the following transformation between two inertial frames:&lt;ref group=R&gt;Ignatowski (1910), pp. 973–974&lt;/ref&gt;&lt;ref group=R&gt;Ignatowski (1910/11ab)&lt;/ref&gt;

:&lt;math&gt;\begin{align}dx' &amp; =p\ dx-pq\ dt\\
dt' &amp; =-pqn\ dx+p\ dt\\
p &amp; =\frac{1}{\sqrt{1-q^{2}n}}
\end{align}
&lt;/math&gt;

The variable ''n'' can be seen as a space-time constant whose value has to be determined by experiment or taken from a known physical law such as electrodynamics. For that purpose, Ignatowski used the above-mentioned Heaviside ellipsoid representing a contraction of electrostatic fields by ''x''/γ in the direction of motion. It can be seen that this is only consistent with Ignatowski's transformation when ''n=1/c''&lt;sup&gt;2&lt;/sup&gt;, resulting in ''p''=γ and the Lorentz transformation ({{equationNote|4b}}). With ''n''=0, no length changes arise and the Galilean transformation follows. Ignatowski's method was further developed and improved by [[w:Philipp Frank]] and [[w:Hermann Rothe]] (1911, 1912),&lt;ref group=R&gt;Frank &amp; Rothe (1911), pp. 825ff; (1912), p. 750ff.&lt;/ref&gt; with various authors developing similar methods in subsequent years.&lt;ref name=baccetti&gt;Baccetti (2011), see references 1–25 therein.&lt;/ref&gt;

==={{anchor|klein}} Klein, Noether, Conway, Silberstein (1908-11) – Biquaternions===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein and Noether}}
{{Main|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Conway|label 1=History of Lorentz transformations via Quaternions § Conway and Silberstein}}

==={{anchor|Herglotz2}} Ignatowski, Herglotz, Tamaki (1910-11) – Vector transformation===
[[w:Vladimir Ignatowski]] (1910, published 1911) defined the vectorial velocity addition ({{equationNote|4g}}) as well as general Lorentz boost ({{equationNote|4f}}) as&lt;ref group=R&gt;Ignatowski (1910/11a), p. 23; (1910/11b), p. 22&lt;/ref&gt; 

:&lt;math&gt;\begin{matrix}\begin{matrix}\mathfrak{v} =\frac{\mathfrak{v}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{v}'+pq\mathfrak{c}_{0}}{p\left(1+nq\mathfrak{c}_{0}\mathfrak{v}'\right)} &amp; \left|\begin{align}\mathfrak{A}' &amp; =\mathfrak{A}+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}-pqb\mathfrak{c}_{0}\\
b' &amp; =pb-pqn\mathfrak{A}\mathfrak{c}_{0}\\
\\
\mathfrak{A} &amp; =\mathfrak{A}'+(p-1)\mathfrak{c}_{0}\cdot\mathfrak{c}_{0}\mathfrak{A}'+pqb'\mathfrak{c}_{0}\\
b &amp; =pb'+pqn\mathfrak{A}'\mathfrak{c}_{0}
\end{align}
\right.\end{matrix}\\
\left[\mathfrak{v}=\mathbf{u},\ \mathfrak{A}=\mathbf{x},\ b=t,\ \mathfrak{c}_{0}=\frac{\mathbf{v}}{v},\ p=\gamma,\ n=\frac{1}{c^{2}}\right]
\end{matrix}&lt;/math&gt;

An equivalent transformation was given by [[w:Gustav Herglotz]] (1911)&lt;ref group=R&gt;Herglotz (1911), p. 497&lt;/ref&gt; using '''v'''=''(v&lt;sub&gt;x&lt;/sub&gt;, v&lt;sub&gt;y&lt;/sub&gt;, v&lt;sub&gt;z&lt;/sub&gt;)'' and '''r'''=''(x, y, z)'':

:&lt;math&gt;\begin{align}x^{0} &amp; =x+\alpha u(ux+vy+wz)-\beta ut\\
y^{0} &amp; =y+\alpha v(ux+vy+wz)-\beta vt\\
z^{0} &amp; =z+\alpha w(ux+vy+wz)-\beta wt\\
t^{0} &amp; =-\beta(ux+vy+wz)+\beta t\\
 &amp; \alpha=\frac{1}{\sqrt{1-s^{2}}\left(1+\sqrt{1-s^{2}}\right)},\ \beta=\frac{1}{\sqrt{1-s^{2}}}
\end{align}
&lt;/math&gt;

Kajuro Tamaki (1911) represented ({{equationNote|4g}}) as follows (as his paper was based on a 4-vector calculus, Tamaki's schematic is not representing a matrix despite looking very similar to the boost matrix in ({{equationNote|4h}})):&lt;ref group=R&gt;Tamaki (1911), pp. 143-144&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{array}{c|c|c|c|c}
 &amp; x'_{1} &amp; x'_{2} &amp; x'_{3} &amp; x'_{4}\\
\hline x_{1} &amp; 1+l^{2}\left(\cos\psi-1\right) &amp; lm\left(\cos\psi-1\right) &amp; ln\left(\cos\psi-1\right) &amp; l\sin\psi\\
\hline x_{2} &amp; lm\left(\cos\psi-1\right) &amp; 1+m^{2}\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; m\sin\psi\\
\hline x_{3} &amp; ln\left(\cos\psi-1\right) &amp; mn\left(\cos\psi-1\right) &amp; 1+n^{2}\left(\cos\psi-1\right) &amp; n\sin\psi\\
\hline x_{4} &amp; -l\sin\psi &amp; -m\sin\psi &amp; -n\sin\psi &amp; \cos\psi
\end{array}\\
\hline \psi=i\varphi,\ -i\tan i\varphi=-\frac{v}{c},\ \cos i\varphi=\frac{1}{\sqrt{1-(v/c)^{2}}}=\beta,\ -\sin i\varphi=\frac{i(v/c)}{\sqrt{1-(v/c)^{2}}}=i\beta(v/c)\\
\hline \mathbf{r}=\mathbf{r}'+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r}')+\beta\mathbf{v}t'\\
\mathbf{r}'=\mathbf{r}+(\beta-1)\mathbf{v}_{1}(\mathbf{v}_{1}\mathbf{r})-\beta\mathbf{v}t
\end{matrix}&lt;/math&gt;

==={{anchor|Hahn}} Hahn (1912) – Matrix transformation ===
{{See|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Hahn|label 1=History of Lorentz transformations via Quaternions § Hahn}}
Elaborating on [[#Minkowski3|Minkowski's (1907/8)]] matrix representation of the Lorentz transformations, Emil Hahn (1912) used matrix calculus in order to define the Lorentz boost for arbitrary directions (including the exponential form of the boost matrix) in line with ({{equationNote|4h}}), using imaginary rapidity &lt;math&gt;i\psi&lt;/math&gt; and imaginary time &lt;math&gt;x_4 =i\omega t&lt;/math&gt;:&lt;ref group=R&gt;Hahn (1912), pp. 30-32 and p. 36.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\boldsymbol{x}'-\boldsymbol{x}'_{0}=\mathbb{I}_{-\mathbf{c}}(u)\boldsymbol{x}\\
\hline \begin{align}\mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{J}\mathbb{G}_{\mathbf{c}}(u)\mathbb{J}^{-1} &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(r-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \frac{iur(u)}{\omega}\mathbf{c}\\
-\frac{iur(u)}{\omega}\overset{\perp}{\mathbf{c}}; &amp; r(u)
\end{matrix}\right) &amp; (7,p.30)\\
 &amp; =\left(\begin{matrix}\mathbf{E}+(\cos i\psi-1)\mathbf{c}\overset{\perp}{\mathbf{c}}; &amp; \sin i\psi\mathbf{c}\\
-\sin i\psi\mathbf{c}; &amp; \cos i\psi
\end{matrix}\right) &amp; (8,p.30)\\
 &amp; =\left(\begin{matrix}1+(\cos i\psi-1)c_{1}c_{1} &amp; (\cos i\psi-1)c_{1}c_{2} &amp; (\cos i\psi-1)c_{1}c_{3} &amp; \sin i\psi\,c_{1}\\
(\cos i\psi-1)c_{2}c_{1} &amp; 1+(\cos i\psi-1)c_{2}c_{2} &amp; (\cos i\psi-1)c_{2}c_{3} &amp; \sin i\psi\,c_{2}\\
(\cos i\psi-1)c_{3}c_{1} &amp; (\cos i\psi-1)c_{3}c_{2} &amp; 1+(\cos i\psi-1)c_{3}c_{3} &amp; \sin i\psi\,c_{3}\\
-\sin i\psi\,c_{1} &amp; -\sin i\psi\,c_{2} &amp; -\sin i\psi\,c_{3} &amp; 1+(\cos i\psi-1)
\end{matrix}\right) &amp; (1,p.36)\\
\hline \mathbb{I}_{-\mathbf{c}}(u) &amp; =\mathbb{E}-\sin i\psi\left\langle \mathbf{0},\mathbf{c}\right\rangle +(1-\cos i\psi)\left\langle \mathbf{0},\mathbf{c}\right\rangle ^{2} &amp; (10,p.30)\\
\mathbb{I}_{\mathbf{c}}(u) &amp; =e^{\left\langle \mathbf{0},\mathbf{c}\right\rangle i\psi} &amp; (p.31)
\end{align}
\\
\hline \boldsymbol{x}=\left(\begin{matrix}x_{1} &amp; 0 &amp; 0 &amp; 0\\
x_{2} &amp; 0 &amp; 0 &amp; 0\\
x_{3} &amp; 0 &amp; 0 &amp; 0\\
x_{4} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \boldsymbol{x}'=\left(\begin{matrix}x_{1}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{2}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{3}^{\prime} &amp; 0 &amp; 0 &amp; 0\\
x_{4}^{\prime} &amp; 0 &amp; 0 &amp; 0
\end{matrix}\right),\ \mathbb{J}=\left(\begin{matrix}1 &amp; 0 &amp; 0 &amp; 0\\
0 &amp; 1 &amp; 0 &amp; 0\\
0 &amp; 0 &amp; 1 &amp; 0\\
0 &amp; 0 &amp; 0 &amp; i\omega
\end{matrix}\right),\ \mathbb{G}=\left(\begin{matrix}r_{11} &amp; r_{12} &amp; r_{13} &amp; p_{1}\\
r_{21} &amp; r_{22} &amp; r_{23} &amp; p_{2}\\
r_{31} &amp; r_{32} &amp; r_{33} &amp; p_{3}\\
q_{1} &amp; q_{2} &amp; q_{3} &amp; r
\end{matrix}\right)\\
r(u)=\frac{1}{\sqrt{1-\frac{u^{2}}{\omega^{2}}}}=\cos i\psi,\ |\mathbb{I}|=1,\ \left\langle \mathbf{0},\mathbf{c}\right\rangle =\left(\begin{matrix}0 &amp; 0 &amp; 0 &amp; c_{1}\\
0 &amp; 0 &amp; 0 &amp; c_{2}\\
0 &amp; 0 &amp; 0 &amp; c_{3}\\
c_{1} &amp; c_{2} &amp; c_{3} &amp; 0
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===
{{Main|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Borel|label 1=History of Lorentz transformations via Cayley–Hermite transformation § Borel}}

==={{anchor|Born}} Born (1921) – Squeeze mapping===
{{Main|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Born|label 1=History of Lorentz transformations via squeeze mappings § Born}}

==References==
===Historical relativity sources===
{{reflist|3|group=R}}
*{{#section:History of Topics in Special Relativity/relsource|abra1905}}
*{{#section:History of Topics in Special Relativity/relsource|buch08}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04a}}
*{{#section:History of Topics in Special Relativity/relsource|cohn04b}}
*{{#section:History of Topics in Special Relativity/relsource|einst05elek}}
*{{#section:History of Topics in Special Relativity/relsource|fra11}}
*{{#section:History of Topics in Special Relativity/relsource|fra12}}
*{{#section:History of Topics in Special Relativity/relsource|gans05}}
*{{#section:History of Topics in Special Relativity/relsource|grun21a}}
*{{#section:History of Topics in Special Relativity/relsource|grun21b}}
*{{#section:History of Topics in Special Relativity/relsource|heav89}}
*{{#section:History of Topics in Special Relativity/relsource|hahn}}
*{{#section:History of Topics in Special Relativity/relsource|herg11ela}}
*{{#section:History of Topics in Special Relativity/relsource|igna10}}
*{{#section:History of Topics in Special Relativity/relsource|ignat10prin2}}
*{{#section:History of Topics in Special Relativity/relsource|lar97}}
*{{#section:History of Topics in Special Relativity/relsource|lar29}}
*{{#section:History of Topics in Special Relativity/relsource|lar00}}
*{{#section:History of Topics in Special Relativity/relsource|lar04a}}
*{{#section:History of Topics in Special Relativity/relsource|lar04b}}
*{{#section:History of Topics in Special Relativity/relsource|lor92elek}}
*{{#section:History of Topics in Special Relativity/relsource|lor92b}}
*{{#section:History of Topics in Special Relativity/relsource|lor95}}
*{{#section:History of Topics in Special Relativity/relsource|lor99}}
*{{#section:History of Topics in Special Relativity/relsource|lor04}}
*{{#section:History of Topics in Special Relativity/relsource|lor16}}
*{{#section:History of Topics in Special Relativity/relsource|mink07a}}
*{{#section:History of Topics in Special Relativity/relsource|mink07b}}
*{{#section:History of Topics in Special Relativity/relsource|mink08}}
*{{#section:History of Topics in Special Relativity/relsource|poi00}}
*{{#section:History of Topics in Special Relativity/relsource|poi04}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05a}}
*{{#section:History of Topics in Special Relativity/relsource|poinc05b}}
*{{#section:History of Topics in Special Relativity/relsource|sea97}}
*{{#section:History of Topics in Special Relativity/relsource|tamaki11b}}
*{{#section:History of Topics in Special Relativity/relsource|thom89}}
*{{#section:History of Topics in Special Relativity/relsource|voi87}}
*{{#section:History of Topics in Special Relativity/relsource|wien04}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L4}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>7q96ntbkj8d10zc0d1epcnxy60t0eh1</sha1>
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    <title>History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)</title>
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    <revision>
      <id>2692669</id>
      <parentid>2521469</parentid>
      <timestamp>2024-12-19T20:27:44Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <comment>/* Lorentz transformation via Cayley–Hermite transformation */</comment>
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      <text bytes="28117" sha1="ka5v2p4t7bjpg0o5ux684z6dw2dfwx6" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Hermite transformation ==

The [[../Lorentz transformation (general)#math_Q1|E:general transformation '''(Q1)''']] of any quadratic form into itself can also be given using ''arbitrary'' parameters based on the [[w:Cayley transform]] ('''I'''-'''T''')&lt;sup&gt;−1&lt;/sup&gt;·('''I'''+'''T'''), where '''I''' is the [[w:identity matrix]], '''T''' an arbitrary [[w:antisymmetric matrix]], and by adding '''A''' as symmetric matrix defining the quadratic form (there is no primed '''A' ''' because the coefficients are assumed to be the same on both sides):&lt;ref&gt;Hawkins (2013), pp. 210–214&lt;/ref&gt;&lt;ref&gt;Meyer (1899), p. 329&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix}q=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}=q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}\cdot\mathbf{x}'\\
\hline \\
\mathbf{x}=(\mathbf{I}-\mathbf{T}\cdot\mathbf{A})^{-1}\cdot(\mathbf{I}+\mathbf{T}\cdot\mathbf{A})\cdot\mathbf{x}'\\
\text{or}\\
\mathbf{x}=\mathbf{A}^{-1}\cdot(\mathbf{A}-\mathbf{T})\cdot(\mathbf{A}+\mathbf{T})^{-1}\cdot\mathbf{A}\cdot\mathbf{x}'
\end{matrix}&lt;/math&gt;|{{equationRef|Q2}}}}

After [[#Cayley|Cayley (1846)]] introduced transformations related to sums of positive squares, [[#Hermite|Hermite (1853/54, 1854)]] derived transformations for arbitrary quadratic forms, whose result was reformulated in terms of matrices by [[#Cayley|Cayley (1855a, 1855b)]]. For instance, the choice '''A'''=diag(1,1,1) gives an orthogonal transformation which can be used to describe spatial rotations corresponding to the [[w:Euler–Rodrigues formula|w:Euler-Rodrigues parameter]]s ''[a,b,c,d]'' discovered by [[#Euler|Euler (1771) and Rodrigues (1840)]], which can be interpreted as the coefficients of [[w:quaternion]]s. Setting ''d=1'', the equations have the form:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\mathbf{A}=\operatorname{diag}(1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1-a^{2}-b^{2}+c^{2} &amp; 2(bc-a) &amp; 2(ac+b)\\
2(bc+a) &amp; 1-a^{2}+b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; 2(ab+c) &amp; 1+a^{2}-b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1+a^{2}+b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|Q3}}}}

Also the Lorentz interval and the general Lorentz transformation in any dimension can be produced by the Cayley–Hermite formalism.&lt;ref group=R&gt;Borel (1914), pp. 39–41&lt;/ref&gt;&lt;ref group=R&gt;Brill (1925)&lt;/ref&gt;&lt;ref&gt;Klein (1928), Chapter III, § 2B&lt;/ref&gt;&lt;ref&gt;Lorente (2003), section 3.3&lt;/ref&gt; For instance, the [[../Lorentz transformation (general)#math_1a|E:most general Lorentz transformation '''(1a)''']] with ''n''=1 follows from ({{equationNote|Q2}}) with:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a\\
-a &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{1-a^{2}}\left[\begin{matrix}1+a^{2} &amp; -2a\\
-2a &amp; 1+a^{2}
\end{matrix}\right]\cdot\mathbf{x}
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \left.\begin{align}x_{0} &amp; =x_{0}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}\left(1+\beta_{0}^{2}\right)+x_{1}^{\prime}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}^{\prime}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}^{\prime}2\beta_{0}+x_{1}^{\prime}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}\\
\\
x_{0}^{\prime} &amp; =x_{0}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}}-x_{1}\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{x_{0}\left(1+\beta_{0}^{2}\right)-x_{1}2\beta_{0}}{1-\beta_{0}^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2\beta_{0}}{1-\beta_{0}^{2}}+x_{1}\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; = &amp; \frac{-x_{0}2\beta_{0}+x_{1}\left(1+\beta_{0}^{2}\right)}{1-\beta_{0}^{2}}
\end{align}
\right|{\scriptstyle \begin{align}\frac{2\beta_{0}}{1+\beta_{0}^{2}} &amp; =\beta\\
\frac{1+\beta_{0}^{2}}{1-\beta_{0}^{2}} &amp; =\gamma\\
\frac{2\beta_{0}}{1-\beta_{0}^{2}} &amp; =\beta\gamma
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|5a}}}}

This becomes [[../Lorentz transformation (velocity)#math_4a|E:Lorentz boost '''(4a)''']] by setting &lt;math&gt;\tfrac{2a}{1+a^{2}}=\tfrac{v}{c}&lt;/math&gt;, which is equivalent to the relation &lt;math&gt;\tfrac{2\beta_{0}}{1+\beta_{0}^{2}}=\tfrac{v}{c}&lt;/math&gt; known from [[w:Loedel diagram]]s, thus ({{equationNote|5a}}) can be interpreted as a Lorentz boost from the viewpoint of a &quot;median frame&quot; in which two other inertial frames are moving with equal speed &lt;math&gt;\beta_0&lt;/math&gt; in opposite directions.

Furthermore, Lorentz transformation [[../Lorentz transformation (general)#math_1a|E:'''(1a)''']]) with ''n''=2 is given by:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b\\
-a &amp; 0 &amp; c\\
b &amp; -c &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}1+a^{2}+b^{2}+c^{2} &amp; -2(bc-a) &amp; -2(ac+b)\\
2(bc+a) &amp; 1+a^{2}-b^{2}-c^{2} &amp; 2(ab-c)\\
2(ac-b) &amp; -2(ab-c) &amp; 1-a^{2}+b^{2}-c^{2}
\end{matrix}\right]\cdot\mathbf{x}\\
\left(\kappa=1-a^{2}-b^{2}+c^{2}\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5b}}}}

or using ''n''=3:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\mathbf{A}=\operatorname{diag}(-1,1,1,1),\quad\mathbf{T}={\scriptstyle \begin{vmatrix}0 &amp; a &amp; -b &amp; c\\
-a &amp; 0 &amp; d &amp; e\\
b &amp; -d &amp; 0 &amp; f\\
-c &amp; -e &amp; -f &amp; 0
\end{vmatrix}}\\
\hline -x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[{\scriptstyle \begin{align} &amp; 1+a^{2}+b^{2}+c^{2}+ &amp;  &amp; 2(-bd+a+ec+pf) &amp;  &amp; 2(-ad-b+fc-pe) &amp;  &amp; 2(pd+fb-ea+c)\\
 &amp; \quad d^{2}+e^{2}+f^{2}+p^{2} &amp;  &amp; 1+a^{2}-b^{2}-c^{2} &amp;  &amp; 2(-d-ab+pc-fe) &amp;  &amp; 2(fd+pb+ca-e)\\
 &amp; 2(bd+a-ec+pf) &amp;  &amp; \quad-d^{2}-e^{2}+f^{2}+p^{2} &amp;  &amp; 1-a^{2}+b^{2}-c^{2} &amp;  &amp; 2(-ed-cb+pa-f)\\
 &amp; 2(ad-b-fc-pe) &amp;  &amp; 2(d-ab-pc-fe) &amp;  &amp; \quad-d^{2}+e^{2}-f^{2}+p^{2} &amp;  &amp; 1-a^{2}-b^{2}+-c^{2}\\
 &amp; 2(pd-fb+ea+c) &amp;  &amp; 2(fd-pb+ca+e) &amp;  &amp; 2(-ed-cb-pa+f) &amp;  &amp; \quad+d^{2}-e^{2}-f^{2}+p^{2}
\end{align}
}\right]\cdot\mathbf{x}\\
\left(\begin{align}\kappa &amp; =1-a^{2}-b^{2}-c^{2}+d^{2}+e^{2}+f^{2}-p^{2}\\
p &amp; =af+be+cd
\end{align}
\right)
\end{matrix}&lt;/math&gt;|{{equationRef|5c}}}}

The transformation of a binary quadratic form of which Lorentz transformation ({{equationNote|5a}}) is a special case was given by [[#Hermite|Hermite (1854)]], equations containing Lorentz transformations ({{equationNote|5a}}, {{equationNote|5b}}, {{equationNote|5c}}) as special cases were given by [[#Cayley|Cayley (1855)]], Lorentz transformation ({{equationNote|5a}}) was given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry, and Lorentz transformation ({{equationNote|5b}}) was given by [[#Bachmann|Bachmann (1869)]]. In relativity, equations similar to ({{equationNote|5b}}, {{equationNote|5c}}) were first employed by [[#Borel|Borel (1913)]] to represent Lorentz transformations.

==Historical notation==
==={{anchor|Euler}} Euler (1771) – Euler-Rodrigues parameter===
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Euler|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Euler}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Euler|label 1=History of Lorentz transformations via hyperbolic functions § Euler}}

Euler (1771) demonstrated the invariance of quadratic forms in terms of sum of squares under a linear substitution and its coefficients, now known as [[w:orthogonal transformation]]. The transformation in three dimensions was given as

:&lt;math&gt;\begin{matrix}X^{2}+Y^{2}+Z^{2}=x^{2}+y^{2}+z^{2}\\
\hline \begin{align}X &amp; =Ax+By+Cz\\
Y &amp; =Dx+Ey+Fz\\
Z &amp; =Gx+Hy+Iz
\end{align}
\begin{matrix}\left|{\scriptstyle \begin{align}1 &amp; =AA+DD+GG\\
1 &amp; =BB+EE+HH\\
1 &amp; =CC+FF+II\\
0 &amp; =AB+DE+GH\\
0 &amp; =AG+DF+GI\\
0 &amp; =BC+EF+HI
\end{align}
}\right.\end{matrix}\end{matrix}&lt;/math&gt;

in which the coefficiens ''A,B,C,D,E,F,G,H,I'' were related by Euler to four arbitrary parameter ''p,q,r,s'', which where rediscovered by [[w:Olinde Rodrigues]] (1840) who related them to rotation angles&lt;ref group=M&gt;Rodrigues (1840), p. 405&lt;/ref&gt;:&lt;ref group=M&gt;Euler (1771), p. 101&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}A &amp; =\frac{pp+qq-rr-ss}{u} &amp; B &amp; =\frac{2pq+2ps}{u} &amp; C &amp; =\frac{2qs-2pr}{u}\\
D &amp; =\frac{2qr-2ps}{u} &amp; E &amp; =\frac{pp-qq+rr-ss}{u} &amp; F &amp; =\frac{2pq+2rs}{u}\\
G &amp; =\frac{2qs+2pr}{u} &amp; H &amp; =\frac{2rs-2pq}{u} &amp; I &amp; =\frac{pp-qq-rr+ss}{u}
\end{align}
\\
(u=pp+qq+rr+ss)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=These are now called (improper) [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] in line with equation ({{equationNote|Q3}}).}}

==={{anchor|Cayley}} Cayley (1846–1855) – Cayley–Hermite transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Cayley2|label 1=History of Lorentz transformations via Möbius transformations § Cayley}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cayley3|label 1=History of Lorentz transformations via Quaternions § Cayley}}

The [[w:Euler–Rodrigues formula|Euler–Rodrigues parameters]] discovered by [[#Euler|Euler (1871) and Rodrigues (1840)]] leaving invariant &lt;math&gt;x_{0}^{2}+x_{1}^{2}+x_{2}^{2}&lt;/math&gt; were extended to &lt;math&gt;x_{0}^{2}+\dots+x_{n}^{2}&lt;/math&gt; by [[w:Arthur Cayley]] (1846) as a byproduct of what is now called the [[w:Cayley transform]] using the method of skew–symmetric coefficients.&lt;ref group=M name=cayl1&gt;Cayley (1846)&lt;/ref&gt; Following Cayley's methods, a general transformation for quadratic forms into themselves in three (1853) and arbitrary (1854) dimensions was provided by [[#Hermite|Hermite (1853, 1854)]]. Hermite's formula was simplified and brought into matrix form equivalent to ({{equationNote|Q2}}) by Cayley (1855a)&lt;ref group=M&gt;Cayley (1855a), p. 288&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle (\mathrm{x,y,z}\dots)=\left(\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h-\nu, &amp; g+\mu &amp; \dots\\
h+\nu, &amp; b, &amp; f-\lambda &amp; \dots\\
g-\mu, &amp; f+\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\left|\begin{matrix}a, &amp; h+\nu, &amp; g-\mu &amp; \dots\\
h-\nu, &amp; b, &amp; f+\lambda &amp; \dots\\
g+\mu, &amp; f-\lambda, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|^{-1}\left|\begin{matrix}a, &amp; h, &amp; g &amp; \dots\\
h, &amp; b, &amp; f &amp; \dots\\
g, &amp; f, &amp; c &amp; \dots\\
\dots &amp; \dots &amp; \dots &amp; \dots
\end{matrix}\right|\right)\ ^{\frown}(x,y,z\dots)}&lt;/math&gt;

which he abbreviated in 1858, where &lt;math&gt;\Upsilon&lt;/math&gt; is any skew-symmetric matrix:&lt;ref group=M name=cayl1858&gt;Cayley (1858), p. 39&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 214&lt;/ref&gt;

:&lt;math&gt;(\mathrm{x,y,z})=\left(\Omega^{-1}(\Omega-\Upsilon)(\Omega+\Upsilon)^{-1}\Omega\right)(x,y,z)&lt;/math&gt;

{{Lorentzbox|Text=The Cayley–Hermite transformation becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) by setting Ω=diag(-1,1) and  ({{equationNote|5b}}) by setting Ω=diag(-1,1,1) and ({{equationNote|5c}}) by setting Ω=diag(-1,1,1,1).}}

Using the parameters of (1855a), Cayley in a subsequent paper (1855b) particularly discussed several special cases, such as:&lt;ref group=M&gt;Cayley (1855b), p. 210&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}=ax^{2}+by^{2}\\
\hline (\mathrm{x,y})=\frac{1}{ab+\nu^{2}}\cdot\left[\begin{matrix}ab-\nu^{2}, &amp; -2\nu b\\
2\nu a, &amp; ab-\nu^{2}
\end{matrix}\right](x,y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This becomes equivalent to the Lorentz transformation ({{equationNote|5a}}) in 1+1 dimensions by setting ''(a,b)''=(-1,1) and Lorentz boost [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] by additionally setting &lt;math&gt;\tfrac{2\nu}{1+\nu^{2}}=\tfrac{v}{c}&lt;/math&gt;.}}

or:&lt;ref group=M&gt;Cayley (1855b), p. 211&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}=ax^{2}+by^{2}+cz^{2}\\
\hline (\mathrm{x,y,z})=\frac{1}{abc+a\lambda^{2}+b\mu^{2}+c\nu^{2}}\times\left[\begin{matrix}abc+a\lambda^{2}-b\mu^{2}-c\nu^{2}, &amp; 2(\lambda\mu-c\nu)b &amp; 2(\nu\lambda+b\mu)c\\
2(\lambda\mu+c\nu)a, &amp; abc-a\lambda^{2}+b\mu^{2}-c\nu^{2} &amp; 2(\mu\nu-a\lambda)c\\
2(\nu\lambda-b\mu)a &amp; 2(\mu\nu+a\lambda)b &amp; abc-a\lambda^{2}-b\mu^{2}-c\nu^{2}
\end{matrix}\right](x,y,z)
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=This becomes equivalent to the Lorentz transformation ({{equationNote|5b}}) by setting ''(a,b,c)''=(-1,1,1).}}

or:&lt;ref group=M&gt;Cayley (1855b), pp. 212–213&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{matrix}a\mathrm{x}^{2}+b\mathrm{y}^{2}+c\mathrm{z}^{2}+d\mathrm{w}^{2}=ax^{2}+by^{2}+cz^{2}+dw^{2}\\
\hline (\mathrm{x,y,z,w})=\frac{1}{k}\cdot\left[\begin{align} &amp; abcd-bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2} &amp;  &amp; 2b\left(-cd\nu-\tau\phi+d\lambda\mu-c\rho\sigma\right),\\
 &amp; \quad-bd\mu^{2}-cd\nu^{2}-\phi^{2}, &amp;  &amp; abcd+bc\rho^{2}-ca\sigma^{2}+ab\tau^{2}-ad\lambda^{2}\\
 &amp; 2a\left(cd\nu+\tau\phi+d\lambda\mu-c\rho\sigma\right), &amp;  &amp; \quad+bd\mu^{2}-cd\nu^{2}-\phi^{2},\\
 &amp; 2a\left(-bd\mu-\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; 2b\left(ad\lambda+\rho\phi+d\mu\nu-a\sigma\tau\right),\\
 &amp; 2a\left(bc\rho+\lambda\phi+c\nu\sigma-b\mu\tau\right), &amp;  &amp; 2b\left(ac\sigma+\mu\phi-c\nu\rho+a\lambda\tau\right),\\
\\
 &amp; \quad2c\left(bd\mu+\sigma\phi+d\lambda\nu-b\rho\tau\right), &amp;  &amp; \quad2d\left(-bc\rho-\lambda\phi+c\nu\sigma-b\mu\tau\right)\\
 &amp; \quad2c\left(-ad\lambda-\rho\phi+d\mu\nu-a\sigma\tau\right), &amp;  &amp; \quad2d\left(-ac\sigma-\mu\phi-c\nu\rho+a\lambda\tau\right)\\
 &amp; \quad abcd+bc\rho^{2}+ca\sigma^{2}-ab\tau^{2}-ad\lambda^{2} &amp;  &amp; \quad2d\left(-ab\tau-\nu\phi+b\mu\rho-a\lambda\sigma\right)\\
 &amp; \quad\quad-bd\mu^{2}+cd\nu^{2}-\phi^{2}, &amp;  &amp; \quad abcd-bc\rho^{2}-ca\sigma^{2}-ab\tau^{2}+ad\lambda^{2}\\
 &amp; \quad2c\left(ab\tau+\nu\phi+b\mu\rho-a\lambda\sigma\right), &amp;  &amp; \quad\quad+bd\mu^{2}+cd\nu^{2}-\phi^{2},
\end{align}
\right]\cdot(x,y,z,w)\\
\left(\begin{align}k &amp; =abcd+bc\rho^{2}+ca\sigma^{2}+ab\tau^{2}+ad\lambda^{2}+bd\mu^{2}+cd\nu^{2}+\phi^{2}\\
\phi &amp; =\lambda\rho+\mu\sigma+\nu\tau
\end{align}
\right)
\end{matrix}}&lt;/math&gt;

{{Lorentzbox|Text=This becomes equivalent to the Lorentz transformation ({{equationNote|5c}}) by setting ''(a,b,c,d)''=(-1,1,1,1).}}

==={{anchor|Hermite}} Hermite (1853, 1854) – Cayley–Hermite transformation===

[[w:Charles Hermite]] (1853) extended the number theoretical work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] and others (including himself) by additionally analyzing ''indefinite'' ternary quadratic forms that can be transformed into the Lorentz interval ±''(x&lt;sup&gt;2&lt;/sup&gt;+y&lt;sup&gt;2&lt;/sup&gt;-z&lt;sup&gt;2&lt;/sup&gt;)'', and by using [[#Cayley|Cayley's (1846)]] method of skew–symmetric coefficients he derived transformations leaving invariant almost all types of ternary quadratic forms.&lt;ref group=M name=herm&gt;Hermite (1853/54a), p. 307ff.&lt;/ref&gt; This was generalized by him in 1854 to ''n'' dimensions:&lt;ref group=M&gt;Hermite (1854b), p. 64&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), p. 212&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f\left(X_{1},X_{2},\dots\right)=f\left(x_{1},x_{2},\dots\right)\\
\hline X_{r}=2\xi_{r}-x_{r}=\xi_{r}-\frac{1}{2}\sum_{s=1}^{n}\lambda_{r,s}\frac{df}{d\xi_{s}}\\
\left(\lambda_{r,s}=-\lambda_{s,r},\ \lambda_{r,r}=0\right)
\end{matrix}&lt;/math&gt;

This result was subsequently expressed in matrix form by [[#Cayley|Cayley (1855)]], while [[w:Ferdinand Georg Frobenius]] (1877) added some modifications in order to include some special cases of quadratic forms that cannot be dealt with by the Cayley–Hermite transformation.&lt;ref group=M&gt;Frobenius (1877)&lt;/ref&gt;&lt;ref&gt;Hawkins (2013), pp. 219ff&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to equation ({{equationNote|Q2}}), and becomes the Lorentz transformation by setting the coefficients of the quadratic form ''f'' to diag(-1,1,...1).}}

For instance, the special case of the transformation of a binary quadratic form into itself was given by Hermite as follows:&lt;ref group=M&gt;Hermite (1854b), pp. 64–65&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}f=ax^{2}+2bxy+cy^{2}\\
\hline \begin{align}X &amp; =\frac{\left(1-2\lambda b+\lambda^{2}D\right)x-2\lambda cy}{1-\lambda^{2}D} &amp;  &amp; =x(t-bu)-cuy\\
Y &amp; =\frac{2\lambda ax+\left(1+2\lambda b+\lambda^{2}D\right)y}{1-\lambda^{2}D} &amp;  &amp; =xau+(t+bu)y
\end{align}
\\
\left(b^{2}-ac=D,\ t=\frac{1+\lambda^{2}D}{1-\lambda^{2}D},\ u=\frac{2\lambda}{1-\lambda^{2}D},\ t^{2}-Du^{2}=1\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This becomes equivalent to Lorentz boost ({{equationNote|5a}}) by setting ''(a,b,c)''=(-1,0,1) and Lorentz boost of velocity by additionally setting &lt;math&gt;\tfrac{2\lambda}{1+\lambda^{2}}=\tfrac{v}{c}&lt;/math&gt; which produces ''t''=γ and ''u''=βγ.}}

=== {{anchor|Bachmann}} Bachmann (1869) – Cayley–Hermite transformation ===

[[w:Paul Gustav Heinrich Bachmann]] (1869) adapted [[#Hermite|Hermite's (1853/54)]] transformation of ternary quadratic forms to the case of integer transformations. He particularly analyzed the Lorentz interval and its transformation, and also alluded to the analogue result of [[../Lorentz transformation (Möbius)#Gauss3|E:Gauss (1800) in terms of Cayley–Klein parameters]], while Bachmann formulated his result in terms of the Cayley–Hermite transformation:&lt;ref group=M&gt;Bachmann (1869), p. 303&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+x^{\prime2}-x^{\prime\prime2}\\
\hline \begin{align}\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X &amp; =\left(p^{2}-q^{2}+q^{\prime2}-q^{\prime\prime2}\right)x-2(pq''+qq')x'-2(pq'+qq'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X' &amp; =2(pq''-qq')x+\left(p^{2}+q^{2}-q^{\prime2}-q^{\prime\prime2}\right)x'+2(pq-q'q'')x''\\
\left(p^{2}-q^{2}-q^{\prime2}+q^{\prime\prime2}\right)X'' &amp; =-2(pq'-qq')x+2(pq+q'q'')x'+\left(p^{2}+q^{2}+q^{\prime2}+q^{\prime\prime2}\right)x''
\end{align}
\end{matrix}&lt;/math&gt;

He described this transformation in 1898 in the first part of his &quot;arithmetics of quadratic forms&quot; as well.&lt;ref&gt;Bachmann (1898), pp. 101–102&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|5b}}), producing the relation &lt;math&gt;X^{2}+X^{\prime2}-X^{\prime\prime2}=x^{2}+x^{\prime2}-x^{\prime\prime2}&lt;/math&gt;.}}

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Laguerre|label 1=History of Lorentz transformations via sphere transformations § Laguerre}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Laguerre|label 1=History of Lorentz transformations via squeeze mappings § Laguerre}}

After previous work by [[w:Albert Ribaucour]] (1870),&lt;ref group=M&gt;Ribaucour (1870)&lt;/ref&gt; a transformation which transforms oriented spheres into oriented spheres, oriented planes into oriented planes, and oriented lines into oriented lines, was explicitly formulated by [[w:Edmond Laguerre]] (1882) as &quot;[[w:Spherical wave transformation#Transformation by reciprocal directions|transformation by reciprocal directions]]&quot; which was later called &quot;Laguerre inversion/transformation&quot;. It can be seen as a special case of the conformal group in terms of [[../Lorentz transformation (conformal)#Lie|E:Lie's transformations of oriented spheres]]. In two dimensions the transformation or oriented lines has the form (''R'' being the radius):&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D^{2}-D^{\prime2} &amp; =R^{2}-R^{\prime2}\\
D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Laguerre didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\alpha}{1+\alpha^{2}}=\tfrac{v}{c}&lt;/math&gt;.}}

==={{anchor|Darboux2}} Darboux (1887) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Darboux|label 1=History of Lorentz transformations via sphere transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Darboux1|label 1=History of Lorentz transformations via trigonometric functions § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Darboux2|label 1=History of Lorentz transformations via squeeze mappings § Darboux}}

Following [[#Laguerre|Laguerre (1882)]], [[w:Gaston Darboux]] (1887) presented the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change for ''R'') to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Darboux didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2k}{1+k^{2}}=\tfrac{v}{c}&lt;/math&gt;.}}

==={{anchor|Smith}} Smith (1900) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Smith|label 1=History of Lorentz transformations via sphere transformations § Smith}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Smith|label 1=History of Lorentz transformations via squeeze mappings § Smith}}
	
[[w:Percey F. Smith]] (1900) followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;
	
:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\	
\hline p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R	
\end{matrix}&lt;/math&gt;
	
{{Lorentzbox|Text=This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|5a}}) in terms of Cayley–Hermite parameters (even though Smith didn't use the Cayley-Hermite transformation ({{equationNote|Q2}})). The Lorentz boost of velocity follows with &lt;math&gt;\tfrac{2\kappa}{1+\kappa^{2}}=\tfrac{v}{c}&lt;/math&gt;.}}

==={{anchor|Borel}} Borel (1913–14) – Cayley–Hermite parameter===

[[w:Émile Borel]] (1913) started by demonstrating Euclidean motions using Euler-Rodrigues parameter in three dimensions, and [[#Cayley|Cayley's (1846)]] parameter in four dimensions. Then he demonstrated the connection to indefinite quadratic forms expressing hyperbolic motions and Lorentz transformations. In three dimensions equivalent to ({{equationNote|5b}}):&lt;ref group=R&gt;Borel (1913/14), p. 39&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}-1=0\\
\hline {\scriptstyle \begin{align}\delta a &amp; =\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta b &amp; =2(\lambda\mu+\nu\rho), &amp; \delta c &amp; =-2(\lambda\nu+\mu\rho),\\
\delta a' &amp; =2(\lambda\mu-\nu\rho), &amp; \delta b' &amp; =-\lambda^{2}+\mu^{2}+\nu^{2}-\rho^{2}, &amp; \delta c' &amp; =2(\lambda\rho-\mu\nu),\\
\delta a'' &amp; =2(\lambda\nu-\mu\rho), &amp; \delta b'' &amp; =2(\lambda\rho+\mu\nu), &amp; \delta c'' &amp; =-\left(\lambda^{2}+\mu^{2}+\nu^{2}+\rho^{2}\right),
\end{align}
}\\
\left(\delta=\lambda^{2}+\mu^{2}-\rho^{2}-\nu^{2}\right)\\
\lambda=\nu=0\rightarrow\text{Hyperbolic rotation}
\end{matrix}&lt;/math&gt;

In four dimensions equivalent to ({{equationNote|5c}}):&lt;ref group=R&gt;Borel (1913/14), p. 41&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-\left(t_{1}-t_{2}\right)^{2}\\
\hline {\scriptstyle \begin{align} &amp; \left(\mu^{2}+\nu^{2}-\alpha^{2}\right)\cos\varphi+\left(\lambda^{2}-\beta^{2}-\gamma^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi-\gamma\operatorname{sh}{\theta}\\
 &amp; -(\alpha\beta+\lambda\mu)(\cos\varphi-\operatorname{ch}{\theta})-\nu\sin\varphi+\gamma\operatorname{sh}{\theta} &amp;  &amp; \left(\mu^{2}+\nu^{2}-\beta^{2}\right)\cos\varphi+\left(\mu^{2}-\alpha^{2}-\gamma^{2}\right)\operatorname{ch}{\theta}\\
 &amp; -(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi-\beta\operatorname{sh}{\theta} &amp;  &amp; -(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\lambda\sin\varphi+\alpha\operatorname{sh}{\theta}\\
 &amp; (\gamma\mu-\beta\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta} &amp;  &amp; -(\alpha\nu-\lambda\gamma)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
\\
 &amp; \quad-(\alpha\gamma+\lambda\nu)(\cos\varphi-\operatorname{ch}{\theta})+\mu\sin\varphi+\beta\operatorname{sh}{\theta} &amp;  &amp; \quad(\beta\nu-\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})+\alpha\sin\varphi-\lambda\operatorname{sh}{\theta}\\
 &amp; \quad-(\beta\mu+\mu\nu)(\cos\varphi-\operatorname{ch}{\theta})-\lambda\sin\varphi-\alpha\operatorname{sh}{\theta} &amp;  &amp; \quad(\lambda\gamma-\alpha\nu)(\cos\varphi-\operatorname{ch}{\theta})+\beta\sin\varphi-\mu\operatorname{sh}{\theta}\\
 &amp; \quad\left(\lambda^{2}+\mu^{2}-\gamma^{2}\right)\cos\varphi+\left(\nu^{2}-\alpha^{2}-\beta^{2}\right)\operatorname{ch}{\theta} &amp;  &amp; \quad(\alpha\mu-\beta\lambda)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta}\\
 &amp; \quad(\beta\gamma-\alpha\mu)(\cos\varphi-\operatorname{ch}{\theta})+\gamma\sin\varphi-\nu\operatorname{sh}{\theta} &amp;  &amp; \quad-\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\cos\varphi+\left(\lambda^{2}+\mu^{2}+\nu^{2}\right)\operatorname{ch}{\theta}
\end{align}
}\\
\left(\alpha^{2}+\beta^{2}+\gamma^{2}-\lambda^{2}-\mu^{2}-\nu^{2}=-1\right)
\end{matrix}&lt;/math&gt;

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bach69}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay46gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay55gau}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay58quad}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eul71}} 
*{{#section:History of Topics in Special Relativity/mathsource|fro77}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm53}} 
*{{#section:History of Topics in Special Relativity/mathsource|herm54}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu81}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|rib70}} 
*{{#section:History of Topics in Special Relativity/mathsource|rod40}} 
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|bor14}}
{{#section:History of Topics in Special Relativity/relsource|brill25}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L6}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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    <title>History of Topics in Special Relativity/Lorentz transformation (Möbius)</title>
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      <comment>/* Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations */</comment>
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      <text bytes="59456" sha1="3xq67kf85jh6cvw2firku7b9q5i4x85" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations==

The previously mentioned Euler-Rodrigues parameter ''a,b,c,d'' (i.e. Cayley-Hermite parameter in [[../Lorentz transformation (Cayley-Hermite)#math_Q3|E:('''Q3''')]] with ''d=1'') are closely related to Cayley–Klein parameter α,β,γ,δ introduced by [[#Cayley2|Helmholtz (1866/67), Cayley (1879)]] and [[#Klein|Klein (1884)]] to connect Möbius transformations &lt;math&gt;\tfrac{\alpha\zeta+\beta}{\gamma\zeta+\delta}&lt;/math&gt; and rotations:&lt;ref group=M&gt;Klein (1896/97), p. 12&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha &amp; =1+ib, &amp; \beta &amp; =-a+ic,\\
\gamma &amp; =a+ic, &amp; \delta &amp; =1-ib.
\end{align}
&lt;/math&gt;

thus [[../Lorentz transformation (Cayley-Hermite)#math_Q3|E:('''Q3''')]] becomes:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}\frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; \beta\delta-\alpha\gamma &amp; \frac{i}{2}\left(-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}\right)\\
\gamma\delta+\alpha\beta &amp; \alpha\delta+\beta\gamma &amp; i(\alpha\beta+\gamma\delta)\\
-\frac{i}{2}\left(-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; -i(\alpha\gamma+\beta\delta) &amp; \frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right)
\end{matrix}\right]\cdot\mathbf{x}\\
(\kappa=\alpha\delta-\beta\gamma)
\end{matrix}&lt;/math&gt;|{{equationRef|Q4}}}}

Also the Lorentz transformation can be expressed with variants of the Cayley–Klein parameters: One relates these parameters to a spin-matrix '''D''', the [[w:spin transformation]]s of variables &lt;math&gt;\xi',\eta',\bar{\xi}',\bar{\eta}'&lt;/math&gt; (the overline denotes [[w:complex conjugate]]), and the [[w:Möbius transformation]] of &lt;math&gt;\zeta',\bar{\zeta}'&lt;/math&gt;. When defined in terms of isometries of hyperblic space (hyperbolic motions), the [[w:Hermitian matrix]] '''u''' associated with these Möbius transformations produces an invariant determinant &lt;math&gt;\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}&lt;/math&gt; identical to the Lorentz interval. Therefore, these transformations were described by [[w:John Lighton Synge]] as being a &quot;factory for the mass production of Lorentz transformations&quot;.&lt;ref name=synge /&gt; It also turns out that the related [[w:spin group]] Spin(3, 1) or [[w:special linear group]] SL(2, C) acts as the [[w:Double cover (topology)|double cover]] of the Lorentz group (one Lorentz transformation corresponds to two spin transformations of different sign), while the [[w:Möbius group]] Con(0,2) or [[w:projective special linear group]] PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.

In space, the Möbius/Spin/Lorentz transformations can be written as:&lt;ref&gt;Klein (1928), § 3A&lt;/ref&gt;&lt;ref name=synge&gt;Synge (1956), ch. IV, 11&lt;/ref&gt;&lt;ref&gt;Penrose &amp; Rindler (1984), section 2.1&lt;/ref&gt;&lt;ref name=&quot;Lorente 2003, section 4&quot;&gt;Lorente (2003), section 4&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\zeta=\frac{x_{1}+ix_{2}}{x_{0}-x_{3}}=\frac{x_{0}+x_{3}}{x_{1}-ix_{2}}\rightarrow\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}\left|\zeta'=\frac{\xi'}{\eta'}\rightarrow\begin{align}\xi' &amp; =\alpha\xi+\beta\eta\\
\eta' &amp; =\gamma\xi+\delta\eta
\end{align}
\right.\\
\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}X_{1} &amp; X_{2}\\
X_{3} &amp; X_{4}
\end{matrix}\right)=\left(\begin{matrix}\bar{\xi}\xi &amp; \xi\bar{\eta}\\
\bar{\xi}\eta &amp; \bar{\eta}\eta
\end{matrix}\right)=\left(\begin{matrix}x_{0}+x_{3} &amp; x_{1}-ix_{2}\\
x_{1}+ix_{2} &amp; x_{0}-x_{3}
\end{matrix}\right)\\
\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}
\end{matrix}\right|\begin{matrix}\mathbf{D}=\left(\begin{matrix}\alpha &amp; \beta\\
\gamma &amp; \delta
\end{matrix}\right)\\
\begin{align}\det\boldsymbol{\mathbf{D}} &amp; =1\end{align}
\end{matrix}\\
\hline \mathbf{u}'=\mathbf{D}\cdot\mathbf{u}\cdot\bar{\mathbf{D}}^{\mathrm{T}}=\begin{align}X_{1}^{\prime} &amp; =X_{1}\alpha\bar{\alpha}+X_{2}\alpha\bar{\beta}+X_{3}\bar{\alpha}\beta+X_{4}\beta\bar{\beta}\\
X_{2}^{\prime} &amp; =X_{1}\bar{\alpha}\gamma+X_{2}\bar{\alpha}\delta+X_{3}\bar{\beta}\gamma+X_{4}\bar{\beta}\delta\\
X_{3}^{\prime} &amp; =X_{1}\alpha\bar{\gamma}+X_{2}\alpha\bar{\delta}+X_{3}\beta\bar{\gamma}+X_{4}\beta\bar{\delta}\\
X_{4}^{\prime} &amp; =X_{1}\gamma\bar{\gamma}+X_{2}\gamma\bar{\delta}+X_{3}\bar{\gamma}\delta+X_{4}\delta\bar{\delta}
\end{align}
\\
\hline \begin{align}X_{3}^{\prime}X_{2}^{\prime}-X_{1}^{\prime}X_{4}^{\prime} &amp; =X_{3}X_{2}-X_{1}X_{4}=0\\
\det\mathbf{u}'=x_{0}^{\prime2}-x_{1}^{\prime2}-x_{2}^{\prime2}-x_{3}^{\prime2} &amp; =\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|6a}}}}

thus:&lt;ref&gt;Penrose &amp; Rindler (1984), p. 17&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{2}\left[{\scriptstyle \begin{align} &amp; \alpha\bar{\alpha}+\beta\bar{\beta}+\gamma\bar{\gamma}+\delta\bar{\delta} &amp;  &amp; \alpha\bar{\beta}+\beta\bar{\alpha}+\gamma\bar{\delta}+\delta\bar{\gamma} &amp;  &amp; i(\alpha\bar{\beta}-\beta\bar{\alpha}+\gamma\bar{\delta}-\delta\bar{\gamma}) &amp;  &amp; \alpha\bar{\alpha}-\beta\bar{\beta}+\gamma\bar{\gamma}-\delta\bar{\delta}\\
 &amp; \alpha\bar{\gamma}+\gamma\bar{\alpha}+\beta\bar{\delta}+\delta\bar{\beta} &amp;  &amp; \alpha\bar{\delta}+\delta\bar{\alpha}+\beta\bar{\gamma}+\gamma\bar{\beta} &amp;  &amp; i(\alpha\bar{\delta}-\delta\bar{\alpha}+\gamma\bar{\beta}-\beta\bar{\gamma}) &amp;  &amp; \alpha\bar{\gamma}+\gamma\bar{\alpha}-\beta\bar{\delta}-\delta\bar{\beta}\\
 &amp; i(\gamma\bar{\alpha}-\alpha\bar{\gamma}+\delta\bar{\beta}-\beta\bar{\delta}) &amp;  &amp; i(\delta\bar{\alpha}-\alpha\bar{\delta}+\gamma\bar{\beta}-\beta\bar{\gamma}) &amp;  &amp; \alpha\bar{\delta}+\delta\bar{\alpha}-\beta\bar{\gamma}-\gamma\bar{\beta} &amp;  &amp; i(\gamma\bar{\alpha}-\alpha\bar{\gamma}+\beta\bar{\delta}-\delta\bar{\beta})\\
 &amp; \alpha\bar{\alpha}+\beta\bar{\beta}-\gamma\bar{\gamma}-\delta\bar{\delta} &amp;  &amp; \alpha\bar{\beta}+\beta\bar{\alpha}-\gamma\bar{\delta}-\delta\bar{\gamma} &amp;  &amp; i(\alpha\bar{\beta}-\beta\bar{\alpha}+\delta\bar{\gamma}-\gamma\bar{\delta}) &amp;  &amp; \alpha\bar{\alpha}-\beta\bar{\beta}-\gamma\bar{\gamma}+\delta\bar{\delta}
\end{align}
}\right]\cdot\mathbf{x}\\
(\alpha\delta-\beta\gamma=1)
\end{matrix}&lt;/math&gt;|{{equationRef|6b}}}}

or in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation ('''1b''')]] one can substitute &lt;math&gt;\left[u_{1},\ u_{2},\ u_{3},\ 1\right]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}},\ \tfrac{x_{0}}{x_{0}}\right]&lt;/math&gt; so that the Möbius/Lorentz transformations become related to the unit sphere:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}+u_{3}^{\prime2}=1\\
\hline \left.\begin{matrix}\zeta=\frac{u_{1}+iu_{2}}{1-u_{3}}=\frac{1+u_{3}}{u_{1}-iu_{2}}\\
\zeta'=\frac{u_{1}^{\prime}+iu_{2}^{\prime}}{1-u_{3}^{\prime}}=\frac{1+u_{3}^{\prime}}{u_{1}^{\prime}-iu_{2}^{\prime}}
\end{matrix}\right|\quad\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}
\end{matrix}&lt;/math&gt;|{{equationRef|6c}}}}

The general transformation '''u′''' in ({{equationNote|6a}}) was given by [[#Cayley2|Cayley (1854)]], while the general relation between Möbius transformations and transformation '''u′''' leaving invariant the [[w:generalized circle]] was pointed out by [[#Poincare2|Poincaré (1883)]] in relation to [[w:Kleinian group]]s. The adaptation to the Lorentz interval by which ({{equationNote|6a}}) becomes a Lorentz transformation was given by [[#Klein2|Klein (1889-1893, 1896/97)]], [[#Bianchi2|Bianchi (1893)]], [[#Fricke|Fricke (1893, 1897)]]. Its reformulation as Lorentz transformation ({{equationNote|6b}}) was provided by [[#Bianchi2|Bianchi (1893)]] and [[#Fricke|Fricke (1893, 1897)]]. Lorentz transformation ({{equationNote|6c}}) was given by [[#Klein2|Klein (1884)]] in relation to surfaces of second degree and the invariance of the unit sphere. In relativity, ({{equationNote|6a}}) was first employed by [[#Herglotz1|Herglotz (1909/10)]].

In the plane, the transformations can be written as:&lt;ref name=k28&gt;Klein (1928), § 2A&lt;/ref&gt;&lt;ref name=&quot;Lorente 2003, section 4&quot;/&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\zeta=\frac{x_{1}}{x_{0}-x_{2}}=\frac{x_{0}+x_{2}}{x_{1}}\rightarrow\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}\left|\zeta'=\frac{\xi'}{\eta'}\rightarrow\begin{align}\xi' &amp; =\alpha\xi+\beta\eta\\
\eta' &amp; =\gamma\xi+\delta\eta
\end{align}
\right.\\
\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}X_{1} &amp; X_{2}\\
X_{2} &amp; X_{3}
\end{matrix}\right)=\left(\begin{matrix}\xi^{2} &amp; \xi\eta\\
\xi\eta &amp; \eta^{2}
\end{matrix}\right)=\left(\begin{matrix}x_{0}+x_{2} &amp; x_{1}\\
x_{1} &amp; x_{0}-x_{2}
\end{matrix}\right)\\
\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}
\end{matrix}\right|\begin{matrix}\mathbf{D}=\left(\begin{matrix}\alpha &amp; \beta\\
\gamma &amp; \delta
\end{matrix}\right)\\
\begin{align}\det\boldsymbol{\mathbf{D}} &amp; =1\end{align}
\end{matrix}\\
\hline \mathbf{u}'=\mathbf{D}\cdot\mathbf{u}\cdot\mathbf{D}^{\mathrm{T}}=\begin{align}X_{1}^{\prime} &amp; =X_{1}\alpha^{2}+X_{2}2\alpha\beta+X_{3}\beta^{2}\\
X_{2}^{\prime} &amp; =X_{1}\alpha\gamma+X_{2}(\alpha\delta+\beta\gamma)+X_{3}\beta\delta\\
X_{3}^{\prime} &amp; =X_{1}\gamma^{2}+X_{2}2\gamma\delta+X_{3}\delta^{2}
\end{align}
\\
\hline \begin{align}X_{2}^{\prime2}-X_{1}^{\prime}X_{3}^{\prime} &amp; =X_{2}^{2}-X_{1}X_{3}=0\\
\det\mathbf{u}'=x_{0}^{\prime2}-x_{1}^{\prime2}-x_{2}^{\prime2} &amp; =\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|6d}}}}

thus

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\left[\begin{matrix}\frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \alpha\beta+\gamma\delta &amp; \frac{1}{2}\left(\alpha^{2}-\beta^{2}+\gamma^{2}-\delta^{2}\right)\\
\alpha\gamma+\beta\delta &amp; \alpha\delta+\beta\gamma &amp; \alpha\gamma-\beta\delta\\
\frac{1}{2}\left(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2}\right) &amp; \alpha\beta-\gamma\delta &amp; \frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right)
\end{matrix}\right]\cdot\mathbf{x}\\
(\alpha\delta-\beta\gamma=1)
\end{matrix}&lt;/math&gt;|{{equationRef|6e}}}}

which includes the special case &lt;math&gt;\beta=\gamma=0&lt;/math&gt; implying &lt;math&gt;\delta=1/\alpha&lt;/math&gt;, reducing the transformation to a Lorentz boost in 1+1 dimensions:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}X_{1}X_{3}=X_{1}^{\prime}X_{3}^{\prime}\quad\Rightarrow\quad-x_{0}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{align}X_{1} &amp; =\alpha^{2}X_{1}^{\prime}\\
X_{2} &amp; =X_{2}^{\prime}\\
X_{3} &amp; =\frac{1}{\alpha^{2}}X_{3}^{\prime}
\end{align}
\quad\Rightarrow\quad\begin{align}x_{0} &amp; =\frac{x_{0}^{\prime}\left(\alpha^{4}+1\right)+x_{2}^{\prime}\left(\alpha^{4}-1\right)}{2\alpha^{2}}\\
x_{1} &amp; =x_{1}^{\prime}\\
x_{2} &amp; =\frac{x_{0}^{\prime}\left(\alpha^{4}-1\right)+x_{2}^{\prime}\left(\alpha^{4}+1\right)}{2\alpha^{2}}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|6f}}}}

Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}=-1\\
\hline \left.\begin{matrix}\zeta=\frac{x_{1}+ix_{2}}{x_{0}+1}=\frac{x_{0}-1}{x_{1}-ix_{2}}\\
\zeta'=\frac{x_{1}^{\prime}+ix_{2}^{\prime}}{x_{0}^{\prime}+1}=\frac{x_{0}^{\prime}-1}{x_{1}^{\prime}-ix_{2}^{\prime}}
\end{matrix}\right|\quad\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}
\end{matrix}&lt;/math&gt;|{{equationRef|6g}}}}

The general transformation '''u′''' and its invariant &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; in ({{equationNote|6d}}) was already used by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]] in the theory of integer binary quadratic forms. The invariant &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; was also studied by [[#Klein|Klein (1871)]] in connection to hyperbolic plane geometry (see [[../Lorentz transformation (hyperbolic)#math_3d|E:('''3d''')]]), while the connection between '''u′''' and &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; with the Möbius transformation was analyzed by [[#Poincare2|Poincaré (1886)]] in relation to [[w:Fuchsian group]]s. The adaptation to the Lorentz interval by which ({{equationNote|6d}}) becomes a Lorentz transformation was given by [[#Bianchi2|Bianchi (1888)]] and [[#Fricke|Fricke (1891)]]. Lorentz Transformation ({{equationNote|6e}}) was stated by [[#Gauss3|Gauss around 1800]] (posthumously published 1863), as well as [[#Selling|Selling (1873)]], [[#Bianchi2|Bianchi (1888)]], [[#Fricke|Fricke (1891)]], [[#Woods|Woods (1895)]] in relation to integer indefinite ternary quadratic forms. Lorentz transformation ({{equationNote|6f}}) was given by [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Eisenhart|Eisenhart (1905)]]. Lorentz transformation ({{equationNote|6g}}) of the hyperboloid was stated by [[#Poincare2|Poincaré (1881)]] and [[#Hausdorff|Hausdorff (1899)]].

==Historical notation==
==={{anchor|Lagrange}} Lagrange (1773) – Binary quadratic forms===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation of coefficients ''(p,q,r)'' is identical to transformation '''u′''' in ({{equationNote|6d}}) and becomes the complete Lorentz transformation by setting
:&lt;math&gt;\begin{align}(p,q,r) &amp; =\left(x_{0}+x_{2},\ x_{1},\ x_{0}-x_{2}\right)\\
(P,Q,R) &amp; =\left(x_{0}^{\prime}+x_{2}^{\prime},\ x_{1}^{\prime},\ x_{0}^{\prime}-x_{2}^{\prime}\right)
\end{align}&lt;/math&gt;.}}

==={{anchor|Gauss}} Gauss (1800)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gauss|label 1=History of Lorentz transformations in general § Gauss}}
==== Binary quadratic form====
The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

{{Lorentzbox|Text=The transformation of coefficients ''(a,b,c)'' is identical to transformation '''u′''' in ({{equationNote|6d}}) and becomes the complete Lorentz transformation by setting
:&lt;math&gt;\begin{align}(a,b,c) &amp; =\left(x_{0}+x_{2},\ x_{1},\ x_{0}-x_{2}\right)\\
(a',b',c') &amp; =\left(x_{0}^{\prime}+x_{2}^{\prime},\ x_{1}^{\prime},\ x_{0}^{\prime}-x_{2}^{\prime}\right)
\end{align}&lt;/math&gt;.}}

===={{anchor|Gauss3}} Cayley–Klein parameter====

After [[../Lorentz transformation (general)#Gauss2|E:Gauss (1798/1801)]] defined the integer ternary quadratic form
:&lt;math&gt;f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)&lt;/math&gt;

he derived around 1800 (posthumously published in 1863) the most general transformation of the Lorentz interval &lt;math&gt;\scriptstyle\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt; into itself, using a coefficient system α,β,γ,δ:&lt;ref group=M&gt;Gauss (1800/1863), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)\\
\hline \begin{matrix}\alpha\delta+\beta\gamma &amp; \alpha\beta-\gamma\delta &amp; \alpha\beta+\gamma\delta\\
\alpha\gamma-\beta\delta &amp; \frac{1}{2}(\alpha\alpha+\delta\delta-\beta\beta-\gamma\gamma) &amp; \frac{1}{2}(\alpha\alpha+\gamma\gamma-\beta\beta-\delta\delta)\\
\alpha\gamma+\beta\delta &amp; \frac{1}{2}(\alpha\alpha+\beta\beta-\gamma\gamma-\delta\delta) &amp; \frac{1}{2}(\alpha\alpha+\beta\beta+\gamma\gamma+\delta\delta)
\end{matrix}\\
(\alpha\delta-\beta\gamma=1)
\end{matrix} &lt;/math&gt;

Gauss' result was cited by [[../Lorentz transformation (Cayley-Hermite)#Bachmann|E:Bachmann (1869)]], [[#Selling|Selling (1873)]], [[#Bianchi2|Bianchi (1888)]], [[w:Leonard Eugene Dickson]] (1923).&lt;ref&gt;Dickson (1923), p. 210&lt;/ref&gt; The parameters α,β,γ,δ, when applied to spatial rotations, were later called Cayley–Klein parameters.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

==={{anchor|Cayley2}} Cayley (1854) – Cayley–Klein parameter===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Cayley|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Cayley}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cayley3|label 1=History of Lorentz transformations via Quaternions § Cayley}}

Already in 1854, Cayley published an alternative method of transforming quadratic forms by using certain parameters α,β,γ,δ in relation to an ''improper'' homographic transformation of a surface of second order into itself:&lt;ref group=M name=cayl1854&gt;Cayley (1854), p. 135&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xy-zw=0\\
x_{2}y_{2}-z_{2}w_{2}=x_{1}y_{1}-z_{1}w_{1}\\
\hline \left.\begin{align}MM'x_{2} &amp; =\gamma'\delta x_{1}+\alpha\alpha'y_{1}-\alpha'\delta z_{1}-\alpha\gamma'w_{1}\\
MM'y_{2} &amp; =\beta\beta'x_{1}+\gamma\delta'y_{1}-\beta\delta'z_{1}-\beta'\gamma w_{1}\\
MM'z_{2} &amp; =\beta\gamma'x_{1}+\gamma\alpha'y_{1}-\beta\alpha'z_{1}-\gamma\gamma'w_{1}\\
MM'w_{2} &amp; =\beta'\delta x_{1}+\alpha\delta'y_{1}-\delta\delta'z_{1}-\alpha\beta'w_{1}
\end{align}
\right|\begin{align}M^{2} &amp; =\alpha\beta-\gamma\delta\\
M^{\prime2} &amp; =\alpha'\beta'-\gamma'\delta'
\end{align}
\end{matrix}&lt;/math&gt;

By setting &lt;math&gt;\left(x_{1},y_{1}\dots\right)\Rightarrow\left(x_{1}+iy_{1},x_{1}-iy_{1}\dots\right)&lt;/math&gt; and rewriting M and M' in terms of four different parameters &lt;math&gt;M^{2}=a^{2}+b^{2}+c^{2}+d^{2}&lt;/math&gt; he demonstrated the invariance of &lt;math&gt;x_1^2+y_1^2+z_1^2+w_1^2&lt;/math&gt;, and subsequently showed the relation to 4D quaternion transformations. Fricke &amp; Klein (1897) credited Cayley by calling the above transformation the most general (real or complex) space collineation of first kind of an absolute surface of second kind into itself.&lt;ref group=M name=fri /&gt; Parameters α,β,γ,δ are similar to what was later called Cayley–Klein parameters in relation to spatial rotations (which was done by Cayley in 1879&lt;ref group=M&gt;Cayley (1879), p. 238f.&lt;/ref&gt; and before by [[w:Hermann von Helmholtz]] (1866/67)&lt;ref group=M&gt;Helmholtz (1866/67), p. 513&lt;/ref&gt;).

{{Lorentzbox|Text=Cayley's improper transformation becomes proper with some sign changes, and becomes equivalent to Lorentz transformation &lt;math&gt;\mathbf{u}'&lt;/math&gt; in ({{equationNote|6a}}) by setting M=M'=1 and:
:&lt;math&gt;(x_1,\ y_1,\ z_1,\ w_1)=\left(x_0+x_3,\ x_0-x_3,\ x_1+ix_2,\ x_1-ix_2 \right)&lt;/math&gt;.
Subsequently solved for &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; it becomes Lorentz transformation ({{equationNote|6b}}).}}

==={{anchor|Klein}} Klein (1871–97)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Klein|label 1=History of Lorentz transformations in general § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Klein|label 1=History of Lorentz transformations via squeeze mappings § Klein}}

===={{anchor|Klein1}} Cayley absolute and non-Euclidean geometry====

Elaborating on Cayley's (1859) definition of an &quot;absolute&quot; ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a &quot;fundamental [[w:conic section]]&quot; in order to discuss motions such as rotation and translation in the non-Euclidean plane,&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt; and another fundamental form by using [[w:homogeneous coordinates]] ''x,y'' related to a circle with radius ''2c'' with measure of curvature &lt;math&gt;-\tfrac{1}{4c^{2}}&lt;/math&gt;. When ''c'' is positive, the measure of curvature is negative and the fundamental conic section is real, thus the geometry becomes hyperbolic ([[w:Beltrami–Klein model]]):&lt;ref group=M&gt;Klein (1871), p. 618&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1}x_{2}-x_{3}^{2} &amp; =0\\
x^{2}+y^{2}-4c^{2} &amp; =0
\end{align}
\left|\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\
\hline \begin{align}x_{1} &amp; =\alpha_{1}y_{1}\\
x_{2} &amp; =\alpha_{2}y_{2}\\
x_{3} &amp; =\alpha_{3}y_{3}
\end{align}
\\
\left(\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)
\end{matrix}\right.&lt;/math&gt;

In (1873) he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to an ellipsoid or two-sheet [[w:hyperboloid]].&lt;ref group=M&gt;Klein (1873), pp. 127-128&lt;/ref&gt;

{{Lorentzbox|Text=Using positive ''c'' in &lt;math&gt;-\tfrac{1}{4c^{2}}&lt;/math&gt; in line with hyperbolic geometry or directly by setting &lt;math&gt;-\tfrac{1}{4c^{2}}=-x_{0}&lt;/math&gt;, Klein's two quadratic forms can be related to expressions &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; and &lt;math&gt;x_{0}^{2}-x_{1}^{2}-x_{2}^{2}&lt;/math&gt; for the Lorentz interval in ({{equationNote|6d}}).}}

===={{anchor|Klein2}} Möbius transformation, spin transformation, Cayley–Klein parameter====

In (1872) while devising the [[w:Erlangen program]], Klein discussed the general relation between projective metrics, [[w:binary form]]s and conformal geometry transforming a sphere into itself in terms of linear transformations of the [[w:complex variable]] ''x+iy''.&lt;ref group=M&gt;Klein (1872), 6&lt;/ref&gt; Following Klein, these relations were discussed by [[w:Ludwig Wedekind]] (1875) using &lt;math&gt;z'=\tfrac{\alpha z+\beta}{\gamma z+\delta}&lt;/math&gt;.&lt;ref group=M&gt;Wedekind (1875), 1&lt;/ref&gt; Klein (1875) then showed that all finite groups of motions follow by determining all finite groups of such linear transformations of ''x+iy'' into itself.&lt;ref group=M&gt;Klein (1875), §1–2&lt;/ref&gt; In (1878),&lt;ref group=M&gt;Klein (1878), 8.&lt;/ref&gt; Klein classified the substitutions of &lt;math&gt;\omega'=\tfrac{\alpha\omega+\beta}{\gamma\omega+\delta}&lt;/math&gt; with αδ-βγ=1 into hyperbolic, elliptic, parabolic, and in (1882)&lt;ref group=M&gt;Klein (1882), p. 173.&lt;/ref&gt; he added the loxodromic substitution as the combination of elliptic and hyperbolic ones. (In 1890, [[w:Robert Fricke]] in his edition of Klein's lectures of [[w:elliptic function]]s and [[w:Modular form]]s, referred to the analogy of this treatment to the theory of quadratic forms as given by Gauss and in particular Dirichlet.)&lt;ref group=M name=fri /&gt;

In (1884) Klein related the linear fractional transformations (interpreted as rotations around the ''x+iy''-sphere) to Cayley–Klein parameters [α,β,γ,δ], to Euler–Rodrigues parameters ''[a,b,c,d]'', and to the [[w:unit sphere]] by means of [[w:stereographic projection]], and also discussed transformations preserving surfaces of second degree equivalent to the transformation given by [[#Cayley2|Cayley (1854)]]:&lt;ref group=M&gt;Klein (1884), Part I, Ch. I, §1–2; Part II, Ch. II, 10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{matrix}z'=\frac{\alpha z+\beta}{\gamma z+\delta}\rightarrow z=z_{1}:z_{2}\rightarrow\begin{align}z_{1}^{\prime} &amp; =\alpha z_{1}+\beta z_{2}\\
z_{2}^{\prime} &amp; =\gamma z_{1}+\delta z_{2}
\end{align}
\\
\xi^{2}+\eta^{2}+\zeta^{2}=1\\
z=x+iy=\frac{\xi+i\eta}{1-\zeta}\\
z'=\frac{(d+ic)z-(b-ia)}{(b+ia)z+(d-ic)}\\
\left(a^{2}+b^{2}+c^{2}+d^{2}=1\right)
\end{matrix}\right| &amp; \begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
\lambda'=\frac{a\lambda+b}{c\lambda+d},\ \mu'=\frac{a'\mu+b'}{c'\mu+d'}\\
\lambda=\lambda_{1}:\lambda_{2},\ \mu=\mu_{1}:\mu_{2}\\
X_{1}:X_{2}:X_{3}:X_{4}=\lambda_{1}\mu_{1}:-\lambda_{2}\mu_{1}:\lambda_{1}\mu_{2}:\lambda_{2}\mu_{2}
\end{matrix}\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The formulas on the left related to the unit sphere are equivalent to Lorentz transformation ({{equationNote|6c}}). The formulas on the right can be related to those on the left by setting
:&lt;math&gt;(X_{1},\ X_{2},\ X_{3},\ X_{4})=\left(1+\zeta,\ -\xi+i\eta,\ \xi+i\eta,\ 1-\zeta\right)&lt;/math&gt;
and become equivalent to Lorentz transformation ({{equationNote|6a}}) by setting
:&lt;math&gt;\left[\xi,\ \eta,\ \zeta,\ 1\right]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}},\ \tfrac{x_{0}}{x_{0}}\right]&lt;/math&gt;
and subsequently solved for ''x''&lt;sub&gt;1&lt;/sub&gt;... it becomes Lorentz transformation ({{equationNote|6b}}).}}

In his lecture in the winter semester of 1889/90 (published 1892–93), he discussed the hyperbolic plane by using (as in 1871) the Lorentz interval in terms of a circle with radius ''2k'' as the basis of hyperbolic geometry, and another quadratic form to discuss the &quot;kinematics of hyperbolic geometry&quot; consisting of motions and congruent displacements of the hyperbolic plane into itself:&lt;ref group=M&gt;Klein (1893a), p. 109ff; pp. 138–140; pp. 249–250&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}x^{2}+y^{2}-4k^{2}t^{2}=0\\
x_{1}x_{3}-x_{2}^{2}=0
\end{matrix} &amp; \left|\begin{matrix}x_{1}x_{3}-x_{2}^{2}=0\\
\frac{x_{1}}{x_{2}}=\frac{x_{2}}{x_{3}}=\lambda=\frac{\lambda_{1}}{\lambda_{2}}\\
\lambda'=\frac{\alpha\lambda+\beta}{\gamma\lambda+\delta}\rightarrow\begin{align}\lambda_{1}^{\prime} &amp; =\alpha\lambda_{1}+\beta\lambda_{2}\\
\lambda_{2}^{\prime} &amp; =\gamma\lambda_{1}+\delta\lambda_{2}
\end{align}
\\
\left(\alpha\delta-\beta\gamma=1\right)\\
\begin{align}x_{1}:x_{2}:x_{3} &amp; =\lambda^{2}:\lambda:1=\lambda_{1}^{2}:\lambda_{1}\lambda_{2}:\lambda_{2}^{2}\\
 &amp; =\lambda^{\prime2}:\lambda':1=\lambda_{1}^{\prime2}:\lambda_{1}^{\prime}\lambda_{2}^{\prime}:\lambda_{2}^{\prime2};
\end{align}
\end{matrix}\right.\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Klein's Lorentz interval &lt;math&gt;x^{2}+y^{2}-4k^{2}t^{2}&lt;/math&gt; can be connected with the other interval &lt;math&gt;x_{1}x_{3}-x_{2}^{2}&lt;/math&gt; by setting
:&lt;math&gt;(x_{1},\ x_{2},\ x_{3})=\left(x+iy,\ 2kt,\ x-iy\right)&lt;/math&gt;,
by which the transformation system on the right becomes equivalent to Lorentz transformation ({{equationNote|6d}}) with ''2k=1'', and subsequently solved for ''x''&lt;sub&gt;1&lt;/sub&gt;... it becomes equivalent to Lorentz transformation ({{equationNote|6e}}).}}

In his lecture in the summer semester of 1890 (published 1892–93), he discussed general surfaces of second degree, including an &quot;oval&quot; surface corresponding to hyperbolic space and its motions:&lt;ref group=M&gt;Klein (1893b); general surface: pp. 61–66, 116–119, hyperbolic space: pp. 82, 86, 143–144&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}\text{General surfaces of second degree}:\\
\begin{align}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2} &amp; \text{(no real parts, elliptic)}\\
z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2} &amp; \text{(oval,hyperbolic)}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2}-z_{4}^{2} &amp; \text{(ring)}\\
z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2} &amp; \text{(oval,hyperbolic)}\\
-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2} &amp; \text{(no real parts,elliptic)}
\end{align}
\\
\text{all of which can be brought into the form:}\\
y_{1}y_{3}+y_{2}y_{4}=0\\
\text{Transformation:}\\
\begin{align}\varrho y_{1} &amp; =\lambda_{1}\mu_{1}, &amp; \varrho y_{1}^{\prime} &amp; =\lambda_{1}^{\prime}\mu_{1}^{\prime}\\
\varrho y_{2} &amp; =\lambda_{2}\mu_{1}, &amp; \varrho y_{2}^{\prime} &amp; =\lambda_{2}^{\prime}\mu_{1}^{\prime}\\
\varrho y_{3} &amp; =\lambda_{2}\mu_{2}, &amp; \varrho y_{3}^{\prime} &amp; =-\lambda_{2}^{\prime}\mu_{2}^{\prime}\\
\varrho y_{4} &amp; =\lambda_{1}\mu_{2}, &amp; \varrho y_{4}^{\prime} &amp; =\lambda_{1}^{\prime}\mu_{2}^{\prime}
\end{align}
\end{matrix}\right|\begin{matrix}\text{Oval (=hyperbolic motions in space):}\\
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0\\
=\left(x_{1}+ix_{3}\right)\left(x_{1}-ix_{3}\right)+\left(x_{2}+x_{4}\right)\left(x_{2}-x_{4}\right)=0\\
=y_{1}y_{3}+y_{2}y_{4}=0\\
\\
x^{2}+y^{2}+z^{2}-1=0\\
\hline \lambda=\frac{x+iy}{1-z},\ \lambda'=\frac{\alpha\lambda+\beta}{\gamma\lambda+\delta},\ \mu'=\frac{\bar{\alpha}\mu+\bar{\beta}}{\bar{\gamma}\mu+\bar{\delta}}\\
\begin{align}\lambda_{1}^{\prime} &amp; =\alpha\lambda_{1}+\beta\lambda_{2}\\
\lambda_{2}^{\prime} &amp; =\gamma\lambda_{1}+\delta\lambda_{2}
\end{align}
,\ \begin{align}\mu_{1}^{\prime} &amp; =\bar{\alpha}\mu_{1}+\bar{\beta}\mu_{2}\\
\mu_{2}^{\prime} &amp; =\bar{\gamma}\mu_{1}+\bar{\delta}\mu_{2}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation of the unit sphere &lt;math&gt;x^{2}+y^{2}+z^{2}-1=0&lt;/math&gt; on the right is equivalent to Lorentz transformation ({{equationNote|6c}}). Plugging the values for λ,μ,λ′,μ′,... from the right into the transformations on the left, and relating them to Klein's homogeneous coordinates &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0&lt;/math&gt; by &lt;math&gt;(x,\ y,\ z,\ 1)=\left(\tfrac{x_{1}}{x_{4}},\ \tfrac{x_{2}}{x_{4}},\ \tfrac{x_{3}}{x_{4}},\ \tfrac{x_{4}}{x_{4}}\right)&lt;/math&gt; leads to Lorentz transformation ({{equationNote|6a}}). Subsequently solved for ''x''&lt;sub&gt;1&lt;/sub&gt;... it becomes Lorentz transformation ({{equationNote|6b}}).}}

In (1896/97), Klein again defined hyperbolic motions and explicitly used ''t'' as time coordinate, even though he added those cautionary remarks: ''&quot;We shall consider t also as capable of complex values, not for the sake of studying the behavior of a fictitious, imaginary time, but because it is only by taking this step that it becomes possible to bring about the intimate association of kinetics and the theory of functions of a complex variable. [..] the non-Euclidean geometry has no meta-physical significance here or in the subsequent discussion&quot;''. Using homogeneous coordinates, Klein defined the sphere x,y,z,t and then another &quot;movable&quot; sphere X,Y,Z,T as follows:&lt;ref group=M&gt;Klein (1896/97), pp. 13–14&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}-t^{2}=0\\
=(x+iy)(x-iy)+(z+t)(z-t)=0\\
x+iy:x-iy:z+t:t-z=\zeta_{1}\zeta_{2}^{\prime}:\zeta_{2}\zeta_{1}^{\prime}:\zeta_{1}\zeta_{1}^{\prime}:\zeta_{2}\zeta_{2}^{\prime}\\
\frac{\zeta_{1}}{\zeta_{2}}=\zeta\quad\Rightarrow\quad\zeta=\frac{x+iy}{t-z}=\frac{t+z}{x-iy};\\
\hline X^{2}+Y^{2}+Z^{2}-T^{2}=0\\
\text{introducing}\ Z,Z_{1},Z_{2}\dots\text{similarly as above}\ \zeta,\zeta_{1},\zeta_{2}\dots
\end{matrix}&lt;/math&gt;

which he related by the following transformation:

:&lt;math&gt;\begin{matrix}\zeta=\frac{\alpha Z+\beta}{\gamma Z+\delta}\rightarrow\begin{align}\zeta_{1} &amp; =\alpha Z_{1}+\beta Z_{2}\\
\zeta_{2} &amp; =\gamma Z_{1}+\delta Z_{2}
\end{align}
,\ \begin{align}\zeta_{1}^{\prime} &amp; =\bar{\alpha}Z_{1}^{\prime}+\bar{\beta}Z_{2}^{\prime}\\
\zeta_{2}^{\prime} &amp; =\bar{\gamma}Z_{1}^{\prime}+\bar{\delta}Z_{2}^{\prime}\text{ }
\end{align}
\\
(\alpha\delta-\beta\gamma=1)\\
\hline \begin{array}{c|c|c|c|c}
 &amp; X+iY &amp; X-iY &amp; T+Z &amp; T-Z\\
\hline x+iy &amp; \alpha\bar{\delta} &amp; \beta\bar{\gamma} &amp; \alpha\bar{\gamma} &amp; \beta\bar{\delta}\\
\hline x-iy &amp; \gamma\bar{\beta} &amp; \delta\bar{\alpha} &amp; \gamma\bar{\alpha} &amp; \delta\bar{\beta}\\
\hline t+z &amp; \alpha\bar{\beta} &amp; \beta\bar{\alpha} &amp; \alpha\bar{\alpha} &amp; \beta\bar{\beta}\\
\hline t-z &amp; \gamma\bar{\delta} &amp; \delta\bar{\gamma} &amp; \gamma\bar{\gamma} &amp; \delta\bar{\delta}
\end{array}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}). Klein's work was summarized and extended by [[#Bianchi2|Bianchi (1888-1893)]] and [[#Fricke|Fricke (1893-1897)]], obtaining equivalent Lorentz transformations.}}

==={{anchor|Selling}} Selling (1873–74) – Quadratic forms===

Continuing the work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] on definite ternary quadratic forms and [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] on indefinite ternary quadratic forms, [[w:Eduard Selling]] (1873) used the auxiliary coefficients ξ,η,ζ by which a definite form &lt;math&gt;\mathfrak{f}&lt;/math&gt; and an indefinite form ''f'' can be rewritten in terms of three squares:&lt;ref group=M&gt;Selling (1873), p. 174 and p. 179&lt;/ref&gt;&lt;ref&gt;Bachmann (1923), chapter 16&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{align}\mathfrak{f} &amp; =\mathfrak{a}x^{2}+\mathfrak{b}y^{2}+\mathfrak{c}z^{2}+2\mathfrak{g}yz+2\mathfrak{h}zx+2\mathfrak{k}xy\\
 &amp; =\left(\xi x+\eta y+\zeta z\right)^{2}+\left(\xi_{1}x+\eta_{1}y+\zeta_{1}z\right)^{2}+\left(\xi_{2}x+\eta_{2}y+\zeta_{2}z\right)^{2}\\
\\
f &amp; =ax^{2}+by^{2}+cz^{2}+2gyz+2hzx+2kxy\\
 &amp; =\left(\xi x+\eta y+\zeta z\right)^{2}-\left(\xi_{1}x+\eta_{1}y+\zeta_{1}z\right)^{2}-\left(\xi_{2}x+\eta_{2}y+\zeta_{2}z\right)^{2}
\end{align}
\left|\begin{align}\xi^{2}+\xi_{1}^{2}+\xi_{2}^{2} &amp; =\mathfrak{a}\\
\eta^{2}+\eta_{1}^{2}+\eta_{2}^{2} &amp; =\mathfrak{b}\\
\zeta^{2}+\zeta_{1}^{2}+\zeta_{2}^{2} &amp; =\mathfrak{c}\\
\eta\zeta+\eta_{1}\zeta_{1}+\eta_{2}\zeta_{2} &amp; =\mathfrak{g}\\
\zeta\xi+\zeta_{1}\xi_{1}+\zeta_{2}\xi_{2} &amp; =\mathfrak{h}\\
\xi\eta+\xi_{1}\eta_{1}+\xi_{2}\eta_{2} &amp; =\mathfrak{k}
\end{align}
\right|\begin{align}\xi^{2}-\xi_{1}^{2}-\xi_{2}^{2} &amp; =a\\
\eta^{2}-\eta_{1}^{2}-\eta_{2}^{2} &amp; =b\\
\zeta^{2}-\zeta_{1}^{2}-\zeta_{2}^{2} &amp; =c\\
\eta\zeta-\eta_{1}\zeta_{1}-\eta_{2}\zeta_{2} &amp; =g\\
\zeta\xi-\zeta_{1}\xi_{1}-\zeta_{2}\xi_{2} &amp; =h\\
\xi\eta-\xi_{1}\eta_{1}-\xi_{2}\eta_{2} &amp; =k
\end{align}
}&lt;/math&gt;

In addition, Selling showed that auxiliary coefficients ξ,η,ζ can be geometrically interpreted as point coordinates which are in motion upon one sheet of a two-sheet hyperboloid, which is related to Selling's formalism for the reduction of indefinite forms by using definite forms.&lt;ref group=M&gt;Selling (1873), pp. 182–183&lt;/ref&gt;

Selling also reproduced the Lorentz transformation given by [[#Gauss3|Gauss (1800/63)]], to whom he gave full credit, and called it the only example of a particular indefinite ternary form known to him that has ever been discussed:&lt;ref group=M&gt;Selling (1873/74), p. 227 (see also p. 225 for citation).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)\\
\hline W=\begin{vmatrix}\frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{2}\left(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2}\right) &amp; \alpha\gamma+\beta\delta\\
\frac{1}{2}\left(\alpha^{2}-\beta^{2}+\gamma^{2}-\delta^{2}\right) &amp; \frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; \alpha\gamma-\beta\delta\\
\alpha\beta+\gamma\delta &amp; \alpha\beta-\gamma\delta &amp; \alpha\delta+\beta\gamma
\end{vmatrix}\\
\left(\begin{vmatrix}\alpha &amp; \beta\\
\gamma &amp; \delta
\end{vmatrix}=1\right)
\end{matrix} &lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) or [[../Lorentz transformation (squeeze)#math_9b|E:('''9b''')]] as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

==={{anchor|Poincare2}} Poincaré (1881-86) – Möbius transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Poincare|label 1=History of Lorentz transformations in general § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Poincare3|label 1=History of Lorentz transformations via velocity § Poincaré}}

[[w:Henri Poincaré]] (1881a) demonstrated the connection of his formulas of the hyperboloid model [see [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]]] to Möbius transformations:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
\left[X=\frac{\xi}{\zeta+1},\ Y=\frac{\eta}{\zeta+1}\right]\rightarrow t=X+iY\\
\hline \xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1\\
\left[X'=\frac{\xi'}{\zeta'+1},\ Y'=\frac{\eta'}{\zeta'+1}\right]\rightarrow t'=X'+iY'\\
\hline t'=\frac{ht+k}{h't+k'}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6g}}).}}

Poincaré (1881b) also used the Möbius transformation &lt;math&gt;\tfrac{az+b}{cz+d}&lt;/math&gt; in relation to [[w:Fuchsian function]]s and the discontinuous [[w:Fuchsian group]], being a special case of the hyperbolic group leaving invariant the &quot;fundamental circle&quot; ([[w:Poincaré disk model]] and [[w:Poincaré half-plane model]] of hyperbolic geometry).&lt;ref group=M&gt;Poincaré (1881b), p. 333&lt;/ref&gt; He then extended [[#Klein2|Klein's (1878-1882)]] study on the relation between Möbius transformations and hyperbolic, elliptic, parabolic, and loxodromic substitutions, and while formulating [[w:Kleinian group]]s (1883) he used the following transformation leaving invariant the [[w:generalized circle]]:&lt;ref group=M&gt;Poincaré (1883), pp. 49–50; 53–54&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(z,\ \frac{\alpha z+\beta}{\gamma z+\delta}\right),\ \left(z_{0},\ \frac{\alpha_{0}z_{0}+\beta_{0}}{\gamma_{0}z_{0}+\delta_{0}}\right)\\
\hline z=\xi+i\eta,\ z_{0}=\xi-i\eta,\ \rho^{2}=\xi^{2}+\eta^{2}+\zeta^{2}\\
A\rho^{\prime2}+Bz^{\prime}+B_{0}z_{0}^{\prime}+C=0\\
\hline \begin{align}\rho^{\prime2} &amp; =\frac{\rho^{2}\alpha\alpha_{0}+z\alpha\beta_{0}+z_{0}\beta\alpha_{0}+\beta\beta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}\\
z^{\prime} &amp; =\frac{\rho^{2}\alpha\gamma_{0}+z\alpha\delta_{0}+z_{0}\beta\gamma_{0}+\beta\delta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}\\
z_{0}^{\prime} &amp; =\frac{\rho^{2}\gamma\alpha_{0}+z\gamma\beta_{0}+z_{0}\delta\alpha_{0}+\delta\beta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting &lt;math&gt;[\rho^{2},\ z,\ z_{0}]=\left[\tfrac{X_{1}}{X_{4}},\ \tfrac{X_{2}}{X_{4}},\ \tfrac{X_{3}}{X_{4}}\right]&lt;/math&gt; this becomes transformation '''u′''' in ({{equationNote|6a}}) and becomes the complete Lorentz transformation by setting &lt;math&gt;{\scriptstyle \left[\begin{matrix}X_{1} &amp; X_{2}\\
X_{3} &amp; X_{4}
\end{matrix}\right]=\left[\begin{matrix}x_{0}+x_{3} &amp; x_{1}-ix_{2}\\
x_{1}+ix_{2} &amp; x_{0}-x_{3}
\end{matrix}\right]}&lt;/math&gt;.}}

In 1886, Poincaré investigated the relation between indefinite ternary quadratic forms and Fuchsian functions and groups:&lt;ref group=M&gt;Poincaré (1886), p. 735ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(z,\ \frac{\alpha z+\beta}{\gamma z+\delta}\right)\\
\hline Y^{\prime2}-X'Z'=Y^{2}-XZ\\
\hline \begin{align}X' &amp; =\alpha^{2}X+2\alpha\gamma Y+\gamma^{2}Z\\
Y' &amp; =\alpha\beta X+(\alpha\delta+\beta\gamma)Y+\gamma\delta Z\\
Z' &amp; =\beta^{2}X+2\beta\gamma Y+\delta^{2}Z
\end{align}
\\
\left[{\scriptstyle \begin{align}X= &amp; ax+by+cz, &amp; Y &amp; =a'x+b'y+c'z, &amp; Z &amp; =a''x+b''y+c''z,\\
X'= &amp; ax'+by'+cz', &amp; Y' &amp; =a'x'+b'y'+c'z', &amp; Z' &amp; =a''x'+b''y'+c''z',
\end{align}
}\right]
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text= This is equivalent to transformation '''u′''' in ({{equationNote|6d}}) and becomes the complete Lorentz transformation by suitibly choosing the coefficients ''a,b,c,...'' so that ''[X,Y,Z]=[x+z, y, -x+z]''.}}

==={{anchor|Bianchi2}} Bianchi (1888-93) – Möbius and spin transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Bianchi1|label 1=History of Lorentz transformations via trigonometric functions § Bianchi}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Bianchi1|label 1=History of Lorentz transformations via squeeze mappings § Bianchi}}

Related to [[#Klein1|Klein's (1871)]] and [[#Poincare2|Poincaré's (1881-1887)]] work on non-Euclidean geometry and indefinite quadratic forms, [[w:Luigi Bianchi]] (1888) analyzed the differential Lorentz interval in term of conic sections and hyperboloids, alluded to the linear fractional transformation of &lt;math&gt;\omega&lt;/math&gt; and its conjugate &lt;math&gt;\omega_{1}&lt;/math&gt; with parameters α,β,γ,δ in order to preserve the Lorentz interval, and gave credit to [[#Gauss3|Gauss (1800/63)]] who obtained the same coefficient system:&lt;ref group=M&gt;Bianchi (1888), pp. 547; 562–563 (especially footnote on p. 563); 571–572&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=dx^{2}+dy^{2}-dz^{2};\ x^{2}+y^{2}-z^{2}=0;\\
\hline X_{3}^{2}+Y_{3}^{2}-Z_{3}^{2}=-1\\
X_{3}=i\frac{1-\omega\omega_{1}}{\omega-\omega_{1}},\ Y_{3}=i\frac{\omega-\omega_{1}}{\omega-\omega_{1}},\ Z_{3}=i\frac{1+\omega\omega_{1}}{\omega-\omega_{1}},\\
\omega=\frac{\alpha\omega'+\beta}{\gamma\omega'+\delta}\quad(\alpha\delta-\beta\gamma=1)\\
\hline \left(\begin{matrix}\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}, &amp; \gamma\delta-\alpha\beta, &amp; \frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}\\
\beta\delta-\alpha\gamma, &amp; \alpha\delta+\beta\gamma, &amp; \beta\delta+\alpha\gamma\\
\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}, &amp; \alpha\beta+\gamma\delta, &amp; \frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}
\end{matrix}\right)\\
\hline \begin{align}x' &amp; =\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\gamma\delta-\alpha\beta)y+\frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}z+c_{1}\\
y' &amp; =(\beta\delta-\alpha\gamma)x+(\alpha\delta+\beta\gamma)y+(\beta\delta+\alpha\gamma)z+c_{2}\\
z' &amp; =\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\alpha\beta+\gamma\delta)y+\frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}z+c_{3}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The is equivalent to Lorentz transformations ({{equationNote|6d}}) and ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) or [[../Lorentz transformation (squeeze)#math_9b|E:('''9b''')]] as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

In 1893, Bianchi gave the coefficients in the case of four dimensions:&lt;ref group=M name=bi&gt;Bianchi (1893), § 3&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}z &amp; =\frac{\alpha z'+\beta}{\gamma z'+\delta}\\
 &amp; (\alpha\delta-\beta\gamma=1)
\end{align}
\rightarrow\begin{align}z &amp; =\frac{\xi}{\eta}\\
z' &amp; =\frac{\xi'}{\eta'}
\end{align}
\rightarrow\begin{align}\xi &amp; =\alpha\xi'+\beta\eta'\\
\eta &amp; =\gamma\xi'+\delta\eta'\\
\\
\xi_{0} &amp; =\alpha_{0}\xi'_{0}+\beta_{0}\eta'_{0}\\
\eta_{0} &amp; =\gamma_{0}\xi'_{0}+\delta_{0}\eta'_{0}
\end{align}
\\
\hline {\scriptstyle F=\left(u_{1}+u{}_{4}\right)\xi\xi_{0}+\left(u_{2}+iu{}_{3}\right)\xi\eta_{0}+\left(u_{2}-iu{}_{3}\right)\xi_{0}\eta+\left(u_{4}-u{}_{1}\right)\eta\eta_{0}}\\
{\scriptstyle F'=\left(u'_{1}+u'{}_{4}\right)\xi'\xi'_{0}+\left(u'_{2}+iu'{}_{3}\right)\xi'\eta'_{0}+\left(u'_{2}-iu'{}_{3}\right)\xi'_{0}\eta'+\left(u'_{4}-u'{}_{1}\right)\eta'\eta'_{0}}\\
{\scriptstyle \left(u_{2}+iu{}_{3}\right)\left(u_{2}-iu{}_{3}\right)+\left(u_{1}-u{}_{4}\right)\left(u_{1}+u{}_{4}\right)=\left(u'_{2}+iu'{}_{3}\right)\left(u'_{2}-iu'{}_{3}\right)+\left(u'_{1}-u'{}_{4}\right)\left(u'_{1}+u'{}_{4}\right)}\\
\hline {\scriptstyle \begin{align}u'_{1}+u'_{4} &amp; =\alpha\alpha_{0}\left(u_{1}+u{}_{4}\right)+\alpha\gamma_{0}\left(u_{2}+iu{}_{3}\right)+\alpha_{0}\gamma\left(u_{2}-iu{}_{3}\right)+\gamma\gamma_{0}\left(u_{4}-u{}_{1}\right)\\
u'_{2}+iu'_{3} &amp; =\alpha\beta_{0}\left(u_{1}+u{}_{4}\right)+\alpha\delta_{0}\left(u_{2}+iu{}_{3}\right)+\beta_{0}\gamma\left(u_{2}-iu{}_{3}\right)+\gamma\delta_{0}\left(u_{4}-u{}_{1}\right)\\
u'_{2}-iu'_{3} &amp; =\alpha_{0}\beta\left(u_{1}+u{}_{4}\right)+\alpha_{0}\delta\left(u_{2}-iu{}_{3}\right)+\beta\gamma_{0}\left(u_{2}+iu{}_{3}\right)+\gamma_{0}\delta\left(u_{4}-u{}_{1}\right)\\
u'_{4}-u'_{1} &amp; =\beta\beta_{0}\left(u_{1}+u{}_{4}\right)+\beta\delta_{0}\left(u_{2}+iu{}_{3}\right)+\beta_{0}\delta\left(u_{2}-iu{}_{3}\right)+\delta\delta_{0}\left(u_{4}-u{}_{1}\right)
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}).}}

Solving for &lt;math&gt;u'_{1}\dots&lt;/math&gt; Bianchi obtained:&lt;ref group=M name=bi /&gt;

:&lt;math&gt;\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}-u_{4}^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}+u_{3}^{\prime2}-u_{4}^{\prime2}\\
\hline {\scriptstyle \begin{align}u'_{1} &amp; =\frac{1}{2}\left(\alpha\alpha_{0}-\beta\beta_{0}-\gamma\gamma_{0}+\delta\delta_{0}\right)u_{1}+\frac{1}{2}\left(\alpha\gamma_{0}+\alpha_{0}\gamma-\beta\delta_{0}-\beta_{0}\delta\right)u_{2}+\\
 &amp; +\frac{i}{2}\left(\alpha\gamma_{0}-\alpha_{0}\gamma+\beta_{0}\delta-\beta\delta_{0}\right)u_{3}+\frac{1}{2}\left(\alpha\alpha_{0}-\beta\beta_{0}+\gamma\gamma_{0}-\delta\delta_{0}\right)u_{4}\\
u'_{2} &amp; =\frac{1}{2}\left(\alpha\beta_{0}+\alpha_{0}\beta-\gamma\delta_{0}-\gamma_{0}\delta\right)u_{1}+\frac{1}{2}\left(\alpha\delta_{0}+\alpha_{0}\delta+\beta\gamma_{0}+\beta_{0}\gamma\right)u_{2}+\\
 &amp; +\frac{i}{2}\left(\alpha\delta_{0}-\alpha_{0}\delta+\beta\gamma_{0}-\beta_{0}\gamma\right)u_{3}+\frac{1}{2}\left(\alpha\beta_{0}+\alpha_{0}\beta+\gamma\delta_{0}+\gamma_{0}\delta\right)u_{4}\\
u'_{3} &amp; =\frac{i}{2}\left(\alpha_{0}\beta-\alpha\beta_{0}+\gamma\delta_{0}-\gamma_{0}\delta\right)u_{1}+\frac{i}{2}\left(\alpha_{0}\delta-\alpha\delta_{0}+\beta\gamma_{0}-\beta_{0}\gamma\right)u_{2}+\\
 &amp; +\frac{1}{2}\left(\alpha\delta_{0}+\alpha_{0}\delta-\beta\gamma_{0}-\beta_{0}\gamma\right)u_{3}+\frac{i}{2}\left(\alpha_{0}\beta-\alpha\beta_{0}+\gamma_{0}\delta-\gamma\delta_{0}\right)u_{4}\\
u'_{4} &amp; =\frac{1}{2}\left(\alpha\alpha_{0}+\beta\beta_{0}-\gamma\gamma_{0}-\delta\delta_{0}\right)u_{1}+\frac{1}{2}\left(\alpha\gamma_{0}+\alpha_{0}\gamma+\beta\delta_{0}+\beta_{0}\delta\right)u_{2}+\\
 &amp; +\frac{i}{2}\left(\alpha\gamma_{0}-\alpha_{0}\gamma+\beta\delta_{0}-\beta_{0}\delta\right)u_{3}+\frac{1}{2}\left(\alpha\alpha_{0}+\beta\beta_{0}+\gamma\gamma_{0}+\delta\delta_{0}\right)u_{4}
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6b}}).}}

==={{anchor|Fricke}} Fricke (1891–97) – Möbius and spin transformations===

[[w:Robert Fricke]] (1891) – following the work of his teacher [[#Klein2|Klein (1878–1882)]] as well as [[#Poincare2|Poincaré (1881–1887)]] on automorphic functions and group theory – obtained the following transformation for an integer ternary quadratic form&lt;ref group=M&gt;Fricke (1891), §§ 1, 6&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 221, 232&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\omega'=\frac{\delta\omega+\beta}{\gamma\omega+\alpha}\ (\alpha\delta-\beta\gamma=1),\ \omega=\frac{\eta}{\xi},\\
\hline \begin{align}\xi' &amp; =\xi\alpha^{2}+2\eta\alpha\gamma+\zeta\gamma^{2}\\
\eta' &amp; =\xi\alpha\beta+\eta(\alpha\delta+\beta\gamma)+\zeta\gamma\delta\\
\zeta' &amp; =\xi\beta^{2}+2\eta\beta\delta+\zeta\delta^{2}
\end{align}
\\
\hline \xi'\zeta'-\eta'^{2}=(\alpha\delta-\beta\gamma)^{2}\left(\xi\zeta-\eta^{2}\right)\\
\xi=x\sqrt{q}-y,\ \eta=z,\ \zeta=x\sqrt{q}+y\\
\hline qx^{\prime2}-y^{\prime2}-z^{\prime2}=qx^{2}-y^{2}-z^{2}\\
\hline \left(\begin{matrix}\frac{1}{2}\left(+\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{2\sqrt{q}}\left(-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{\sqrt{q}}(\alpha\gamma+\beta\delta)\\
\frac{1}{2}\sqrt{q}\left(-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{2}\left(+\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; (-\alpha\gamma+\beta\delta)\\
\sqrt{q}(\alpha\beta+\gamma\delta) &amp; (-\alpha\beta+\gamma\delta) &amp; (\alpha\delta+\beta\gamma)
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By setting ''q=1'', the first part is equivalent to Lorentz transformation ({{equationNote|6d}}) and the second part is equivalent to ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) or [[../Lorentz transformation (squeeze)#math_9b|E:('''9b''')]] as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

And the general case of four dimensions in 1893:&lt;ref group=M&gt;Fricke (1893), pp. 706, 710–711&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}y'_{2}y'_{3}-y'_{1}y'_{4}=y_{2}y_{3}-y_{1}y_{4}\\
\hline \begin{align}y_{1}^{\prime} &amp; =\alpha\bar{\alpha}y_{1}+\alpha\bar{\beta}y_{2}+\beta\bar{\alpha}y_{3}+\beta\bar{\beta}y_{4}\\
y_{2}^{\prime} &amp; =\alpha\bar{\gamma}y_{1}+\alpha\bar{\delta}y_{2}+\beta\bar{\gamma}y_{3}+\beta\bar{\delta}y_{4}\\
y_{3}^{\prime} &amp; =\gamma\bar{\alpha}y_{1}+\gamma\bar{\beta}y_{2}+\delta\bar{\alpha}y_{3}+\delta\bar{\beta}y_{4}\\
y_{4}^{\prime} &amp; =\gamma\bar{\gamma}y_{1}+\gamma\bar{\delta}y_{2}+\delta\bar{\gamma}y_{3}+\delta\bar{\delta}y_{4}
\end{align}
\\
\hline \begin{align}y_{1} &amp; =z_{4}\sqrt{s}+z_{3}\sqrt{r}, &amp; y_{2} &amp; =z_{1}\sqrt{p}+iz_{2}\sqrt{q}\\
y_{3} &amp; =z_{1}\sqrt{p}-iz_{2}\sqrt{q}, &amp; y_{4} &amp; =z_{4}\sqrt{s}-z_{3}\sqrt{r}
\end{align}
\\
\hline pz_{1}^{\prime2}+qz{}_{2}^{\prime2}+rz{}_{3}^{\prime2}-sz{}_{4}^{\prime2}=pz_{1}^{2}+qz_{2}^{2}+rz_{3}^{2}-sz_{4}^{2}\\
\hline z'_{i}=\alpha_{i1}z_{1}+\alpha_{i2}z_{2}+\alpha_{i3}z_{3}+\alpha_{i4}z_{4}\\
{\scriptstyle \begin{align}2\alpha_{11}\ \text{or}\ 2\alpha_{22} &amp; =\alpha\bar{\delta}+\delta\bar{\alpha}\pm\beta\bar{\gamma}\pm\gamma\bar{\beta}, &amp; 2\alpha_{33}\ \text{or}\ 2\alpha_{44} &amp; =\alpha\bar{\alpha}+\delta\bar{\delta}\pm\beta\bar{\beta}\pm\gamma\bar{\gamma}\\
\frac{2\alpha_{12}\sqrt{p}}{i\sqrt{p}}\ \text{or}\ \frac{2\alpha_{21}i\sqrt{p}}{\sqrt{p}} &amp; =\alpha\bar{\delta}-\bar{\delta}\alpha\mp\beta\bar{\gamma}\pm\gamma\bar{\beta}, &amp; \frac{2\alpha_{34}\sqrt{r}}{\sqrt{s}}\ \text{or}\ \frac{2\alpha_{43}\sqrt{s}}{\sqrt{r}} &amp; =\alpha\bar{\alpha}-\delta\bar{\delta}\pm\beta\bar{\beta}\pm\gamma\bar{\gamma}\\
\frac{2\alpha_{13}\sqrt{p}}{\sqrt{r}}\ \text{or}\ \frac{2\alpha_{24}i\sqrt{p}}{\sqrt{s}} &amp; =\alpha\bar{\gamma}-\delta\bar{\beta}\pm\gamma\bar{\alpha}\pm\beta\bar{\delta}, &amp; \frac{2\alpha_{14}\sqrt{p}}{\sqrt{s}}\ \text{or}\ \frac{2\alpha_{23}i\sqrt{q}}{\sqrt{r}} &amp; =\alpha\bar{\gamma}+\delta\bar{\beta}\pm\gamma\bar{\alpha}\pm\beta\bar{\delta}\\
\frac{2\alpha_{31}\sqrt{r}}{\sqrt{p}}\ \text{or}\ \frac{2\alpha_{43}\sqrt{s}}{i\sqrt{q}} &amp; =\alpha\bar{\beta}-\delta\bar{\gamma}\pm\beta\bar{\alpha}\mp\gamma\bar{\delta}, &amp; \frac{2\alpha_{41}\sqrt{s}}{\sqrt{p}}\ \text{or}\ \frac{2\alpha_{32}\sqrt{r}}{i\sqrt{q}} &amp; =\alpha\bar{\beta}+\delta\bar{\gamma}\pm\beta\bar{\alpha}\pm\gamma\bar{\delta}
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By setting ''p=q=r=s=1'', the first part is equivalent to Lorentz transformation ({{equationNote|6a}}) and the second part to ({{equationNote|6b}}).}}

Supported by Felix Klein, Fricke summarized his and Klein's work in a treatise concerning [[w:automorphic function]]s (1897). Using a sphere as the absolute, in which the interior of the sphere is denoted as hyperbolic space, they defined hyperbolic motions, and stressed that any hyperbolic motion corresponds to &quot;circle relations&quot; (now called Möbius transformations):&lt;ref group=M name=fri&gt;Fricke &amp; Klein (1897), §12–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
=(z_{4}+z_{3})(z_{4}-z_{3})-(z_{1}+iz_{2})(z_{1}-iz_{2})=0\\
=y_{1}y_{4}-y_{2}y_{3}=0\\
\left(y_{1}=z_{4}+z_{3},\ y_{2}=z_{1}+iz_{2},\ y_{3}=z_{1}-iz_{2},\ y_{4}=z_{4}-z_{3}\right)\\
\zeta=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}},\ \bar{\zeta}=\frac{z_{1}-iz_{2}}{z_{4}-z_{3}}\\
\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta},\ \bar{\zeta}'=\frac{\overline{\alpha\zeta}+\bar{\beta}}{\overline{\gamma\zeta}+\bar{\delta}}\quad(\alpha\delta-\beta\gamma\ne0)\\
z_{1}:z_{2}:z_{3}z_{4}=(\zeta+\bar{\zeta}):-i(\zeta-\bar{\zeta}):(\zeta\bar{\zeta}-1):(\zeta\bar{\zeta}+1)\\
y_{1}:y_{2}:y_{3}y_{4}=\zeta\bar{\zeta}:\zeta:\bar{\zeta}:1=\zeta_{1}\bar{\zeta}_{1}:\zeta_{1}\bar{\zeta}_{2}:\zeta_{2}\bar{\zeta}_{1}:\zeta_{2}\bar{\zeta}_{2}\\
\left(\zeta=\zeta_{1}:\zeta_{2},\ \bar{\zeta}=\bar{\zeta}_{1}:\bar{\zeta}_{2}\right)\\
\hline \begin{align}y_{1}^{\prime} &amp; =\alpha\bar{\alpha}y_{1}+\alpha\bar{\beta}y_{2}+\beta\bar{\alpha}y_{3}+\beta\bar{\beta}y_{4}\\
y_{2}^{\prime} &amp; =\alpha\bar{\gamma}y_{1}+\alpha\bar{\delta}y_{2}+\beta\bar{\gamma}y_{3}+\beta\bar{\delta}y_{4}\\
y_{3}^{\prime} &amp; =\gamma\bar{\alpha}y_{1}+\gamma\bar{\beta}y_{2}+\delta\bar{\alpha}y_{3}+\delta\bar{\beta}y_{4}\\
y_{4}^{\prime} &amp; =\gamma\bar{\gamma}y_{1}+\gamma\bar{\delta}y_{2}+\delta\bar{\gamma}y_{3}+\delta\bar{\delta}y_{4}
\end{align}
\end{matrix}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}).}}

==={{anchor|Woods}} Woods (1895) – Spin transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Woods2|label 1=History of Lorentz transformations via hyperbolic functions § Woods}}

In a thesis supervised by Felix Klein, [[w:Frederick S. Woods]] (1895) further developed [[#Bianchi2|Bianchi's (1888)]] treatment of surfaces satisfying the Lorentz interval (pseudominimal surface), and used the transformation of [[#Gauss3|Gauss (1800/63)]] and [[#Bianchi2|Bianchi (1888)]] while discussing automorphisms of that surface:&lt;ref group=M&gt;Woods (1895), pp. 2–3; 10–11; 34–35&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}=0;\quad x^{2}+y^{2}-z^{2}=-1\\
\hline \left(x,y,z\right)\Rightarrow\omega\\
\begin{align}\omega_{1}^{\prime} &amp; =\alpha\omega_{1}+\beta\omega_{2}\\
\omega_{2}^{\prime} &amp; =\gamma\omega_{1}+\delta\omega_{2}
\end{align}
\quad(\alpha\delta-\beta\gamma=1)\\
\hline \begin{align}x' &amp; =(-1)^{k}\left[\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\gamma\delta-\alpha\beta)y+\frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}z\right]+c_{1}\\
y' &amp; =(-1)^{k}\left[(\beta\delta-\alpha\gamma)x+(\alpha\delta+\beta\gamma)y+(\beta\delta+\alpha\gamma)z\right]+c_{2}\\
z' &amp; =(-1)^{k}\left[\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\alpha\beta+\gamma\delta)y+\frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}z\right]+c_{3}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The expressions within the brackets are equivalent to Lorentz transformations ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

Already in the context of special relativity, [[w:Gustav Herglotz]] (1909/10) followed [[#Klein2|Klein (1889–1897)]] as well as [[#Fricke|Fricke &amp; Klein (1897)]] concerning the Cayley absolute, hyperbolic motion and its transformation, and classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. He provided the general case (on the left) and the hyperbolic substitution (on the right) as follows:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\
Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\
Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}
\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\
\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bia88diff}} 
*{{#section:History of Topics in Special Relativity/mathsource|bia93quat}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay54hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay79hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|fri91}} 
*{{#section:History of Topics in Special Relativity/mathsource|fri93}} 
*{{#section:History of Topics in Special Relativity/mathsource|fri97}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau00}} 
*{{#section:History of Topics in Special Relativity/mathsource|helm66}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei72a}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei72b}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei73}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei75}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei79}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei82}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei84}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei90a}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei90b}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei93a}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei93b}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei96}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin83}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin86}} 
*{{#section:History of Topics in Special Relativity/mathsource|sel73}} 
*{{#section:History of Topics in Special Relativity/mathsource|wed75}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo95}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|herg10}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L7}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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      <comment>/* Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations */</comment>
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      <text bytes="59434" sha1="jwlknkalqlrx8ao5p1k9bjxar14r938" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via Cayley–Klein parameters, Möbius and spin transformations==

The previously mentioned Euler-Rodrigues parameter ''a,b,c,d'' (i.e. Cayley-Hermite parameter in [[../Lorentz transformation (Cayley-Hermite)#math_Q3|E:('''Q3''')]] with ''d=1'') are closely related to Cayley–Klein parameter α,β,γ,δ introduced by [[#Cayley2|Helmholtz (1866/67), Cayley (1879)]] and [[#Klein|Klein (1884)]] to connect Möbius transformations &lt;math&gt;\tfrac{\alpha\zeta+\beta}{\gamma\zeta+\delta}&lt;/math&gt; and rotations:&lt;ref group=M&gt;Klein (1896/97), p. 12&lt;/ref&gt;

:&lt;math&gt;\begin{align}\alpha &amp; =1+ib, &amp; \beta &amp; =-a+ic,\\
\gamma &amp; =a+ic, &amp; \delta &amp; =1-ib.
\end{align}
&lt;/math&gt;

thus [[../Lorentz transformation (Cayley-Hermite)#math_Q3|E:('''Q3''')]] becomes:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{\kappa}\left[\begin{matrix}\frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; \beta\delta-\alpha\gamma &amp; \frac{i}{2}\left(-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}\right)\\
\gamma\delta+\alpha\beta &amp; \alpha\delta+\beta\gamma &amp; i(\alpha\beta+\gamma\delta)\\
-\frac{i}{2}\left(-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; -i(\alpha\gamma+\beta\delta) &amp; \frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right)
\end{matrix}\right]\cdot\mathbf{x}\\
(\kappa=\alpha\delta-\beta\gamma)
\end{matrix}&lt;/math&gt;|{{equationRef|Q4}}}}

Also the Lorentz transformation can be expressed with variants of the Cayley–Klein parameters: One relates these parameters to a spin-matrix '''D''', the [[w:spin transformation]]s of variables &lt;math&gt;\xi',\eta',\bar{\xi}',\bar{\eta}'&lt;/math&gt; (the overline denotes [[w:complex conjugate]]), and the [[w:Möbius transformation]] of &lt;math&gt;\zeta',\bar{\zeta}'&lt;/math&gt;. When defined in terms of isometries of hyperblic space (hyperbolic motions), the [[w:Hermitian matrix]] '''u''' associated with these Möbius transformations produces an invariant determinant &lt;math&gt;\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}&lt;/math&gt; identical to the Lorentz interval. Therefore, these transformations were described by [[w:John Lighton Synge]] as being a &quot;factory for the mass production of Lorentz transformations&quot;.&lt;ref name=synge /&gt; It also turns out that the related [[w:spin group]] Spin(3, 1) or [[w:special linear group]] SL(2, C) acts as the [[w:Double cover (topology)|double cover]] of the Lorentz group (one Lorentz transformation corresponds to two spin transformations of different sign), while the [[w:Möbius group]] Con(0,2) or [[w:projective special linear group]] PSL(2, C) is isomorphic to both the Lorentz group and the group of isometries of hyperbolic space.

In space, the Möbius/Spin/Lorentz transformations can be written as:&lt;ref&gt;Klein (1928), § 3A&lt;/ref&gt;&lt;ref name=synge&gt;Synge (1956), ch. IV, 11&lt;/ref&gt;&lt;ref&gt;Penrose &amp; Rindler (1984), section 2.1&lt;/ref&gt;&lt;ref name=&quot;Lorente 2003, section 4&quot;&gt;Lorente (2003), section 4&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\zeta=\frac{x_{1}+ix_{2}}{x_{0}-x_{3}}=\frac{x_{0}+x_{3}}{x_{1}-ix_{2}}\rightarrow\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}\left|\zeta'=\frac{\xi'}{\eta'}\rightarrow\begin{align}\xi' &amp; =\alpha\xi+\beta\eta\\
\eta' &amp; =\gamma\xi+\delta\eta
\end{align}
\right.\\
\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}X_{1} &amp; X_{2}\\
X_{3} &amp; X_{4}
\end{matrix}\right)=\left(\begin{matrix}\bar{\xi}\xi &amp; \xi\bar{\eta}\\
\bar{\xi}\eta &amp; \bar{\eta}\eta
\end{matrix}\right)=\left(\begin{matrix}x_{0}+x_{3} &amp; x_{1}-ix_{2}\\
x_{1}+ix_{2} &amp; x_{0}-x_{3}
\end{matrix}\right)\\
\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}
\end{matrix}\right|\begin{matrix}\mathbf{D}=\left(\begin{matrix}\alpha &amp; \beta\\
\gamma &amp; \delta
\end{matrix}\right)\\
\begin{align}\det\boldsymbol{\mathbf{D}} &amp; =1\end{align}
\end{matrix}\\
\hline \mathbf{u}'=\mathbf{D}\cdot\mathbf{u}\cdot\bar{\mathbf{D}}^{\mathrm{T}}=\begin{align}X_{1}^{\prime} &amp; =X_{1}\alpha\bar{\alpha}+X_{2}\alpha\bar{\beta}+X_{3}\bar{\alpha}\beta+X_{4}\beta\bar{\beta}\\
X_{2}^{\prime} &amp; =X_{1}\bar{\alpha}\gamma+X_{2}\bar{\alpha}\delta+X_{3}\bar{\beta}\gamma+X_{4}\bar{\beta}\delta\\
X_{3}^{\prime} &amp; =X_{1}\alpha\bar{\gamma}+X_{2}\alpha\bar{\delta}+X_{3}\beta\bar{\gamma}+X_{4}\beta\bar{\delta}\\
X_{4}^{\prime} &amp; =X_{1}\gamma\bar{\gamma}+X_{2}\gamma\bar{\delta}+X_{3}\bar{\gamma}\delta+X_{4}\delta\bar{\delta}
\end{align}
\\
\hline \begin{align}X_{3}^{\prime}X_{2}^{\prime}-X_{1}^{\prime}X_{4}^{\prime} &amp; =X_{3}X_{2}-X_{1}X_{4}=0\\
\det\mathbf{u}'=x_{0}^{\prime2}-x_{1}^{\prime2}-x_{2}^{\prime2}-x_{3}^{\prime2} &amp; =\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|6a}}}}

thus:&lt;ref&gt;Penrose &amp; Rindler (1984), p. 17&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}+x_{3}^{\prime2}\\
\hline \mathbf{x}'=\frac{1}{2}\left[{\scriptstyle \begin{align} &amp; \alpha\bar{\alpha}+\beta\bar{\beta}+\gamma\bar{\gamma}+\delta\bar{\delta} &amp;  &amp; \alpha\bar{\beta}+\beta\bar{\alpha}+\gamma\bar{\delta}+\delta\bar{\gamma} &amp;  &amp; i(\alpha\bar{\beta}-\beta\bar{\alpha}+\gamma\bar{\delta}-\delta\bar{\gamma}) &amp;  &amp; \alpha\bar{\alpha}-\beta\bar{\beta}+\gamma\bar{\gamma}-\delta\bar{\delta}\\
 &amp; \alpha\bar{\gamma}+\gamma\bar{\alpha}+\beta\bar{\delta}+\delta\bar{\beta} &amp;  &amp; \alpha\bar{\delta}+\delta\bar{\alpha}+\beta\bar{\gamma}+\gamma\bar{\beta} &amp;  &amp; i(\alpha\bar{\delta}-\delta\bar{\alpha}+\gamma\bar{\beta}-\beta\bar{\gamma}) &amp;  &amp; \alpha\bar{\gamma}+\gamma\bar{\alpha}-\beta\bar{\delta}-\delta\bar{\beta}\\
 &amp; i(\gamma\bar{\alpha}-\alpha\bar{\gamma}+\delta\bar{\beta}-\beta\bar{\delta}) &amp;  &amp; i(\delta\bar{\alpha}-\alpha\bar{\delta}+\gamma\bar{\beta}-\beta\bar{\gamma}) &amp;  &amp; \alpha\bar{\delta}+\delta\bar{\alpha}-\beta\bar{\gamma}-\gamma\bar{\beta} &amp;  &amp; i(\gamma\bar{\alpha}-\alpha\bar{\gamma}+\beta\bar{\delta}-\delta\bar{\beta})\\
 &amp; \alpha\bar{\alpha}+\beta\bar{\beta}-\gamma\bar{\gamma}-\delta\bar{\delta} &amp;  &amp; \alpha\bar{\beta}+\beta\bar{\alpha}-\gamma\bar{\delta}-\delta\bar{\gamma} &amp;  &amp; i(\alpha\bar{\beta}-\beta\bar{\alpha}+\delta\bar{\gamma}-\gamma\bar{\delta}) &amp;  &amp; \alpha\bar{\alpha}-\beta\bar{\beta}-\gamma\bar{\gamma}+\delta\bar{\delta}
\end{align}
}\right]\cdot\mathbf{x}\\
(\alpha\delta-\beta\gamma=1)
\end{matrix}&lt;/math&gt;|{{equationRef|6b}}}}

or in line with [[../Lorentz transformation (general)#math_1b|E:general Lorentz transformation ('''1b''')]] one can substitute &lt;math&gt;\left[u_{1},\ u_{2},\ u_{3},\ 1\right]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}},\ \tfrac{x_{0}}{x_{0}}\right]&lt;/math&gt; so that the Möbius/Lorentz transformations become related to the unit sphere:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}+u_{3}^{\prime2}=1\\
\hline \left.\begin{matrix}\zeta=\frac{u_{1}+iu_{2}}{1-u_{3}}=\frac{1+u_{3}}{u_{1}-iu_{2}}\\
\zeta'=\frac{u_{1}^{\prime}+iu_{2}^{\prime}}{1-u_{3}^{\prime}}=\frac{1+u_{3}^{\prime}}{u_{1}^{\prime}-iu_{2}^{\prime}}
\end{matrix}\right|\quad\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}
\end{matrix}&lt;/math&gt;|{{equationRef|6c}}}}

The general transformation '''u′''' in ({{equationNote|6a}}) was given by [[#Cayley2|Cayley (1854)]], while the general relation between Möbius transformations and transformation '''u′''' leaving invariant the [[w:generalized circle]] was pointed out by [[#Poincare2|Poincaré (1883)]] in relation to [[w:Kleinian group]]s. The adaptation to the Lorentz interval by which ({{equationNote|6a}}) becomes a Lorentz transformation was given by [[#Klein2|Klein (1889-1893, 1896/97)]], [[#Bianchi2|Bianchi (1893)]], [[#Fricke|Fricke (1893, 1897)]]. Its reformulation as Lorentz transformation ({{equationNote|6b}}) was provided by [[#Bianchi2|Bianchi (1893)]] and [[#Fricke|Fricke (1893, 1897)]]. Lorentz transformation ({{equationNote|6c}}) was given by [[#Klein2|Klein (1884)]] in relation to surfaces of second degree and the invariance of the unit sphere. In relativity, ({{equationNote|6a}}) was first employed by [[#Herglotz1|Herglotz (1909/10)]].

In the plane, the transformations can be written as:&lt;ref name=k28&gt;Klein (1928), § 2A&lt;/ref&gt;&lt;ref name=&quot;Lorente 2003, section 4&quot;/&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\zeta=\frac{x_{1}}{x_{0}-x_{2}}=\frac{x_{0}+x_{2}}{x_{1}}\rightarrow\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}\left|\zeta'=\frac{\xi'}{\eta'}\rightarrow\begin{align}\xi' &amp; =\alpha\xi+\beta\eta\\
\eta' &amp; =\gamma\xi+\delta\eta
\end{align}
\right.\\
\hline \left.\begin{matrix}\mathbf{u}=\left(\begin{matrix}X_{1} &amp; X_{2}\\
X_{2} &amp; X_{3}
\end{matrix}\right)=\left(\begin{matrix}\xi^{2} &amp; \xi\eta\\
\xi\eta &amp; \eta^{2}
\end{matrix}\right)=\left(\begin{matrix}x_{0}+x_{2} &amp; x_{1}\\
x_{1} &amp; x_{0}-x_{2}
\end{matrix}\right)\\
\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}
\end{matrix}\right|\begin{matrix}\mathbf{D}=\left(\begin{matrix}\alpha &amp; \beta\\
\gamma &amp; \delta
\end{matrix}\right)\\
\begin{align}\det\boldsymbol{\mathbf{D}} &amp; =1\end{align}
\end{matrix}\\
\hline \mathbf{u}'=\mathbf{D}\cdot\mathbf{u}\cdot\mathbf{D}^{\mathrm{T}}=\begin{align}X_{1}^{\prime} &amp; =X_{1}\alpha^{2}+X_{2}2\alpha\beta+X_{3}\beta^{2}\\
X_{2}^{\prime} &amp; =X_{1}\alpha\gamma+X_{2}(\alpha\delta+\beta\gamma)+X_{3}\beta\delta\\
X_{3}^{\prime} &amp; =X_{1}\gamma^{2}+X_{2}2\gamma\delta+X_{3}\delta^{2}
\end{align}
\\
\hline \begin{align}X_{2}^{\prime2}-X_{1}^{\prime}X_{3}^{\prime} &amp; =X_{2}^{2}-X_{1}X_{3}=0\\
\det\mathbf{u}'=x_{0}^{\prime2}-x_{1}^{\prime2}-x_{2}^{\prime2} &amp; =\det\mathbf{u}=x_{0}^{2}-x_{1}^{2}-x_{2}^{2}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|6d}}}}

thus

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \mathbf{x}'=\left[\begin{matrix}\frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \alpha\beta+\gamma\delta &amp; \frac{1}{2}\left(\alpha^{2}-\beta^{2}+\gamma^{2}-\delta^{2}\right)\\
\alpha\gamma+\beta\delta &amp; \alpha\delta+\beta\gamma &amp; \alpha\gamma-\beta\delta\\
\frac{1}{2}\left(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2}\right) &amp; \alpha\beta-\gamma\delta &amp; \frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right)
\end{matrix}\right]\cdot\mathbf{x}\\
(\alpha\delta-\beta\gamma=1)
\end{matrix}&lt;/math&gt;|{{equationRef|6e}}}}

which includes the special case &lt;math&gt;\beta=\gamma=0&lt;/math&gt; implying &lt;math&gt;\delta=1/\alpha&lt;/math&gt;, reducing the transformation to a Lorentz boost in 1+1 dimensions:

{{NumBlk|:|&lt;math&gt;\begin{matrix}X_{1}X_{3}=X_{1}^{\prime}X_{3}^{\prime}\quad\Rightarrow\quad-x_{0}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{2}^{\prime2}\\
\hline \begin{align}X_{1} &amp; =\alpha^{2}X_{1}^{\prime}\\
X_{2} &amp; =X_{2}^{\prime}\\
X_{3} &amp; =\frac{1}{\alpha^{2}}X_{3}^{\prime}
\end{align}
\quad\Rightarrow\quad\begin{align}x_{0} &amp; =\frac{x_{0}^{\prime}\left(\alpha^{4}+1\right)+x_{2}^{\prime}\left(\alpha^{4}-1\right)}{2\alpha^{2}}\\
x_{1} &amp; =x_{1}^{\prime}\\
x_{2} &amp; =\frac{x_{0}^{\prime}\left(\alpha^{4}-1\right)+x_{2}^{\prime}\left(\alpha^{4}+1\right)}{2\alpha^{2}}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|6f}}}}

Finally, by using the Lorentz interval related to a hyperboloid, the Möbius/Lorentz transformations can be written

{{NumBlk|:|&lt;math&gt;\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}=-1\\
\hline \left.\begin{matrix}\zeta=\frac{x_{1}+ix_{2}}{x_{0}+1}=\frac{x_{0}-1}{x_{1}-ix_{2}}\\
\zeta'=\frac{x_{1}^{\prime}+ix_{2}^{\prime}}{x_{0}^{\prime}+1}=\frac{x_{0}^{\prime}-1}{x_{1}^{\prime}-ix_{2}^{\prime}}
\end{matrix}\right|\quad\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta}
\end{matrix}\,&lt;/math&gt;|{{equationRef|6g}}}}

The general transformation '''u′''' and its invariant &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; in ({{equationNote|6d}}) was already used by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]] in the theory of integer binary quadratic forms. The invariant &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; was also studied by [[#Klein|Klein (1871)]] in connection to hyperbolic plane geometry (see [[../Lorentz transformation (hyperbolic)#math_3d|E:('''3d''')]]), while the connection between '''u′''' and &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; with the Möbius transformation was analyzed by [[#Poincare2|Poincaré (1886)]] in relation to [[w:Fuchsian group]]s. The adaptation to the Lorentz interval by which ({{equationNote|6d}}) becomes a Lorentz transformation was given by [[#Bianchi2|Bianchi (1888)]] and [[#Fricke|Fricke (1891)]]. Lorentz Transformation ({{equationNote|6e}}) was stated by [[#Gauss3|Gauss around 1800]] (posthumously published 1863), as well as [[#Selling|Selling (1873)]], [[#Bianchi2|Bianchi (1888)]], [[#Fricke|Fricke (1891)]], [[#Woods|Woods (1895)]] in relation to integer indefinite ternary quadratic forms. Lorentz transformation ({{equationNote|6f}}) was given by [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Eisenhart|Eisenhart (1905)]]. Lorentz transformation ({{equationNote|6g}}) of the hyperboloid was stated by [[#Poincare2|Poincaré (1881)]] and [[#Hausdorff|Hausdorff (1899)]].

==Historical notation==
==={{anchor|Lagrange}} Lagrange (1773) – Binary quadratic forms===
After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows&lt;ref group=M&gt;Lagrange (1773/75), section 22&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\
\hline \begin{align}y &amp; =Ms+Nx\\
z &amp; =ms+nx
\end{align}
\left|\begin{matrix}\begin{align}P &amp; =pM^{2}+2qMm+rm^{2}\\
Q &amp; =pMN+q(Mn+Nm)+rmn\\
R &amp; =pN^{2}+2qNn+rn^{2}
\end{align}
\\
\downarrow\\
PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation of coefficients ''(p,q,r)'' is identical to transformation '''u′''' in ({{equationNote|6d}}) and becomes the complete Lorentz transformation by setting
:&lt;math&gt;\begin{align}(p,q,r) &amp; =\left(x_{0}+x_{2},\ x_{1},\ x_{0}-x_{2}\right)\\
(P,Q,R) &amp; =\left(x_{0}^{\prime}+x_{2}^{\prime},\ x_{1}^{\prime},\ x_{0}^{\prime}-x_{2}^{\prime}\right)
\end{align}&lt;/math&gt;.}}

==={{anchor|Gauss}} Gauss (1800)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Gauss|label 1=History of Lorentz transformations in general § Gauss}}
==== Binary quadratic form====
The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:&lt;ref group=M&gt;Gauss (1798/1801), articles 157–158;&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\
F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\
\hline \begin{align}x &amp; =\alpha x'+\beta y'\\
y &amp; =\gamma x'+\delta y'\\
\\
x' &amp; =\delta x-\beta y\\
y' &amp; =-\gamma x+\alpha y
\end{align}
\left|\begin{matrix}\begin{align}a' &amp; =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\
b' &amp; =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\
c' &amp; =a\beta^{2}+2b\beta\delta+c\delta^{2}
\end{align}
\\
\downarrow\\
b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2}
\end{matrix}\right.
\end{matrix}&lt;/math&gt;

As pointed out by Gauss, ''F'' and ''F′'' are called &quot;proper equivalent&quot; if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.&lt;ref group=M&gt;Gauss (1798/1801), section 159&lt;/ref&gt;

{{Lorentzbox|Text=The transformation of coefficients ''(a,b,c)'' is identical to transformation '''u′''' in ({{equationNote|6d}}) and becomes the complete Lorentz transformation by setting
:&lt;math&gt;\begin{align}(a,b,c) &amp; =\left(x_{0}+x_{2},\ x_{1},\ x_{0}-x_{2}\right)\\
(a',b',c') &amp; =\left(x_{0}^{\prime}+x_{2}^{\prime},\ x_{1}^{\prime},\ x_{0}^{\prime}-x_{2}^{\prime}\right)
\end{align}&lt;/math&gt;.}}

===={{anchor|Gauss3}} Cayley–Klein parameter====

After [[../Lorentz transformation (general)#Gauss2|E:Gauss (1798/1801)]] defined the integer ternary quadratic form
:&lt;math&gt;f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)&lt;/math&gt;

he derived around 1800 (posthumously published in 1863) the most general transformation of the Lorentz interval &lt;math&gt;\scriptstyle\left(\begin{matrix}a, &amp; a', &amp; a''\\
b, &amp; b', &amp; b''
\end{matrix}\right)=\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)&lt;/math&gt; into itself, using a coefficient system α,β,γ,δ:&lt;ref group=M&gt;Gauss (1800/1863), p. 311&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(\begin{matrix}1, &amp; 1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)\\
\hline \begin{matrix}\alpha\delta+\beta\gamma &amp; \alpha\beta-\gamma\delta &amp; \alpha\beta+\gamma\delta\\
\alpha\gamma-\beta\delta &amp; \frac{1}{2}(\alpha\alpha+\delta\delta-\beta\beta-\gamma\gamma) &amp; \frac{1}{2}(\alpha\alpha+\gamma\gamma-\beta\beta-\delta\delta)\\
\alpha\gamma+\beta\delta &amp; \frac{1}{2}(\alpha\alpha+\beta\beta-\gamma\gamma-\delta\delta) &amp; \frac{1}{2}(\alpha\alpha+\beta\beta+\gamma\gamma+\delta\delta)
\end{matrix}\\
(\alpha\delta-\beta\gamma=1)
\end{matrix} &lt;/math&gt;

Gauss' result was cited by [[../Lorentz transformation (Cayley-Hermite)#Bachmann|E:Bachmann (1869)]], [[#Selling|Selling (1873)]], [[#Bianchi2|Bianchi (1888)]], [[w:Leonard Eugene Dickson]] (1923).&lt;ref&gt;Dickson (1923), p. 210&lt;/ref&gt; The parameters α,β,γ,δ, when applied to spatial rotations, were later called Cayley–Klein parameters.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

==={{anchor|Cayley2}} Cayley (1854) – Cayley–Klein parameter===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Cayley|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Cayley}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Cayley3|label 1=History of Lorentz transformations via Quaternions § Cayley}}

Already in 1854, Cayley published an alternative method of transforming quadratic forms by using certain parameters α,β,γ,δ in relation to an ''improper'' homographic transformation of a surface of second order into itself:&lt;ref group=M name=cayl1854&gt;Cayley (1854), p. 135&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}xy-zw=0\\
x_{2}y_{2}-z_{2}w_{2}=x_{1}y_{1}-z_{1}w_{1}\\
\hline \left.\begin{align}MM'x_{2} &amp; =\gamma'\delta x_{1}+\alpha\alpha'y_{1}-\alpha'\delta z_{1}-\alpha\gamma'w_{1}\\
MM'y_{2} &amp; =\beta\beta'x_{1}+\gamma\delta'y_{1}-\beta\delta'z_{1}-\beta'\gamma w_{1}\\
MM'z_{2} &amp; =\beta\gamma'x_{1}+\gamma\alpha'y_{1}-\beta\alpha'z_{1}-\gamma\gamma'w_{1}\\
MM'w_{2} &amp; =\beta'\delta x_{1}+\alpha\delta'y_{1}-\delta\delta'z_{1}-\alpha\beta'w_{1}
\end{align}
\right|\begin{align}M^{2} &amp; =\alpha\beta-\gamma\delta\\
M^{\prime2} &amp; =\alpha'\beta'-\gamma'\delta'
\end{align}
\end{matrix}&lt;/math&gt;

By setting &lt;math&gt;\left(x_{1},y_{1}\dots\right)\Rightarrow\left(x_{1}+iy_{1},x_{1}-iy_{1}\dots\right)&lt;/math&gt; and rewriting M and M' in terms of four different parameters &lt;math&gt;M^{2}=a^{2}+b^{2}+c^{2}+d^{2}&lt;/math&gt; he demonstrated the invariance of &lt;math&gt;x_1^2+y_1^2+z_1^2+w_1^2&lt;/math&gt;, and subsequently showed the relation to 4D quaternion transformations. Fricke &amp; Klein (1897) credited Cayley by calling the above transformation the most general (real or complex) space collineation of first kind of an absolute surface of second kind into itself.&lt;ref group=M name=fri /&gt; Parameters α,β,γ,δ are similar to what was later called Cayley–Klein parameters in relation to spatial rotations (which was done by Cayley in 1879&lt;ref group=M&gt;Cayley (1879), p. 238f.&lt;/ref&gt; and before by [[w:Hermann von Helmholtz]] (1866/67)&lt;ref group=M&gt;Helmholtz (1866/67), p. 513&lt;/ref&gt;).

{{Lorentzbox|Text=Cayley's improper transformation becomes proper with some sign changes, and becomes equivalent to Lorentz transformation &lt;math&gt;\mathbf{u}'&lt;/math&gt; in ({{equationNote|6a}}) by setting M=M'=1 and:
:&lt;math&gt;(x_1,\ y_1,\ z_1,\ w_1)=\left(x_0+x_3,\ x_0-x_3,\ x_1+ix_2,\ x_1-ix_2 \right)&lt;/math&gt;.
Subsequently solved for &lt;math&gt;x_{0},x_{1},x_{2},x_{3}&lt;/math&gt; it becomes Lorentz transformation ({{equationNote|6b}}).}}

==={{anchor|Klein}} Klein (1871–97)===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Klein|label 1=History of Lorentz transformations in general § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Klein|label 1=History of Lorentz transformations via squeeze mappings § Klein}}

===={{anchor|Klein1}} Cayley absolute and non-Euclidean geometry====

Elaborating on Cayley's (1859) definition of an &quot;absolute&quot; ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a &quot;fundamental [[w:conic section]]&quot; in order to discuss motions such as rotation and translation in the non-Euclidean plane,&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt; and another fundamental form by using [[w:homogeneous coordinates]] ''x,y'' related to a circle with radius ''2c'' with measure of curvature &lt;math&gt;-\tfrac{1}{4c^{2}}&lt;/math&gt;. When ''c'' is positive, the measure of curvature is negative and the fundamental conic section is real, thus the geometry becomes hyperbolic ([[w:Beltrami–Klein model]]):&lt;ref group=M&gt;Klein (1871), p. 618&lt;/ref&gt;

:&lt;math&gt;\begin{align}x_{1}x_{2}-x_{3}^{2} &amp; =0\\
x^{2}+y^{2}-4c^{2} &amp; =0
\end{align}
\left|\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\
\hline \begin{align}x_{1} &amp; =\alpha_{1}y_{1}\\
x_{2} &amp; =\alpha_{2}y_{2}\\
x_{3} &amp; =\alpha_{3}y_{3}
\end{align}
\\
\left(\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)
\end{matrix}\right.&lt;/math&gt;

In (1873) he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to an ellipsoid or two-sheet [[w:hyperboloid]].&lt;ref group=M&gt;Klein (1873), pp. 127-128&lt;/ref&gt;

{{Lorentzbox|Text=Using positive ''c'' in &lt;math&gt;-\tfrac{1}{4c^{2}}&lt;/math&gt; in line with hyperbolic geometry or directly by setting &lt;math&gt;-\tfrac{1}{4c^{2}}=-x_{0}&lt;/math&gt;, Klein's two quadratic forms can be related to expressions &lt;math&gt;X_{2}^{2}-X_{1}X_{3}&lt;/math&gt; and &lt;math&gt;x_{0}^{2}-x_{1}^{2}-x_{2}^{2}&lt;/math&gt; for the Lorentz interval in ({{equationNote|6d}}).}}

===={{anchor|Klein2}} Möbius transformation, spin transformation, Cayley–Klein parameter====

In (1872) while devising the [[w:Erlangen program]], Klein discussed the general relation between projective metrics, [[w:binary form]]s and conformal geometry transforming a sphere into itself in terms of linear transformations of the [[w:complex variable]] ''x+iy''.&lt;ref group=M&gt;Klein (1872), 6&lt;/ref&gt; Following Klein, these relations were discussed by [[w:Ludwig Wedekind]] (1875) using &lt;math&gt;z'=\tfrac{\alpha z+\beta}{\gamma z+\delta}&lt;/math&gt;.&lt;ref group=M&gt;Wedekind (1875), 1&lt;/ref&gt; Klein (1875) then showed that all finite groups of motions follow by determining all finite groups of such linear transformations of ''x+iy'' into itself.&lt;ref group=M&gt;Klein (1875), §1–2&lt;/ref&gt; In (1878),&lt;ref group=M&gt;Klein (1878), 8.&lt;/ref&gt; Klein classified the substitutions of &lt;math&gt;\omega'=\tfrac{\alpha\omega+\beta}{\gamma\omega+\delta}&lt;/math&gt; with αδ-βγ=1 into hyperbolic, elliptic, parabolic, and in (1882)&lt;ref group=M&gt;Klein (1882), p. 173.&lt;/ref&gt; he added the loxodromic substitution as the combination of elliptic and hyperbolic ones. (In 1890, [[w:Robert Fricke]] in his edition of Klein's lectures of [[w:elliptic function]]s and [[w:Modular form]]s, referred to the analogy of this treatment to the theory of quadratic forms as given by Gauss and in particular Dirichlet.)&lt;ref group=M name=fri /&gt;

In (1884) Klein related the linear fractional transformations (interpreted as rotations around the ''x+iy''-sphere) to Cayley–Klein parameters [α,β,γ,δ], to Euler–Rodrigues parameters ''[a,b,c,d]'', and to the [[w:unit sphere]] by means of [[w:stereographic projection]], and also discussed transformations preserving surfaces of second degree equivalent to the transformation given by [[#Cayley2|Cayley (1854)]]:&lt;ref group=M&gt;Klein (1884), Part I, Ch. I, §1–2; Part II, Ch. II, 10&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left.\begin{matrix}z'=\frac{\alpha z+\beta}{\gamma z+\delta}\rightarrow z=z_{1}:z_{2}\rightarrow\begin{align}z_{1}^{\prime} &amp; =\alpha z_{1}+\beta z_{2}\\
z_{2}^{\prime} &amp; =\gamma z_{1}+\delta z_{2}
\end{align}
\\
\xi^{2}+\eta^{2}+\zeta^{2}=1\\
z=x+iy=\frac{\xi+i\eta}{1-\zeta}\\
z'=\frac{(d+ic)z-(b-ia)}{(b+ia)z+(d-ic)}\\
\left(a^{2}+b^{2}+c^{2}+d^{2}=1\right)
\end{matrix}\right| &amp; \begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
\lambda'=\frac{a\lambda+b}{c\lambda+d},\ \mu'=\frac{a'\mu+b'}{c'\mu+d'}\\
\lambda=\lambda_{1}:\lambda_{2},\ \mu=\mu_{1}:\mu_{2}\\
X_{1}:X_{2}:X_{3}:X_{4}=\lambda_{1}\mu_{1}:-\lambda_{2}\mu_{1}:\lambda_{1}\mu_{2}:\lambda_{2}\mu_{2}
\end{matrix}\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The formulas on the left related to the unit sphere are equivalent to Lorentz transformation ({{equationNote|6c}}). The formulas on the right can be related to those on the left by setting
:&lt;math&gt;(X_{1},\ X_{2},\ X_{3},\ X_{4})=\left(1+\zeta,\ -\xi+i\eta,\ \xi+i\eta,\ 1-\zeta\right)&lt;/math&gt;
and become equivalent to Lorentz transformation ({{equationNote|6a}}) by setting
:&lt;math&gt;\left[\xi,\ \eta,\ \zeta,\ 1\right]=\left[\tfrac{x_{1}}{x_{0}},\ \tfrac{x_{2}}{x_{0}},\ \tfrac{x_{3}}{x_{0}},\ \tfrac{x_{0}}{x_{0}}\right]&lt;/math&gt;
and subsequently solved for ''x''&lt;sub&gt;1&lt;/sub&gt;... it becomes Lorentz transformation ({{equationNote|6b}}).}}

In his lecture in the winter semester of 1889/90 (published 1892–93), he discussed the hyperbolic plane by using (as in 1871) the Lorentz interval in terms of a circle with radius ''2k'' as the basis of hyperbolic geometry, and another quadratic form to discuss the &quot;kinematics of hyperbolic geometry&quot; consisting of motions and congruent displacements of the hyperbolic plane into itself:&lt;ref group=M&gt;Klein (1893a), p. 109ff; pp. 138–140; pp. 249–250&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{matrix}x^{2}+y^{2}-4k^{2}t^{2}=0\\
x_{1}x_{3}-x_{2}^{2}=0
\end{matrix} &amp; \left|\begin{matrix}x_{1}x_{3}-x_{2}^{2}=0\\
\frac{x_{1}}{x_{2}}=\frac{x_{2}}{x_{3}}=\lambda=\frac{\lambda_{1}}{\lambda_{2}}\\
\lambda'=\frac{\alpha\lambda+\beta}{\gamma\lambda+\delta}\rightarrow\begin{align}\lambda_{1}^{\prime} &amp; =\alpha\lambda_{1}+\beta\lambda_{2}\\
\lambda_{2}^{\prime} &amp; =\gamma\lambda_{1}+\delta\lambda_{2}
\end{align}
\\
\left(\alpha\delta-\beta\gamma=1\right)\\
\begin{align}x_{1}:x_{2}:x_{3} &amp; =\lambda^{2}:\lambda:1=\lambda_{1}^{2}:\lambda_{1}\lambda_{2}:\lambda_{2}^{2}\\
 &amp; =\lambda^{\prime2}:\lambda':1=\lambda_{1}^{\prime2}:\lambda_{1}^{\prime}\lambda_{2}^{\prime}:\lambda_{2}^{\prime2};
\end{align}
\end{matrix}\right.\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Klein's Lorentz interval &lt;math&gt;x^{2}+y^{2}-4k^{2}t^{2}&lt;/math&gt; can be connected with the other interval &lt;math&gt;x_{1}x_{3}-x_{2}^{2}&lt;/math&gt; by setting
:&lt;math&gt;(x_{1},\ x_{2},\ x_{3})=\left(x+iy,\ 2kt,\ x-iy\right)&lt;/math&gt;,
by which the transformation system on the right becomes equivalent to Lorentz transformation ({{equationNote|6d}}) with ''2k=1'', and subsequently solved for ''x''&lt;sub&gt;1&lt;/sub&gt;... it becomes equivalent to Lorentz transformation ({{equationNote|6e}}).}}

In his lecture in the summer semester of 1890 (published 1892–93), he discussed general surfaces of second degree, including an &quot;oval&quot; surface corresponding to hyperbolic space and its motions:&lt;ref group=M&gt;Klein (1893b); general surface: pp. 61–66, 116–119, hyperbolic space: pp. 82, 86, 143–144&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}\text{General surfaces of second degree}:\\
\begin{align}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2} &amp; \text{(no real parts, elliptic)}\\
z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2} &amp; \text{(oval,hyperbolic)}\\
z_{1}^{2}+z_{2}^{2}-z_{3}^{2}-z_{4}^{2} &amp; \text{(ring)}\\
z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2} &amp; \text{(oval,hyperbolic)}\\
-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}-z_{4}^{2} &amp; \text{(no real parts,elliptic)}
\end{align}
\\
\text{all of which can be brought into the form:}\\
y_{1}y_{3}+y_{2}y_{4}=0\\
\text{Transformation:}\\
\begin{align}\varrho y_{1} &amp; =\lambda_{1}\mu_{1}, &amp; \varrho y_{1}^{\prime} &amp; =\lambda_{1}^{\prime}\mu_{1}^{\prime}\\
\varrho y_{2} &amp; =\lambda_{2}\mu_{1}, &amp; \varrho y_{2}^{\prime} &amp; =\lambda_{2}^{\prime}\mu_{1}^{\prime}\\
\varrho y_{3} &amp; =\lambda_{2}\mu_{2}, &amp; \varrho y_{3}^{\prime} &amp; =-\lambda_{2}^{\prime}\mu_{2}^{\prime}\\
\varrho y_{4} &amp; =\lambda_{1}\mu_{2}, &amp; \varrho y_{4}^{\prime} &amp; =\lambda_{1}^{\prime}\mu_{2}^{\prime}
\end{align}
\end{matrix}\right|\begin{matrix}\text{Oval (=hyperbolic motions in space):}\\
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0\\
=\left(x_{1}+ix_{3}\right)\left(x_{1}-ix_{3}\right)+\left(x_{2}+x_{4}\right)\left(x_{2}-x_{4}\right)=0\\
=y_{1}y_{3}+y_{2}y_{4}=0\\
\\
x^{2}+y^{2}+z^{2}-1=0\\
\hline \lambda=\frac{x+iy}{1-z},\ \lambda'=\frac{\alpha\lambda+\beta}{\gamma\lambda+\delta},\ \mu'=\frac{\bar{\alpha}\mu+\bar{\beta}}{\bar{\gamma}\mu+\bar{\delta}}\\
\begin{align}\lambda_{1}^{\prime} &amp; =\alpha\lambda_{1}+\beta\lambda_{2}\\
\lambda_{2}^{\prime} &amp; =\gamma\lambda_{1}+\delta\lambda_{2}
\end{align}
,\ \begin{align}\mu_{1}^{\prime} &amp; =\bar{\alpha}\mu_{1}+\bar{\beta}\mu_{2}\\
\mu_{2}^{\prime} &amp; =\bar{\gamma}\mu_{1}+\bar{\delta}\mu_{2}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The transformation of the unit sphere &lt;math&gt;x^{2}+y^{2}+z^{2}-1=0&lt;/math&gt; on the right is equivalent to Lorentz transformation ({{equationNote|6c}}). Plugging the values for λ,μ,λ′,μ′,... from the right into the transformations on the left, and relating them to Klein's homogeneous coordinates &lt;math&gt;x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=0&lt;/math&gt; by &lt;math&gt;(x,\ y,\ z,\ 1)=\left(\tfrac{x_{1}}{x_{4}},\ \tfrac{x_{2}}{x_{4}},\ \tfrac{x_{3}}{x_{4}},\ \tfrac{x_{4}}{x_{4}}\right)&lt;/math&gt; leads to Lorentz transformation ({{equationNote|6a}}). Subsequently solved for ''x''&lt;sub&gt;1&lt;/sub&gt;... it becomes Lorentz transformation ({{equationNote|6b}}).}}

In (1896/97), Klein again defined hyperbolic motions and explicitly used ''t'' as time coordinate, even though he added those cautionary remarks: ''&quot;We shall consider t also as capable of complex values, not for the sake of studying the behavior of a fictitious, imaginary time, but because it is only by taking this step that it becomes possible to bring about the intimate association of kinetics and the theory of functions of a complex variable. [..] the non-Euclidean geometry has no meta-physical significance here or in the subsequent discussion&quot;''. Using homogeneous coordinates, Klein defined the sphere x,y,z,t and then another &quot;movable&quot; sphere X,Y,Z,T as follows:&lt;ref group=M&gt;Klein (1896/97), pp. 13–14&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}+z^{2}-t^{2}=0\\
=(x+iy)(x-iy)+(z+t)(z-t)=0\\
x+iy:x-iy:z+t:t-z=\zeta_{1}\zeta_{2}^{\prime}:\zeta_{2}\zeta_{1}^{\prime}:\zeta_{1}\zeta_{1}^{\prime}:\zeta_{2}\zeta_{2}^{\prime}\\
\frac{\zeta_{1}}{\zeta_{2}}=\zeta\quad\Rightarrow\quad\zeta=\frac{x+iy}{t-z}=\frac{t+z}{x-iy};\\
\hline X^{2}+Y^{2}+Z^{2}-T^{2}=0\\
\text{introducing}\ Z,Z_{1},Z_{2}\dots\text{similarly as above}\ \zeta,\zeta_{1},\zeta_{2}\dots
\end{matrix}&lt;/math&gt;

which he related by the following transformation:

:&lt;math&gt;\begin{matrix}\zeta=\frac{\alpha Z+\beta}{\gamma Z+\delta}\rightarrow\begin{align}\zeta_{1} &amp; =\alpha Z_{1}+\beta Z_{2}\\
\zeta_{2} &amp; =\gamma Z_{1}+\delta Z_{2}
\end{align}
,\ \begin{align}\zeta_{1}^{\prime} &amp; =\bar{\alpha}Z_{1}^{\prime}+\bar{\beta}Z_{2}^{\prime}\\
\zeta_{2}^{\prime} &amp; =\bar{\gamma}Z_{1}^{\prime}+\bar{\delta}Z_{2}^{\prime}\text{ }
\end{align}
\\
(\alpha\delta-\beta\gamma=1)\\
\hline \begin{array}{c|c|c|c|c}
 &amp; X+iY &amp; X-iY &amp; T+Z &amp; T-Z\\
\hline x+iy &amp; \alpha\bar{\delta} &amp; \beta\bar{\gamma} &amp; \alpha\bar{\gamma} &amp; \beta\bar{\delta}\\
\hline x-iy &amp; \gamma\bar{\beta} &amp; \delta\bar{\alpha} &amp; \gamma\bar{\alpha} &amp; \delta\bar{\beta}\\
\hline t+z &amp; \alpha\bar{\beta} &amp; \beta\bar{\alpha} &amp; \alpha\bar{\alpha} &amp; \beta\bar{\beta}\\
\hline t-z &amp; \gamma\bar{\delta} &amp; \delta\bar{\gamma} &amp; \gamma\bar{\gamma} &amp; \delta\bar{\delta}
\end{array}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}). Klein's work was summarized and extended by [[#Bianchi2|Bianchi (1888-1893)]] and [[#Fricke|Fricke (1893-1897)]], obtaining equivalent Lorentz transformations.}}

==={{anchor|Selling}} Selling (1873–74) – Quadratic forms===

Continuing the work of [[../Lorentz transformation (general)#Gauss2|E:Gauss (1801)]] on definite ternary quadratic forms and [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] on indefinite ternary quadratic forms, [[w:Eduard Selling]] (1873) used the auxiliary coefficients ξ,η,ζ by which a definite form &lt;math&gt;\mathfrak{f}&lt;/math&gt; and an indefinite form ''f'' can be rewritten in terms of three squares:&lt;ref group=M&gt;Selling (1873), p. 174 and p. 179&lt;/ref&gt;&lt;ref&gt;Bachmann (1923), chapter 16&lt;/ref&gt;

:&lt;math&gt;{\scriptstyle \begin{align}\mathfrak{f} &amp; =\mathfrak{a}x^{2}+\mathfrak{b}y^{2}+\mathfrak{c}z^{2}+2\mathfrak{g}yz+2\mathfrak{h}zx+2\mathfrak{k}xy\\
 &amp; =\left(\xi x+\eta y+\zeta z\right)^{2}+\left(\xi_{1}x+\eta_{1}y+\zeta_{1}z\right)^{2}+\left(\xi_{2}x+\eta_{2}y+\zeta_{2}z\right)^{2}\\
\\
f &amp; =ax^{2}+by^{2}+cz^{2}+2gyz+2hzx+2kxy\\
 &amp; =\left(\xi x+\eta y+\zeta z\right)^{2}-\left(\xi_{1}x+\eta_{1}y+\zeta_{1}z\right)^{2}-\left(\xi_{2}x+\eta_{2}y+\zeta_{2}z\right)^{2}
\end{align}
\left|\begin{align}\xi^{2}+\xi_{1}^{2}+\xi_{2}^{2} &amp; =\mathfrak{a}\\
\eta^{2}+\eta_{1}^{2}+\eta_{2}^{2} &amp; =\mathfrak{b}\\
\zeta^{2}+\zeta_{1}^{2}+\zeta_{2}^{2} &amp; =\mathfrak{c}\\
\eta\zeta+\eta_{1}\zeta_{1}+\eta_{2}\zeta_{2} &amp; =\mathfrak{g}\\
\zeta\xi+\zeta_{1}\xi_{1}+\zeta_{2}\xi_{2} &amp; =\mathfrak{h}\\
\xi\eta+\xi_{1}\eta_{1}+\xi_{2}\eta_{2} &amp; =\mathfrak{k}
\end{align}
\right|\begin{align}\xi^{2}-\xi_{1}^{2}-\xi_{2}^{2} &amp; =a\\
\eta^{2}-\eta_{1}^{2}-\eta_{2}^{2} &amp; =b\\
\zeta^{2}-\zeta_{1}^{2}-\zeta_{2}^{2} &amp; =c\\
\eta\zeta-\eta_{1}\zeta_{1}-\eta_{2}\zeta_{2} &amp; =g\\
\zeta\xi-\zeta_{1}\xi_{1}-\zeta_{2}\xi_{2} &amp; =h\\
\xi\eta-\xi_{1}\eta_{1}-\xi_{2}\eta_{2} &amp; =k
\end{align}
}&lt;/math&gt;

In addition, Selling showed that auxiliary coefficients ξ,η,ζ can be geometrically interpreted as point coordinates which are in motion upon one sheet of a two-sheet hyperboloid, which is related to Selling's formalism for the reduction of indefinite forms by using definite forms.&lt;ref group=M&gt;Selling (1873), pp. 182–183&lt;/ref&gt;

Selling also reproduced the Lorentz transformation given by [[#Gauss3|Gauss (1800/63)]], to whom he gave full credit, and called it the only example of a particular indefinite ternary form known to him that has ever been discussed:&lt;ref group=M&gt;Selling (1873/74), p. 227 (see also p. 225 for citation).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(\begin{matrix}1, &amp; -1, &amp; -1\\
0, &amp; 0, &amp; 0
\end{matrix}\right)\\
\hline W=\begin{vmatrix}\frac{1}{2}\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{2}\left(\alpha^{2}+\beta^{2}-\gamma^{2}-\delta^{2}\right) &amp; \alpha\gamma+\beta\delta\\
\frac{1}{2}\left(\alpha^{2}-\beta^{2}+\gamma^{2}-\delta^{2}\right) &amp; \frac{1}{2}\left(\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; \alpha\gamma-\beta\delta\\
\alpha\beta+\gamma\delta &amp; \alpha\beta-\gamma\delta &amp; \alpha\delta+\beta\gamma
\end{vmatrix}\\
\left(\begin{vmatrix}\alpha &amp; \beta\\
\gamma &amp; \delta
\end{vmatrix}=1\right)
\end{matrix} &lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) or [[../Lorentz transformation (squeeze)#math_9b|E:('''9b''')]] as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

==={{anchor|Poincare2}} Poincaré (1881-86) – Möbius transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Poincare|label 1=History of Lorentz transformations in general § Poincaré}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (velocity)#Poincare3|label 1=History of Lorentz transformations via velocity § Poincaré}}

[[w:Henri Poincaré]] (1881a) demonstrated the connection of his formulas of the hyperboloid model [see [[../Lorentz transformation (general)#Poincare|E:Poincaré (1881)]]] to Möbius transformations:&lt;ref group=M name=p1&gt;Poincaré (1881a), pp. 133–134&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\xi^{2}+\eta^{2}-\zeta^{2}=-1\\
\left[X=\frac{\xi}{\zeta+1},\ Y=\frac{\eta}{\zeta+1}\right]\rightarrow t=X+iY\\
\hline \xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1\\
\left[X'=\frac{\xi'}{\zeta'+1},\ Y'=\frac{\eta'}{\zeta'+1}\right]\rightarrow t'=X'+iY'\\
\hline t'=\frac{ht+k}{h't+k'}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6g}}).}}

Poincaré (1881b) also used the Möbius transformation &lt;math&gt;\tfrac{az+b}{cz+d}&lt;/math&gt; in relation to [[w:Fuchsian function]]s and the discontinuous [[w:Fuchsian group]], being a special case of the hyperbolic group leaving invariant the &quot;fundamental circle&quot; ([[w:Poincaré disk model]] and [[w:Poincaré half-plane model]] of hyperbolic geometry).&lt;ref group=M&gt;Poincaré (1881b), p. 333&lt;/ref&gt; He then extended [[#Klein2|Klein's (1878-1882)]] study on the relation between Möbius transformations and hyperbolic, elliptic, parabolic, and loxodromic substitutions, and while formulating [[w:Kleinian group]]s (1883) he used the following transformation leaving invariant the [[w:generalized circle]]:&lt;ref group=M&gt;Poincaré (1883), pp. 49–50; 53–54&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(z,\ \frac{\alpha z+\beta}{\gamma z+\delta}\right),\ \left(z_{0},\ \frac{\alpha_{0}z_{0}+\beta_{0}}{\gamma_{0}z_{0}+\delta_{0}}\right)\\
\hline z=\xi+i\eta,\ z_{0}=\xi-i\eta,\ \rho^{2}=\xi^{2}+\eta^{2}+\zeta^{2}\\
A\rho^{\prime2}+Bz^{\prime}+B_{0}z_{0}^{\prime}+C=0\\
\hline \begin{align}\rho^{\prime2} &amp; =\frac{\rho^{2}\alpha\alpha_{0}+z\alpha\beta_{0}+z_{0}\beta\alpha_{0}+\beta\beta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}\\
z^{\prime} &amp; =\frac{\rho^{2}\alpha\gamma_{0}+z\alpha\delta_{0}+z_{0}\beta\gamma_{0}+\beta\delta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}\\
z_{0}^{\prime} &amp; =\frac{\rho^{2}\gamma\alpha_{0}+z\gamma\beta_{0}+z_{0}\delta\alpha_{0}+\delta\beta_{0}}{\rho^{2}\gamma\gamma_{0}+z\gamma\delta_{0}+z_{0}\delta\gamma_{0}+\delta\delta_{0}}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Setting &lt;math&gt;[\rho^{2},\ z,\ z_{0}]=\left[\tfrac{X_{1}}{X_{4}},\ \tfrac{X_{2}}{X_{4}},\ \tfrac{X_{3}}{X_{4}}\right]&lt;/math&gt; this becomes transformation '''u′''' in ({{equationNote|6a}}) and becomes the complete Lorentz transformation by setting &lt;math&gt;{\scriptstyle \left[\begin{matrix}X_{1} &amp; X_{2}\\
X_{3} &amp; X_{4}
\end{matrix}\right]=\left[\begin{matrix}x_{0}+x_{3} &amp; x_{1}-ix_{2}\\
x_{1}+ix_{2} &amp; x_{0}-x_{3}
\end{matrix}\right]}&lt;/math&gt;.}}

In 1886, Poincaré investigated the relation between indefinite ternary quadratic forms and Fuchsian functions and groups:&lt;ref group=M&gt;Poincaré (1886), p. 735ff.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\left(z,\ \frac{\alpha z+\beta}{\gamma z+\delta}\right)\\
\hline Y^{\prime2}-X'Z'=Y^{2}-XZ\\
\hline \begin{align}X' &amp; =\alpha^{2}X+2\alpha\gamma Y+\gamma^{2}Z\\
Y' &amp; =\alpha\beta X+(\alpha\delta+\beta\gamma)Y+\gamma\delta Z\\
Z' &amp; =\beta^{2}X+2\beta\gamma Y+\delta^{2}Z
\end{align}
\\
\left[{\scriptstyle \begin{align}X= &amp; ax+by+cz, &amp; Y &amp; =a'x+b'y+c'z, &amp; Z &amp; =a''x+b''y+c''z,\\
X'= &amp; ax'+by'+cz', &amp; Y' &amp; =a'x'+b'y'+c'z', &amp; Z' &amp; =a''x'+b''y'+c''z',
\end{align}
}\right]
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text= This is equivalent to transformation '''u′''' in ({{equationNote|6d}}) and becomes the complete Lorentz transformation by suitibly choosing the coefficients ''a,b,c,...'' so that ''[X,Y,Z]=[x+z, y, -x+z]''.}}

==={{anchor|Bianchi2}} Bianchi (1888-93) – Möbius and spin transformations===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Bianchi1|label 1=History of Lorentz transformations via trigonometric functions § Bianchi}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Bianchi1|label 1=History of Lorentz transformations via squeeze mappings § Bianchi}}

Related to [[#Klein1|Klein's (1871)]] and [[#Poincare2|Poincaré's (1881-1887)]] work on non-Euclidean geometry and indefinite quadratic forms, [[w:Luigi Bianchi]] (1888) analyzed the differential Lorentz interval in term of conic sections and hyperboloids, alluded to the linear fractional transformation of &lt;math&gt;\omega&lt;/math&gt; and its conjugate &lt;math&gt;\omega_{1}&lt;/math&gt; with parameters α,β,γ,δ in order to preserve the Lorentz interval, and gave credit to [[#Gauss3|Gauss (1800/63)]] who obtained the same coefficient system:&lt;ref group=M&gt;Bianchi (1888), pp. 547; 562–563 (especially footnote on p. 563); 571–572&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}ds^{2}=dx^{2}+dy^{2}-dz^{2};\ x^{2}+y^{2}-z^{2}=0;\\
\hline X_{3}^{2}+Y_{3}^{2}-Z_{3}^{2}=-1\\
X_{3}=i\frac{1-\omega\omega_{1}}{\omega-\omega_{1}},\ Y_{3}=i\frac{\omega-\omega_{1}}{\omega-\omega_{1}},\ Z_{3}=i\frac{1+\omega\omega_{1}}{\omega-\omega_{1}},\\
\omega=\frac{\alpha\omega'+\beta}{\gamma\omega'+\delta}\quad(\alpha\delta-\beta\gamma=1)\\
\hline \left(\begin{matrix}\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}, &amp; \gamma\delta-\alpha\beta, &amp; \frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}\\
\beta\delta-\alpha\gamma, &amp; \alpha\delta+\beta\gamma, &amp; \beta\delta+\alpha\gamma\\
\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}, &amp; \alpha\beta+\gamma\delta, &amp; \frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}
\end{matrix}\right)\\
\hline \begin{align}x' &amp; =\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\gamma\delta-\alpha\beta)y+\frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}z+c_{1}\\
y' &amp; =(\beta\delta-\alpha\gamma)x+(\alpha\delta+\beta\gamma)y+(\beta\delta+\alpha\gamma)z+c_{2}\\
z' &amp; =\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\alpha\beta+\gamma\delta)y+\frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}z+c_{3}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The is equivalent to Lorentz transformations ({{equationNote|6d}}) and ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) or [[../Lorentz transformation (squeeze)#math_9b|E:('''9b''')]] as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

In 1893, Bianchi gave the coefficients in the case of four dimensions:&lt;ref group=M name=bi&gt;Bianchi (1893), § 3&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}z &amp; =\frac{\alpha z'+\beta}{\gamma z'+\delta}\\
 &amp; (\alpha\delta-\beta\gamma=1)
\end{align}
\rightarrow\begin{align}z &amp; =\frac{\xi}{\eta}\\
z' &amp; =\frac{\xi'}{\eta'}
\end{align}
\rightarrow\begin{align}\xi &amp; =\alpha\xi'+\beta\eta'\\
\eta &amp; =\gamma\xi'+\delta\eta'\\
\\
\xi_{0} &amp; =\alpha_{0}\xi'_{0}+\beta_{0}\eta'_{0}\\
\eta_{0} &amp; =\gamma_{0}\xi'_{0}+\delta_{0}\eta'_{0}
\end{align}
\\
\hline {\scriptstyle F=\left(u_{1}+u{}_{4}\right)\xi\xi_{0}+\left(u_{2}+iu{}_{3}\right)\xi\eta_{0}+\left(u_{2}-iu{}_{3}\right)\xi_{0}\eta+\left(u_{4}-u{}_{1}\right)\eta\eta_{0}}\\
{\scriptstyle F'=\left(u'_{1}+u'{}_{4}\right)\xi'\xi'_{0}+\left(u'_{2}+iu'{}_{3}\right)\xi'\eta'_{0}+\left(u'_{2}-iu'{}_{3}\right)\xi'_{0}\eta'+\left(u'_{4}-u'{}_{1}\right)\eta'\eta'_{0}}\\
{\scriptstyle \left(u_{2}+iu{}_{3}\right)\left(u_{2}-iu{}_{3}\right)+\left(u_{1}-u{}_{4}\right)\left(u_{1}+u{}_{4}\right)=\left(u'_{2}+iu'{}_{3}\right)\left(u'_{2}-iu'{}_{3}\right)+\left(u'_{1}-u'{}_{4}\right)\left(u'_{1}+u'{}_{4}\right)}\\
\hline {\scriptstyle \begin{align}u'_{1}+u'_{4} &amp; =\alpha\alpha_{0}\left(u_{1}+u{}_{4}\right)+\alpha\gamma_{0}\left(u_{2}+iu{}_{3}\right)+\alpha_{0}\gamma\left(u_{2}-iu{}_{3}\right)+\gamma\gamma_{0}\left(u_{4}-u{}_{1}\right)\\
u'_{2}+iu'_{3} &amp; =\alpha\beta_{0}\left(u_{1}+u{}_{4}\right)+\alpha\delta_{0}\left(u_{2}+iu{}_{3}\right)+\beta_{0}\gamma\left(u_{2}-iu{}_{3}\right)+\gamma\delta_{0}\left(u_{4}-u{}_{1}\right)\\
u'_{2}-iu'_{3} &amp; =\alpha_{0}\beta\left(u_{1}+u{}_{4}\right)+\alpha_{0}\delta\left(u_{2}-iu{}_{3}\right)+\beta\gamma_{0}\left(u_{2}+iu{}_{3}\right)+\gamma_{0}\delta\left(u_{4}-u{}_{1}\right)\\
u'_{4}-u'_{1} &amp; =\beta\beta_{0}\left(u_{1}+u{}_{4}\right)+\beta\delta_{0}\left(u_{2}+iu{}_{3}\right)+\beta_{0}\delta\left(u_{2}-iu{}_{3}\right)+\delta\delta_{0}\left(u_{4}-u{}_{1}\right)
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}).}}

Solving for &lt;math&gt;u'_{1}\dots&lt;/math&gt; Bianchi obtained:&lt;ref group=M name=bi /&gt;

:&lt;math&gt;\begin{matrix}u_{1}^{2}+u_{2}^{2}+u_{3}^{2}-u_{4}^{2}=u_{1}^{\prime2}+u_{2}^{\prime2}+u_{3}^{\prime2}-u_{4}^{\prime2}\\
\hline {\scriptstyle \begin{align}u'_{1} &amp; =\frac{1}{2}\left(\alpha\alpha_{0}-\beta\beta_{0}-\gamma\gamma_{0}+\delta\delta_{0}\right)u_{1}+\frac{1}{2}\left(\alpha\gamma_{0}+\alpha_{0}\gamma-\beta\delta_{0}-\beta_{0}\delta\right)u_{2}+\\
 &amp; +\frac{i}{2}\left(\alpha\gamma_{0}-\alpha_{0}\gamma+\beta_{0}\delta-\beta\delta_{0}\right)u_{3}+\frac{1}{2}\left(\alpha\alpha_{0}-\beta\beta_{0}+\gamma\gamma_{0}-\delta\delta_{0}\right)u_{4}\\
u'_{2} &amp; =\frac{1}{2}\left(\alpha\beta_{0}+\alpha_{0}\beta-\gamma\delta_{0}-\gamma_{0}\delta\right)u_{1}+\frac{1}{2}\left(\alpha\delta_{0}+\alpha_{0}\delta+\beta\gamma_{0}+\beta_{0}\gamma\right)u_{2}+\\
 &amp; +\frac{i}{2}\left(\alpha\delta_{0}-\alpha_{0}\delta+\beta\gamma_{0}-\beta_{0}\gamma\right)u_{3}+\frac{1}{2}\left(\alpha\beta_{0}+\alpha_{0}\beta+\gamma\delta_{0}+\gamma_{0}\delta\right)u_{4}\\
u'_{3} &amp; =\frac{i}{2}\left(\alpha_{0}\beta-\alpha\beta_{0}+\gamma\delta_{0}-\gamma_{0}\delta\right)u_{1}+\frac{i}{2}\left(\alpha_{0}\delta-\alpha\delta_{0}+\beta\gamma_{0}-\beta_{0}\gamma\right)u_{2}+\\
 &amp; +\frac{1}{2}\left(\alpha\delta_{0}+\alpha_{0}\delta-\beta\gamma_{0}-\beta_{0}\gamma\right)u_{3}+\frac{i}{2}\left(\alpha_{0}\beta-\alpha\beta_{0}+\gamma_{0}\delta-\gamma\delta_{0}\right)u_{4}\\
u'_{4} &amp; =\frac{1}{2}\left(\alpha\alpha_{0}+\beta\beta_{0}-\gamma\gamma_{0}-\delta\delta_{0}\right)u_{1}+\frac{1}{2}\left(\alpha\gamma_{0}+\alpha_{0}\gamma+\beta\delta_{0}+\beta_{0}\delta\right)u_{2}+\\
 &amp; +\frac{i}{2}\left(\alpha\gamma_{0}-\alpha_{0}\gamma+\beta\delta_{0}-\beta_{0}\delta\right)u_{3}+\frac{1}{2}\left(\alpha\alpha_{0}+\beta\beta_{0}+\gamma\gamma_{0}+\delta\delta_{0}\right)u_{4}
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6b}}).}}

==={{anchor|Fricke}} Fricke (1891–97) – Möbius and spin transformations===

[[w:Robert Fricke]] (1891) – following the work of his teacher [[#Klein2|Klein (1878–1882)]] as well as [[#Poincare2|Poincaré (1881–1887)]] on automorphic functions and group theory – obtained the following transformation for an integer ternary quadratic form&lt;ref group=M&gt;Fricke (1891), §§ 1, 6&lt;/ref&gt;&lt;ref&gt;Dickson (1923), pp. 221, 232&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\omega'=\frac{\delta\omega+\beta}{\gamma\omega+\alpha}\ (\alpha\delta-\beta\gamma=1),\ \omega=\frac{\eta}{\xi},\\
\hline \begin{align}\xi' &amp; =\xi\alpha^{2}+2\eta\alpha\gamma+\zeta\gamma^{2}\\
\eta' &amp; =\xi\alpha\beta+\eta(\alpha\delta+\beta\gamma)+\zeta\gamma\delta\\
\zeta' &amp; =\xi\beta^{2}+2\eta\beta\delta+\zeta\delta^{2}
\end{align}
\\
\hline \xi'\zeta'-\eta'^{2}=(\alpha\delta-\beta\gamma)^{2}\left(\xi\zeta-\eta^{2}\right)\\
\xi=x\sqrt{q}-y,\ \eta=z,\ \zeta=x\sqrt{q}+y\\
\hline qx^{\prime2}-y^{\prime2}-z^{\prime2}=qx^{2}-y^{2}-z^{2}\\
\hline \left(\begin{matrix}\frac{1}{2}\left(+\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{2\sqrt{q}}\left(-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{\sqrt{q}}(\alpha\gamma+\beta\delta)\\
\frac{1}{2}\sqrt{q}\left(-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; \frac{1}{2}\left(+\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}\right) &amp; (-\alpha\gamma+\beta\delta)\\
\sqrt{q}(\alpha\beta+\gamma\delta) &amp; (-\alpha\beta+\gamma\delta) &amp; (\alpha\delta+\beta\gamma)
\end{matrix}\right)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By setting ''q=1'', the first part is equivalent to Lorentz transformation ({{equationNote|6d}}) and the second part is equivalent to ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) or [[../Lorentz transformation (squeeze)#math_9b|E:('''9b''')]] as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

And the general case of four dimensions in 1893:&lt;ref group=M&gt;Fricke (1893), pp. 706, 710–711&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}y'_{2}y'_{3}-y'_{1}y'_{4}=y_{2}y_{3}-y_{1}y_{4}\\
\hline \begin{align}y_{1}^{\prime} &amp; =\alpha\bar{\alpha}y_{1}+\alpha\bar{\beta}y_{2}+\beta\bar{\alpha}y_{3}+\beta\bar{\beta}y_{4}\\
y_{2}^{\prime} &amp; =\alpha\bar{\gamma}y_{1}+\alpha\bar{\delta}y_{2}+\beta\bar{\gamma}y_{3}+\beta\bar{\delta}y_{4}\\
y_{3}^{\prime} &amp; =\gamma\bar{\alpha}y_{1}+\gamma\bar{\beta}y_{2}+\delta\bar{\alpha}y_{3}+\delta\bar{\beta}y_{4}\\
y_{4}^{\prime} &amp; =\gamma\bar{\gamma}y_{1}+\gamma\bar{\delta}y_{2}+\delta\bar{\gamma}y_{3}+\delta\bar{\delta}y_{4}
\end{align}
\\
\hline \begin{align}y_{1} &amp; =z_{4}\sqrt{s}+z_{3}\sqrt{r}, &amp; y_{2} &amp; =z_{1}\sqrt{p}+iz_{2}\sqrt{q}\\
y_{3} &amp; =z_{1}\sqrt{p}-iz_{2}\sqrt{q}, &amp; y_{4} &amp; =z_{4}\sqrt{s}-z_{3}\sqrt{r}
\end{align}
\\
\hline pz_{1}^{\prime2}+qz{}_{2}^{\prime2}+rz{}_{3}^{\prime2}-sz{}_{4}^{\prime2}=pz_{1}^{2}+qz_{2}^{2}+rz_{3}^{2}-sz_{4}^{2}\\
\hline z'_{i}=\alpha_{i1}z_{1}+\alpha_{i2}z_{2}+\alpha_{i3}z_{3}+\alpha_{i4}z_{4}\\
{\scriptstyle \begin{align}2\alpha_{11}\ \text{or}\ 2\alpha_{22} &amp; =\alpha\bar{\delta}+\delta\bar{\alpha}\pm\beta\bar{\gamma}\pm\gamma\bar{\beta}, &amp; 2\alpha_{33}\ \text{or}\ 2\alpha_{44} &amp; =\alpha\bar{\alpha}+\delta\bar{\delta}\pm\beta\bar{\beta}\pm\gamma\bar{\gamma}\\
\frac{2\alpha_{12}\sqrt{p}}{i\sqrt{p}}\ \text{or}\ \frac{2\alpha_{21}i\sqrt{p}}{\sqrt{p}} &amp; =\alpha\bar{\delta}-\bar{\delta}\alpha\mp\beta\bar{\gamma}\pm\gamma\bar{\beta}, &amp; \frac{2\alpha_{34}\sqrt{r}}{\sqrt{s}}\ \text{or}\ \frac{2\alpha_{43}\sqrt{s}}{\sqrt{r}} &amp; =\alpha\bar{\alpha}-\delta\bar{\delta}\pm\beta\bar{\beta}\pm\gamma\bar{\gamma}\\
\frac{2\alpha_{13}\sqrt{p}}{\sqrt{r}}\ \text{or}\ \frac{2\alpha_{24}i\sqrt{p}}{\sqrt{s}} &amp; =\alpha\bar{\gamma}-\delta\bar{\beta}\pm\gamma\bar{\alpha}\pm\beta\bar{\delta}, &amp; \frac{2\alpha_{14}\sqrt{p}}{\sqrt{s}}\ \text{or}\ \frac{2\alpha_{23}i\sqrt{q}}{\sqrt{r}} &amp; =\alpha\bar{\gamma}+\delta\bar{\beta}\pm\gamma\bar{\alpha}\pm\beta\bar{\delta}\\
\frac{2\alpha_{31}\sqrt{r}}{\sqrt{p}}\ \text{or}\ \frac{2\alpha_{43}\sqrt{s}}{i\sqrt{q}} &amp; =\alpha\bar{\beta}-\delta\bar{\gamma}\pm\beta\bar{\alpha}\mp\gamma\bar{\delta}, &amp; \frac{2\alpha_{41}\sqrt{s}}{\sqrt{p}}\ \text{or}\ \frac{2\alpha_{32}\sqrt{r}}{i\sqrt{q}} &amp; =\alpha\bar{\beta}+\delta\bar{\gamma}\pm\beta\bar{\alpha}\pm\gamma\bar{\delta}
\end{align}
}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=By setting ''p=q=r=s=1'', the first part is equivalent to Lorentz transformation ({{equationNote|6a}}) and the second part to ({{equationNote|6b}}).}}

Supported by Felix Klein, Fricke summarized his and Klein's work in a treatise concerning [[w:automorphic function]]s (1897). Using a sphere as the absolute, in which the interior of the sphere is denoted as hyperbolic space, they defined hyperbolic motions, and stressed that any hyperbolic motion corresponds to &quot;circle relations&quot; (now called Möbius transformations):&lt;ref group=M name=fri&gt;Fricke &amp; Klein (1897), §12–13&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
=(z_{4}+z_{3})(z_{4}-z_{3})-(z_{1}+iz_{2})(z_{1}-iz_{2})=0\\
=y_{1}y_{4}-y_{2}y_{3}=0\\
\left(y_{1}=z_{4}+z_{3},\ y_{2}=z_{1}+iz_{2},\ y_{3}=z_{1}-iz_{2},\ y_{4}=z_{4}-z_{3}\right)\\
\zeta=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}},\ \bar{\zeta}=\frac{z_{1}-iz_{2}}{z_{4}-z_{3}}\\
\zeta'=\frac{\alpha\zeta+\beta}{\gamma\zeta+\delta},\ \bar{\zeta}'=\frac{\overline{\alpha\zeta}+\bar{\beta}}{\overline{\gamma\zeta}+\bar{\delta}}\quad(\alpha\delta-\beta\gamma\ne0)\\
z_{1}:z_{2}:z_{3}z_{4}=(\zeta+\bar{\zeta}):-i(\zeta-\bar{\zeta}):(\zeta\bar{\zeta}-1):(\zeta\bar{\zeta}+1)\\
y_{1}:y_{2}:y_{3}y_{4}=\zeta\bar{\zeta}:\zeta:\bar{\zeta}:1=\zeta_{1}\bar{\zeta}_{1}:\zeta_{1}\bar{\zeta}_{2}:\zeta_{2}\bar{\zeta}_{1}:\zeta_{2}\bar{\zeta}_{2}\\
\left(\zeta=\zeta_{1}:\zeta_{2},\ \bar{\zeta}=\bar{\zeta}_{1}:\bar{\zeta}_{2}\right)\\
\hline \begin{align}y_{1}^{\prime} &amp; =\alpha\bar{\alpha}y_{1}+\alpha\bar{\beta}y_{2}+\beta\bar{\alpha}y_{3}+\beta\bar{\beta}y_{4}\\
y_{2}^{\prime} &amp; =\alpha\bar{\gamma}y_{1}+\alpha\bar{\delta}y_{2}+\beta\bar{\gamma}y_{3}+\beta\bar{\delta}y_{4}\\
y_{3}^{\prime} &amp; =\gamma\bar{\alpha}y_{1}+\gamma\bar{\beta}y_{2}+\delta\bar{\alpha}y_{3}+\delta\bar{\beta}y_{4}\\
y_{4}^{\prime} &amp; =\gamma\bar{\gamma}y_{1}+\gamma\bar{\delta}y_{2}+\delta\bar{\gamma}y_{3}+\delta\bar{\delta}y_{4}
\end{align}
\end{matrix}
&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}).}}

==={{anchor|Woods}} Woods (1895) – Spin transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Woods2|label 1=History of Lorentz transformations in general § Woods}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Woods2|label 1=History of Lorentz transformations via hyperbolic functions § Woods}}

In a thesis supervised by Felix Klein, [[w:Frederick S. Woods]] (1895) further developed [[#Bianchi2|Bianchi's (1888)]] treatment of surfaces satisfying the Lorentz interval (pseudominimal surface), and used the transformation of [[#Gauss3|Gauss (1800/63)]] and [[#Bianchi2|Bianchi (1888)]] while discussing automorphisms of that surface:&lt;ref group=M&gt;Woods (1895), pp. 2–3; 10–11; 34–35&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{2}+y^{2}-z^{2}=0;\quad x^{2}+y^{2}-z^{2}=-1\\
\hline \left(x,y,z\right)\Rightarrow\omega\\
\begin{align}\omega_{1}^{\prime} &amp; =\alpha\omega_{1}+\beta\omega_{2}\\
\omega_{2}^{\prime} &amp; =\gamma\omega_{1}+\delta\omega_{2}
\end{align}
\quad(\alpha\delta-\beta\gamma=1)\\
\hline \begin{align}x' &amp; =(-1)^{k}\left[\frac{\alpha^{2}-\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\gamma\delta-\alpha\beta)y+\frac{-\alpha^{2}-\beta^{2}+\gamma^{2}+\delta^{2}}{2}z\right]+c_{1}\\
y' &amp; =(-1)^{k}\left[(\beta\delta-\alpha\gamma)x+(\alpha\delta+\beta\gamma)y+(\beta\delta+\alpha\gamma)z\right]+c_{2}\\
z' &amp; =(-1)^{k}\left[\frac{-\alpha^{2}+\beta^{2}-\gamma^{2}+\delta^{2}}{2}x+(\alpha\beta+\gamma\delta)y+\frac{\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}}{2}z\right]+c_{3}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The expressions within the brackets are equivalent to Lorentz transformations ({{equationNote|6e}}), containing Lorentz boost ({{equationNote|6f}}) as a special case with &lt;math&gt;\beta=\gamma=0&lt;/math&gt; and &lt;math&gt;\delta=1/\alpha&lt;/math&gt;.}}

=== {{anchor|Herglotz1}} Herglotz (1909/10) – Special relativity ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Herglotz|label 1=History of Lorentz transformations via squeeze mappings § Herglotz}}

Already in the context of special relativity, [[w:Gustav Herglotz]] (1909/10) followed [[#Klein2|Klein (1889–1897)]] as well as [[#Fricke|Fricke &amp; Klein (1897)]] concerning the Cayley absolute, hyperbolic motion and its transformation, and classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic. He provided the general case (on the left) and the hyperbolic substitution (on the right) as follows:&lt;ref group=R&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\left.\begin{matrix}z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{4}^{2}=0\\
z_{1}=x,\ z_{2}=y,\ z_{3}=z,\ z_{4}=t\\
Z=\frac{z_{1}+iz_{2}}{z_{4}-z_{3}}=\frac{x+iy}{t-z},\ Z'=\frac{x'+iy'}{t'-z'}\\
Z=\frac{\alpha Z'+\beta}{\gamma Z'+\delta}
\end{matrix}\right|\begin{matrix}Z=Z'e^{\vartheta}\\
\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|6a}}).}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bia88diff}} 
*{{#section:History of Topics in Special Relativity/mathsource|bia93quat}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay54hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|cay79hom}} 
*{{#section:History of Topics in Special Relativity/mathsource|fri91}} 
*{{#section:History of Topics in Special Relativity/mathsource|fri93}} 
*{{#section:History of Topics in Special Relativity/mathsource|fri97}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau98}} 
*{{#section:History of Topics in Special Relativity/mathsource|gau00}} 
*{{#section:History of Topics in Special Relativity/mathsource|helm66}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei72a}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei72b}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei73}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei75}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei79}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei82}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei84}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei90a}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei90b}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei93a}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei93b}} 
*{{#section:History of Topics in Special Relativity/mathsource|klei96}} 
*{{#section:History of Topics in Special Relativity/mathsource|lag73}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81a}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin81b}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin83}} 
*{{#section:History of Topics in Special Relativity/mathsource|poin86}} 
*{{#section:History of Topics in Special Relativity/mathsource|sel73}} 
*{{#section:History of Topics in Special Relativity/mathsource|wed75}} 
*{{#section:History of Topics in Special Relativity/mathsource|woo95}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|herg10}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L7}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>jwlknkalqlrx8ao5p1k9bjxar14r938</sha1>
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    <title>History of Topics in Special Relativity/Lorentz transformation (trigonometric)</title>
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    <revision>
      <id>2692674</id>
      <parentid>2541688</parentid>
      <timestamp>2024-12-19T20:30:30Z</timestamp>
      <contributor>
        <username>D.H</username>
        <id>52339</id>
      </contributor>
      <comment>/* Lorentz transformation via trigonometric functions */</comment>
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      <text bytes="13486" sha1="5v406kh9ir407z1m0nbx8cvz6b865xs" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via trigonometric functions==

The following general relation connects the speed of light and the relative velocity to hyperbolic and trigonometric functions, where &lt;math&gt;\eta&lt;/math&gt; is the rapidity in [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']], &lt;math&gt;\theta&lt;/math&gt; is equivalent to the [[w:Gudermannian function]] &lt;math&gt;{\rm gd}(\eta)=2\arctan(e^{\eta})-\pi/2&lt;/math&gt;, and &lt;math&gt;\vartheta&lt;/math&gt; is equivalent to the Lobachevskian [[w:angle of parallelism]] &lt;math&gt;\Pi(\eta)=2\arctan(e^{-\eta})&lt;/math&gt;:

:&lt;math&gt;\frac{v}{c}=\tanh\eta=\sin\theta=\cos\vartheta&lt;/math&gt;

This relation was first defined by [[#Varicak|Varićak (1910)]].

a) Using &lt;math&gt;\sin\theta=\tfrac{v}{c}&lt;/math&gt; one obtains the relations &lt;math&gt;\sec\theta=\gamma&lt;/math&gt; and &lt;math&gt;\tan\theta=\beta\gamma&lt;/math&gt;, and the Lorentz boost takes the form:&lt;ref name=maj&gt;Majerník (1986), 536–538&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \left.\begin{align}x_{0}^{\prime} &amp; =x_{0}\sec\theta-x_{1}\tan\theta &amp;  &amp; =\frac{x_{0}-x_{1}\sin\theta}{\cos\theta}\\
x_{1}^{\prime} &amp; =-x_{0}\tan\theta+x_{1}\sec\theta &amp;  &amp; =\frac{-x_{0}\sin\theta+x_{1}}{\cos\theta}\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\sec\theta+x_{1}^{\prime}\tan\theta &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\sin\theta}{\cos\theta}\\
x_{1} &amp; =x_{0}^{\prime}\tan\theta+x_{1}^{\prime}\sec\theta &amp;  &amp; =\frac{x_{0}^{\prime}\sin\theta+x_{1}^{\prime}}{\cos\theta}\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\tan^{2}\theta-\sec^{2}\theta &amp; =-1\\
\frac{\tan\theta}{\sec\theta} &amp; =\sin\theta\\
\frac{1}{\sqrt{1-\sin^{2}\theta}} &amp; =\sec\theta\\
\frac{\sin\theta}{\sqrt{1-\sin^{2}\theta}} &amp; =\tan\theta
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|8a}}}}

This Lorentz transformation was derived by [[#Bianchi1|Bianchi (1886)]] and [[#Darboux1|Darboux (1891/94)]] while transforming pseudospherical surfaces, and by [[#Scheffers|Scheffers (1899)]] as a special case of [[w:contact transformation]] in the plane (Laguerre geometry). In special relativity, it was first used by [[#Plummer|Plummer (1910)]], by [[#Gruner|Gruner (1921)]] while developing [[w:Loedel diagram]]s, and by [[w:Vladimir Karapetoff]] in the 1920s.

b) Using &lt;math&gt;\cos\vartheta=\tfrac{v}{c}&lt;/math&gt; one obtains the relations &lt;math&gt;\csc\vartheta=\gamma&lt;/math&gt; and &lt;math&gt;\cot\vartheta=\beta\gamma&lt;/math&gt;, and the Lorentz boost takes the form:&lt;ref name=maj /&gt;

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}+x_{2}^{\prime2}\\
\hline \left.\begin{align}x_{0}^{\prime} &amp; =x_{0}\csc\vartheta-x_{1}\cot\vartheta &amp;  &amp; =\frac{x_{0}-x_{1}\cos\vartheta}{\sin\vartheta}\\
x_{1}^{\prime} &amp; =-x_{0}\cot\vartheta+x_{1}\csc\vartheta &amp;  &amp; =\frac{-x_{0}\cos\vartheta+x_{1}}{\sin\vartheta}\\
x_{2}^{\prime} &amp; =x_{2}\\
\\
x_{0} &amp; =x_{0}^{\prime}\csc\vartheta+x_{1}^{\prime}\cot\vartheta &amp;  &amp; =\frac{x_{0}^{\prime}+x_{1}^{\prime}\cos\vartheta}{\sin\vartheta}\\
x_{1} &amp; =x_{0}^{\prime}\cot\vartheta+x_{1}^{\prime}\csc\vartheta &amp;  &amp; =\frac{x_{0}^{\prime}\cos\vartheta+x_{1}^{\prime}}{\sin\vartheta}\\
x_{2} &amp; =x_{2}^{\prime}
\end{align}
\right|{\scriptstyle \begin{align}\cot^{2}\vartheta-\csc^{2}\vartheta &amp; =-1\\
\frac{\cot\vartheta}{\csc\vartheta} &amp; =\cos\vartheta\\
\frac{1}{\sqrt{1-\cos^{2}\vartheta}} &amp; =\csc\vartheta\\
\frac{\cos\vartheta}{\sqrt{1-\cos^{2}\vartheta}} &amp; =\cot\vartheta
\end{align}
}
\end{matrix}&lt;/math&gt;|{{equationRef|8b}}}}

This Lorentz transformation was derived by [[#Eisenhart|Eisenhart (1905)]] while transforming pseudospherical surfaces. In special relativity it was first used by [[#Gruner|Gruner (1921)]] while developing [[w:Loedel diagram]]s.

==Historical notation==

==={{anchor|Bianchi1}} Bianchi (1886) – Pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Bianchi2|label 1=History of Lorentz transformations via Möbius transformations § Bianchi}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Bianchi1|label 1=History of Lorentz transformations via squeeze mappings § Bianchi}}

[[w:Luigi Bianchi]] (1886) investigated [[../Lorentz_transformation_(squeeze)#Lie2|E:Lie's transformation (1880)]] of pseudospherical surfaces, obtaining the result:&lt;ref group=M&gt;Bianchi (1886), eq. 1 can be found on p. 226, eq. (2) on p. 240, eq. (3) on pp. 240–241, and for eq. (4) see the footnote on p. 240.&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; u+v=2\alpha,\ u-v=2\beta;\\
(2)\quad &amp; \Omega\left(\alpha,\beta\right)\Rightarrow\Omega\left(k\alpha,\ \frac{\beta}{k}\right);\\
(3)\quad &amp; \theta(u,v)\Rightarrow\theta\left(\frac{u+v\sin\sigma}{\cos\sigma},\ \frac{u\sin\sigma+v}{\cos\sigma}\right)=\Theta_{\sigma}(u,v);\\
 &amp; \text{Inverse:}\left(\frac{u-v\sin\sigma}{\cos\sigma},\ \frac{-u\sin\sigma+v}{\cos\sigma}\right)\\
(4)\quad &amp; \frac{1}{2}\left(k+\frac{1}{k}\right)=\frac{1}{\cos\sigma},\ \frac{1}{2}\left(k-\frac{1}{k}\right)=\frac{\sin\sigma}{\cos\sigma}
\end{align}
&lt;/math&gt;.

{{Lorentzbox|Text=Transformation (3) and its inverse are equivalent to trigonometric Lorentz boost ({{equationNote|8a}}), and becomes Lorentz boost of velocity with &lt;math&gt;\sin\sigma=\tfrac{v}{c}&lt;/math&gt;.}}

==={{anchor|Darboux1}} Darboux (1891/94) – Pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Darboux|label 1=History of Lorentz transformations via sphere transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Darboux2|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Darboux2|label 1=History of Lorentz transformations via squeeze mappings § Darboux}}

Similar to [[#Bianchi1|Bianchi (1886)]], [[w:Gaston Darboux]] (1891/94) showed that the [[../Lorentz_transformation_(squeeze)#Lie2|E:Lie's transformation (1880)]] gives rise to the following relations:&lt;ref group=M&gt;Darboux (1891/94), pp. 381–382&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; u+v=2\alpha,\ u-v=2\beta;\\
(2)\quad &amp; \omega=\varphi\left(\alpha,\beta\right)\Rightarrow\omega=\varphi\left(\alpha m,\ \frac{\beta}{m}\right)\\
(3)\quad &amp; \omega=\psi(u,v)\Rightarrow\omega=\psi\left(\frac{u+v\sin h}{\cos h},\ \frac{v+u\sin h}{\cos h}\right)
\end{align}
&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost [[../Lorentz transformation (squeeze)#math_9a|E:'''(9a)''']] in terms of null coordinates. Transformation (3) is equivalent to trigonometric Lorentz boost ({{equationNote|8a}}), and becomes Lorentz boost [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']] with &lt;math&gt;\sin h=\tfrac{v}{c}&lt;/math&gt;.}}

==={{anchor|Scheffers}} Scheffers (1899) – Contact transformation===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Scheffers|label 1=History of Lorentz transformations via sphere transformations § Scheffers}}

[[w:Georg Scheffers]] (1899) synthetically determined all ''finite'' [[w:contact transformation]]s preserving circles in the plane, consisting of dilatations, inversions, and the following one preserving circles and lines (compare with Laguerre inversion by [[../Lorentz transformation (conformal)#Laguerre|E:Laguerre (1882)]] and [[../Lorentz transformation (conformal)#Darboux2|Darboux (1887)]]):&lt;ref group=M&gt;Scheffers (1899), p. 158&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\sigma^{\prime2}-\rho^{\prime2}=\sigma^{2}-\rho^{2}\\
\hline \rho'=\frac{\rho}{\cos\omega}+\sigma\tan\omega,\quad\sigma'=\rho\tan\omega+\frac{\sigma}{\cos\omega}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|8a}}) by the identity &lt;math&gt;\sec\omega=\tfrac{1}{\cos\omega}&lt;/math&gt;.}}

=== {{anchor|Eisenhart}} Eisenhart (1905) – Pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (squeeze)#Eisenhart|label 1=History of Lorentz transformations via squeeze mappings § Eisenhart}}

[[w:Luther Pfahler Eisenhart]] (1905) followed [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Darboux1|Darboux (1891/94)]] by writing the [[../Lorentz_transformation_(squeeze)#Lie2|E:Lie's transformation (1880)]] of pseudospherical surfaces:&lt;ref group=M&gt;Eisenhart (1905), p. 126&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; \alpha=\frac{u+v}{2},\ \beta=\frac{u-v}{2}\\
(2)\quad &amp; \omega\left(\alpha,\beta\right)\Rightarrow\omega\left(m\alpha,\ \frac{\beta}{m}\right)\\
(3)\quad &amp; \omega(u,v)\Rightarrow\omega(\alpha+\beta,\ \alpha-\beta)\Rightarrow\omega\left(\alpha m+\frac{\beta}{m},\ \alpha m-\frac{\beta}{m}\right)\\
 &amp; \Rightarrow\omega\left[\frac{\left(m^{2}+1\right)u+\left(m^{2}-1\right)v}{2m},\ \frac{\left(m^{2}-1\right)u+\left(m^{2}+1\right)v}{2m}\right]\\
(4)\quad &amp; m=\frac{1-\cos\sigma}{\sin\sigma}\Rightarrow\omega\left(\frac{u-v\cos\sigma}{\sin\sigma},\ \frac{v-u\cos\sigma}{\sin\sigma}\right)
\end{align}&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost [[../Lorentz transformation (squeeze)#math_9a|E:'''(9a)''']] in terms of null coordinates. Transformation (3) is equivalent to Lorentz boost [[../Lorentz transformation (squeeze)#math_9b|E:'''(9b)''']] in terms of Bondi's ''k'' factor, as well as Lorentz boost [[../Lorentz transformation (Möbius)#math_6f|E:'''(6f)''']] with &lt;math&gt;m=\alpha^2&lt;/math&gt;. Transformation (4) is equivalent to trigonometric Lorentz boost ({{equationNote|8b}}), and becomes Lorentz boost [[../Lorentz transformation (velocity)#math_4b|E:'''(4b)''']] with &lt;math&gt;\cos\sigma=\tfrac{v}{c}&lt;/math&gt;. Eisenhart's angle σ corresponds to ϑ of Lorentz boost [[../Lorentz transformation (squeeze)#math_9d|E:'''(9d)''']].}}

==={{anchor|Varicak}} Varićak (1910) – Circular and Hyperbolic functions===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Varicak|label 1=History of Lorentz transformations via hyperbolic functions § Varicak}}

Relativistic velocity in terms of trigonometric functions and its relation to hyperbolic functions was demonstrated by [[w:Vladimir Varićak]] in several papers starting from 1910, who represented the equations of special relativity on the basis of [[w:hyperbolic geometry]] in terms of Weierstrass coordinates. For instance, he showed the relation of rapidity to the [[w:Gudermannian function]] and the [[w:angle of parallelism]]:&lt;ref group=R name=var1&gt;Varićak (1910), p. 93&lt;/ref&gt;

:&lt;math&gt;\frac{v}{c}=\operatorname{th}u=\operatorname{tg}\psi=\sin\operatorname{gd}(u)=\cos\Pi(u)&lt;/math&gt;

{{Lorentzbox|Text=This is the foundation of Lorentz transformation ({{equationNote|8a}}) and ({{equationNote|8b}}).}}

==={{anchor|Plummer}} Plummer (1910) – Trigonometric Lorentz boosts===

[[w:Henry Crozier Keating Plummer]] (1910) defined the following relations&lt;ref group=R&gt;Plummer (1910), p. 256&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\tau=t\sec\beta-x\tan\beta/U\\
\xi=x\sec\beta-Ut\tan\beta\\
\eta=y,\ \zeta=z,\\
\hline \sin\beta=v/U
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|8a}}).}}

==={{anchor|Gruner}} Gruner (1921) – Trigonometric Lorentz boosts===

In order to simplify the graphical representation of Minkowski space, [[w:Paul Gruner]] (1921) (with the aid of Josef Sauter) developed what is now called [[w:Loedel diagram]]s, using the following relations:&lt;ref group=R&gt;Gruner (1921a)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}v=\alpha\cdot c;\quad\beta=\frac{1}{\sqrt{1-\alpha^{2}}}\\
\sin\varphi=\alpha;\quad\beta=\frac{1}{\cos\varphi};\quad\alpha\beta=\tan\varphi\\
\hline x'=\frac{x}{\cos\varphi}-t\cdot\tan\varphi,\quad t'=\frac{t}{\cos\varphi}-x\cdot\tan\varphi
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|8a}}) by the identity &lt;math&gt;\sec\varphi=\tfrac{1}{\cos\varphi}&lt;/math&gt;}}

In another paper Gruner used the alternative relations:&lt;ref group=R&gt;Gruner (1921b)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\alpha=\frac{v}{c};\ \beta=\frac{1}{\sqrt{1-\alpha^{2}}};\\
\cos\theta=\alpha=\frac{v}{c};\ \sin\theta=\frac{1}{\beta};\ \cot\theta=\alpha\cdot\beta\\
\hline x'=\frac{x}{\sin\theta}-t\cdot\cot\theta,\quad t'=\frac{t}{\sin\theta}-x\cdot\cot\theta
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz Lorentz boost ({{equationNote|8b}}) by the identity &lt;math&gt;\csc\theta=\tfrac{1}{\sin\theta}&lt;/math&gt;.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|bia86lez}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar94cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eis05}} 
*{{#section:History of Topics in Special Relativity/mathsource|schef99}}

===Historical relativity sources===
{{reflist|3|group=R}}
{{#section:History of Topics in Special Relativity/relsource|grun21a}}
{{#section:History of Topics in Special Relativity/relsource|grun21b}}
{{#section:History of Topics in Special Relativity/relsource|plum10}}
{{#section:History of Topics in Special Relativity/relsource|var10}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L9}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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==Lorentz transformation via squeeze mappings==
[[File:Hyperbolic sector squeeze mapping.svg|250px|right|thumb|A squeeze mapping relates blue and green parallelograms.]]
As already indicated in [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']] in exponential form or [[../Lorentz transformation (Möbius)#math_6f|E:'''(6f)''']] in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as [[w:squeeze mapping]]s. Using [[w:hyperbola#Hyperbola with equation y = A/x|w:asymptotic coordinates of a hyperbola]] (''u,v''), in relativity also known as [[w:light-cone coordinates]], they have the general form (some authors alternatively add a factor of 2 or &lt;math&gt;\sqrt{2}&lt;/math&gt;):&lt;ref name=terng&gt;Terng &amp; Uhlenbeck (2000), p. 21&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{array}{c}
u=x_{0}-x_{1},\ v=x_{0}+x_{1}\\
u'=x_{0}^{\prime}-x_{1}^{\prime},\ v'=x_{0}^{\prime}+x_{1}^{\prime}
\end{array}\\
\hline (1) &amp; (u',v')=\left(ku,\ \frac{1}{k}v\right)\\
(2) &amp; (u',v')=\left(\frac{1}{k}v,\ ku\right)\\
\hline  &amp; u'v'=uv
\end{matrix}&lt;/math&gt;|{{equationRef|9a}}}}

with arbitrary ''k''. This geometrically corresponds to the transformation of one parallelogram to other ones of same area, whose sides touch a hyperbola and both asymptotes. While equation system (1) corresponds to proper Lorentz boosts, equation system (2) produces improper ones. For instance, solving (1) for &lt;math&gt;x'_0, x'_1&lt;/math&gt; gives:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =\frac{1}{2}\left(k+\frac{1}{k}\right)x_{0}-\frac{1}{2}\left(k-\frac{1}{k}\right)x_{1} &amp;  &amp; =\frac{x_{0}\left(k^{2}+1\right)-x_{1}\left(k^{2}-1\right)}{2k}\\
x_{1}^{\prime} &amp; =-\frac{1}{2}\left(k-\frac{1}{k}\right)x_{0}+\frac{1}{2}\left(k+\frac{1}{k}\right)x_{1} &amp;  &amp; =\frac{-x_{0}\left(k^{2}-1\right)+x_{1}\left(k^{2}+1\right)}{2k}\\
\\
x_{0} &amp; =\frac{1}{2}\left(k+\frac{1}{k}\right)x_{0}^{\prime}+\frac{1}{2}\left(k-\frac{1}{k}\right)x_{1}^{\prime} &amp;  &amp; =\frac{x_{0}^{\prime}\left(k^{2}+1\right)+x_{1}^{\prime}\left(k^{2}-1\right)}{2k}\\
x_{1} &amp; =\frac{1}{2}\left(k-\frac{1}{k}\right)x_{0}^{\prime}+\frac{1}{2}\left(k+\frac{1}{k}\right)x_{1}^{\prime} &amp;  &amp; =\frac{x_{0}^{\prime}\left(k^{2}-1\right)+x_{1}^{\prime}\left(k^{2}+1\right)}{2k}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|9b}}}}

The geometrical foundation of squeeze mapping ({{equationNote|9a}}) was known for a long time since [[#Apo|Apollonius (BC)]] and was used to generate hyperbolas by [[#spei|Speidell (1688) and Whiston (1710)]]. Equation ({{equationNote|9a}}-1) was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Laisant1|Laisant (1874)]] and [[#Gunther1|Günther (1880/81)]] in relation to elliptic trigonometry, or by [[#Lie2|Lie (1879-81)]], [[#Bianchi1|Bianchi (1886, 1894)]], [[#Darboux1|Darboux (1891/94)]], [[#Eisenhart|Eisenhart (1905)]] as [[w:squeeze mapping#Lie transform|Lie transform]]&lt;ref name=terng /&gt; of [[w:pseudospherical surface]]s in terms of the [[w:Sine-Gordon equation]], or by [[#Lipschitz1|Lipschitz (1885/86)]] in transformation theory. Equation ({{equationNote|9a}}-2) was given by [[#Reynaud|Reynaud (1819)]].

From that, different forms of Lorentz transformation were derived: ({{equationNote|9b}}) by [[#Lipschitz1|Lipschitz (1885/86)]], [[#Bianchi1|Bianchi (1886, 1894)]], [[#Eisenhart|Eisenhart (1905)]], trigonometric Lorentz boost [[../Lorentz transformation (trigonometric)#math_8a|E:'''(8a)''']] by [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Darboux1|Darboux (1891/94)]], and trigonometric Lorentz boost [[../Lorentz transformation (trigonometric)#math_8b|E:'''(8b)''']] by [[#Eisenhart|Eisenhart (1905)]]. Lorentz boost ({{equationNote|9b}}) was rediscovered in the framework of special relativity by [[w:Hermann Bondi]] (1964)&lt;ref&gt;Bondi (1964), p. 118&lt;/ref&gt; in terms of [[w:Bondi k-calculus]], by which ''k'' can be physically interpreted as Doppler factor. Since ({{equationNote|9b}}) is equivalent to [[../Lorentz transformation (Möbius)#math_6f|E:'''(6f)''']] in terms of Cayley–Klein parameter by setting &lt;math&gt;k=\alpha^2&lt;/math&gt;, it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation [[../Lorentz transformation (Möbius)#math_6e|E:'''(6e)''']] stated by [[../Lorentz transformation (Möbius)#Gauss3|Gauss around 1800]] (posthumously published 1863), [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]], [[../Lorentz transformation (Möbius)#Bianchi2|E:Bianchi (1888)]], [[../Lorentz transformation (Möbius)#Fricke|E:Fricke (1891)]] and [[../Lorentz transformation (Möbius)#Woods|E:Woods (1895)]].

Rewriting ({{equationNote|9a}}) in terms of [[w:homogeneous coordinates]] signifies squeeze mappings of the unit hyperbola in terms of a [[w:Quadric#Projective quadric|w:projective conic]]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\left[u,v\right]=\left[\frac{y_{1}}{y_{3}},\frac{y_{2}}{y_{3}}\right]\quad\left(uv=1\quad\Rightarrow\quad y_{1}y_{2}-y_{3}^{2}=0\right)\\
\left[k,\frac{1}{k}\right]=\left[\frac{\alpha_{1}}{\alpha_{3}},\frac{\alpha_{2}}{\alpha_{3}}\right]\quad\left(k\frac{1}{k}=1\quad\Rightarrow\quad\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)\\
\hline y'_{1}=\alpha_{1}y_{1}\\
y'_{2}=\alpha_{2}y_{2}\\
y'_{3}=\alpha_{3}y_{3}\\
\hline y_{1}y_{2}-y_{3}^{2}=y'_{1}y'_{2}-y_{3}^{\prime2}=0\\
\hline uv=\frac{y_{1}y_{2}}{y_{3}^{2}}=u'v'=\frac{y'_{1}y'_{2}}{y_{3}^{\prime2}}
\end{matrix}&lt;/math&gt;|{{equationRef|9c}}}}

Such transformations were given by [[#Klein|Klein (1871)]] to express motions in non-Euclidean space.

Furthermore, variables ''u, v'' in ({{equationNote|9a}}) can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation [[../Lorentz transformation (Cayley-Hermite)#math_5b|E:'''(5b)''']] in terms of Cayley-Hermite parameter:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{matrix}u=x_{0}-x_{1}\\
v=x_{0}+x_{1}\\
u'=x_{0}^{\prime}-x_{1}^{\prime}\\
v'=x_{0}^{\prime}+x_{1}^{\prime}
\end{matrix}\Rightarrow\begin{matrix}u_{1}=x_{0}+x_{0}^{\prime}\\
v_{1}=x_{0}-x_{0}^{\prime}\\
u_{2}=x_{1}-x_{1}^{\prime}\\
v_{2}=x_{1}+x_{1}^{\prime}
\end{matrix}\\
\hline (u_{2},v_{2})=\left(au_{1},\ \frac{1}{a}v_{1}\right)\Rightarrow u_{2}v_{2}=u_{1}v_{1}\\
(u',v')=\left(\frac{1+a}{1-a}u,\ \frac{1-a}{1+a}v\right)\Rightarrow u'v'=uv
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\frac{1+a^{2}}{1-a^{2}}-x_{1}\frac{2a}{1-a^{2}} &amp;  &amp; =\frac{x_{0}\left(1+a^{2}\right)-x_{1}2a}{1-a^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2a}{1-a^{2}}+x_{1}\frac{1+a^{2}}{1-a^{2}} &amp;  &amp; =\frac{-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\frac{1+a^{2}}{1-a^{2}}+x_{1}^{\prime}\frac{2a}{1-a^{2}} &amp;  &amp; =\frac{x_{0}^{\prime}\left(1+a^{2}\right)+x_{1}^{\prime}2a}{1-a^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2a}{1-a^{2}}+x_{1}^{\prime}\frac{1+a^{2}}{1-a^{2}} &amp;  &amp; =\frac{x_{0}^{\prime}2a+x_{1}^{\prime}\left(1+a^{2}\right)}{1-a^{2}}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|9d}}}}

These Lorentz transformations were given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry.

On the basis of factors ''k'' or ''a'', all previous Lorentz boosts [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']], [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']], [[../Lorentz transformation (trigonometric)#math_8a|E:'''(8a)''']], [[../Lorentz transformation (trigonometric)#math_8b|E:'''(8b)''']], can be expressed as squeeze mappings as well:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{r|c|c|c|c|c|c}
 &amp; (9a) &amp; (9d) &amp; (3b) &amp; (4a) &amp; (8a) &amp; (8b)\\
\hline \frac{u'}{u}=\frac{x_{0}^{\prime}-x_{1}^{\prime}}{x_{0}-x_{1}}= &amp; k &amp; \frac{1+a}{1-a} &amp; e^{\eta} &amp; \sqrt{\tfrac{1+\beta}{1-\beta}} &amp; \frac{1+\sin\theta}{\cos\theta} &amp; \frac{1+\cos\vartheta}{\sin\vartheta}=\cot\frac{\vartheta}{2}\\
\hline \frac{u_{2}}{u_{1}}=\frac{x_{1}-x_{1}^{\prime}}{x_{0}+x_{0}^{\prime}}= &amp; \frac{k-1}{k+1} &amp; a &amp; \tanh\frac{\eta}{2} &amp; \frac{\gamma-1}{\beta\gamma} &amp; \frac{1-\cos\theta}{\sin\theta}=\tan\frac{\theta}{2} &amp; \frac{1-\sin\vartheta}{\cos\vartheta}\\
\hline  &amp; \frac{k^{2}-1}{k^{2}+1} &amp; \frac{2a}{1+a^{2}} &amp; \tanh\eta &amp; \beta &amp; \sin\theta &amp; \cos\vartheta\\
\hline  &amp; \frac{k^{2}+1}{2k} &amp; \frac{1+a^{2}}{1-a^{2}} &amp; \cosh\eta &amp; \gamma &amp; \sec\theta &amp; \csc\vartheta\\
\hline  &amp; \frac{k^{2}-1}{2k} &amp; \frac{2a}{1-a^{2}} &amp; \sinh\eta &amp; \beta\gamma &amp; \tan\theta &amp; \cot\vartheta
\end{array}&lt;/math&gt;|{{equationRef|9e}}}}

Squeeze mappings in terms of &lt;math&gt;\theta&lt;/math&gt; were used by [[#Darboux1|Darboux (1891/94)]] and [[#Bianchi1|Bianchi (1894)]], in terms of &lt;math&gt;\eta&lt;/math&gt; implicitly by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1891)]], [[#Elliott|Elliott (1903)]], [[#Herglotz|Herglotz (1909/10)]], in terms of &lt;math&gt;\vartheta&lt;/math&gt; by [[#Eisenhart|Eisenhart (1905)]], in terms of &lt;math&gt;\beta&lt;/math&gt; by [[#Born|Born (1921)]], [[w:Bondi k-calculus|w:Milne (1935) and Bondi (1964)]].

==Historical notation==
==={{anchor|Apo}} Apollonius (BC) – Hyperbola mapping ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Apo|label 1=History of Lorentz transformations in general § Apollonius}}
[[File:Apollonius-Halley-XII.png|thumb|&lt;small&gt;Halley's (1710) illustration of Apollonius prop. XII, showing ΔΕ·ΔΖ=HK·ΘΗ.&lt;/small&gt;]]
[[File:Apollonius-Halley-XIII.png|thumb|&lt;small&gt;Halley's (1710) illustration of Apollonius prop. XIII, showing ΓH·HΘ=ΛK·KΔ (also equal to AΛ·ΛK and EM·MΞ), whereas AE·EZ is smaller.&lt;/small&gt;]]
[[w:Apollonius of Perga]] (c. 240–190 BC, and maybe other Greek geometers such as [[w:Menaechmus]] even earlier) defined the following proposition Nr. XII in his second book on conic sections, which was translated into Latin several times by Giovanni Battista Memmo (1537), [[w:Federico Commandino]] (1566), [[w:Isaac Barrow]] (1675), and in particular by [[w:Edmond Halley]] (1710), with the Halley translation reading as follows:&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XII of book II on p. 114; Latin: &quot;Si ab aliquo puncto eorum, qua sunt in sectione, ad asymptotos duæ rectæ lineæ in quibuslibet angulis ducantur, &amp; ab alio quovis puncto in sectione sumpto ducantur aliæ rectæ his ipsis parallelæ : rectangulum sub parallelis contentum æquale erit contento sub rectis ipsis quibus ductæ fuerant parallelae.&lt;br&gt;
Sit hyperbola, cujus asymptoti AB, BΓ sumatur in sectione aliquod punctum Δ, atque ab eo ad ΑΒ, ΒΓ, ducantur ΔΕ, ΔΓ; sumatur autem &amp; alterum punctum H in sectione, per quod ducantur HΘ, HK ipsis ΔΕ, ΔΖ parallelæ: dico rectangulum EΔZ rectangulo ΘHK æquale esse.&lt;br&gt;
Jungatur enim ΔH, &amp; ad A, Γ producatur. itaque quoniam rectangulum AΔΓ aequatur rectangulo AHΓ; erit ut AH ad AΔ ita ΔΓ ad ΓΗ. sed ut AH ad ΑΔ ita ΗΘ ad ΕΔ, &amp; ut ΔΓ ad ΓΗ ita ΔZ ad ΗΚ; quare ut ΘΗ ad ΔΕ ita ΔZ ad ΗK: rectangulum igitur ΕΔZ rectangulo ΘHK est æquale.&quot;&lt;/ref&gt;

:Let there be a hyperbola whose asymptotes are AB, BΓ, and let some point Δ be taken in that section, from which ΔΕ, ΔΓ are drawn to ΑΒ, ΒΓ; and let another point H be taken in that section, through which HΘ, HK are drawn parallel to ΔΕ, ΔΖ: I say that the rectangle EΔZ is equal to the rectangle ΘHK.
:Let ΔH be joined, and A is connected to Γ. Therefore, since the rectangle AΔΓ is equal to the rectangle AHΓ, it follows that AH is to AΔ as ΔΓ is to ΓΗ. But AH is to ΑΔ as ΗΘ is to ΕΔ, and ΔΓ is to ΓΗ as ΔZ is to ΗΚ; wherefore as ΘΗ is to ΔΕ, so ΔZ is to ΗK: therefore the rectangle EΔZ is equal to the rectangle ΘHK. 

A modernized translation was given by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 34 (Apollonius, Book II, Prop. 12).&lt;/ref&gt;

:If ''Q, q'' be any two points on a hyperbola, and parallel straight lines ''QH, qh'' be drawn to meet one asymptote at any angle, and ''QK, qk'' (also parallel to one another) meet the other asymptote at any angle, then ''HQ·QK = hq·qk''. Let Qq meet the asymptotes in R,r. We have RQ.Qr=Rq.qr; therefore RQ:Rq=qr:Qr. But RQ:Rq=HQ:hq, and qr:Qr=qk:QK; therefore HQ:hq=qk:QK, or HQ.QK=hq.qk.

In the next proposition XIII, Apollonius showed that if a line is drawn parallel to the asymptotes, within the space between asymptotes and hyperbola, it must meet the hyperbola exactly once. In his demonstration, Apollonius used the previous proposition XII when comparing the area of several parallelograms whose sides are drawn parallel to the asymptotes.&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XIII of book II on p. 114-115&lt;/ref&gt;&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 35 (Apollonius, Book II, Prop. 13).&lt;/ref&gt;

{{Lorentzbox|Text=The ratios given by Apollonius:
:&lt;math&gt;\frac{\Delta\Gamma}{\Gamma H}=\frac{\Delta Z}{HK},\quad\frac{\Theta H}{\Delta E}=\frac{\Delta Z}{HK}&lt;/math&gt;
represent an equation system that can be solved for ΔΕ, ΔΖ, resulting in the squeeze mapping:
:&lt;math&gt;\Delta E=\frac{\Gamma H}{\Delta\Gamma}\Theta H,\quad\Delta Z=\frac{\Delta\Gamma}{\Gamma H}HK&lt;/math&gt;
producing &lt;math&gt;\Delta E\cdot\Delta Z=HK\cdot\Theta H&lt;/math&gt; in line with Apollonius result that rect. EΔZ is equal to rect. ΘHK.
In case ΔΕ, ΔΖ, HK, ΘΗ are all drawn parallel to the respective asymptotes, it follows u'=ΔΕ, v'=ΔΖ, u=HK, v=ΘΗ, k=&lt;math&gt;\tfrac{\Gamma H}{\Delta\Gamma}&lt;/math&gt; and therefore Apollonius result becomes equivalent to Lorentz boost ({{equationNote|9a}}), signifying squeezed parallelograms located between the asymptotes and the hyperbola.
In general, the identity &lt;math&gt;\Delta E\cdot\Delta Z=HK\cdot\Theta H&lt;/math&gt; demonstrates the invariance of the area of all parallelograms that are constructed in line with the proposition XII, thereby representing all points of a hyperbola defined by ''HK·ΘΗ = const''. That is, the invariant area ''HK·ΘΗ = const.'' together with ''const''=1 gives ''HK''=1/''ΘΗ'', which implies that ''ΘΗ'' is inverse proportional to ''HK''. Thus when ''HK'' is increased into ''k·HK'' using some factor ''k'', it follows that ''ΘΗ'' must be proportionally diminished into ''ΘΗ/k'' in order to preserve  invariance of area.}}

==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#mercator|label 1=History of Lorentz transformations via hyperbolic functions § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated Apollonius' proposition on a rectangular hyperbola algebraically as follows:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations ''c'' and ''d'' implicitly correspond to hyperbolic functions ''cosh'' and ''sinh'' (which were explicitly introduced by [[../Lorentz transformation (hyperbolic)#Riccati|E:Riccati (1757)]] much later). In particular, his result AH·FH=AI·BI and AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of &amp;eta; because:
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}=k&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta} AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta} BI&lt;/math&gt;.}}

==={{anchor|spei}} Speidell (1688), Whiston (1710) – Hyperbola generation ===
[[File:Whiston-hyperbola.png|thumb|&lt;small&gt;Whiston's (1710) illustration of generating a hyperbola by parallelograms of equal area.&lt;/small&gt;]]
The case of squeezing a given square or parallelogram as a means to ''generate'' hyperbolas was discussed by [[w:Euclid Speidell]] (1688):&lt;ref group=M&gt;Speidell (1688), pp. 4-5&lt;/ref&gt;
:[..] from a Square and an infinite company of Oblongs on a Superficies, each Equal to that Square, how a Curve is begotten which shall have the same properties and affections of an Hyperbola inscribed within a Right Angled Cone
:[..] There is a Square ''ABCD'', whose Side or Root is 10, let ''DB'' be prolonged in ''infinitum'', and continually divided equally by the Root, or ''DB'', and those Equal Divisions numbered by 10, 20, 30, 40, 50, 60, 70, &amp;c. in ''infinitum'': Upon these Numbers let Perpendiculars be erected, which call Ordinates, and each of those Perpendiculars of that length, that Perpendiculars let fall from the aforesaid Perpendiculars to the Side or Base ''CD'' (which call Complement Ordinates) the Oblongs made of the Ordinate Perpendiculars, and Complement Ordinate Perpendiculars may be ever Equal to the Square ''AD'', which is easily done thus, for it is &lt;math&gt;\tfrac{100}{20},\tfrac{100}{30},\tfrac{100}{40},\tfrac{100}{50}&lt;/math&gt; &amp;c. produces the Length of the Ordinate Perpendiculars
:[..] all the Oblongs made of the Ordinates, and Complement Ordinates are each of them equal to the Square ''AD'', which is here 100
:[..] the like Demonstration serves for all the Oblongs or Parallelograms standing upon the Base ''CD'', by the Tips or Angular Points of those Parallelograms, or from the Ends of all the Ordinates standing upon 20, 30, 40, 50, 60, 70, in ''infinitum'', draw the Curve Line from ''A'' towards ''E'', so shall you describe the Curve ''AEFGS'' [..].

{{Lorentzbox|Text=This corresponds to squeeze mappings ({{equationNote|9a}}) with ''u=v''=10 and ''k''=1,2,3,4,5,6,7,..., thus ''u'v'=uv''=100.}}

In similar terms, [[w:William Whiston]] (1710/16) wrote:&lt;ref group=M&gt;Whiston (1710/16). In the English version (1716) see pp. 16-17. In the original Latin version (1710) see pp. 16-18&lt;/ref&gt;
:But it is to be acknowledg'd, that many Properties of an Hyperbola are better known from another manner of generating the Figure; which Way is this: Let ''LL'' and ''MM'' be infinite Right Lines intersecting each other in any Angle whatever in the Point ''C'': From any Point whatever, as ''D'' or ''e'', let ''Dc, Dd,'' be drawn parallel to the first Lines, or (''ec, ed''), which with the Lines first drawn make the Parallelograms as ''DcCd'', or ''ecCd''; Now conceive two sides of the Parallelogram as ''Dc, Dd,'' or ''ec, ed'', to be so mov'd this way and that way, that they always keep the same Parallelism, and that at the same time the Area's always remain equal: That is to say, that ''Dc'' and ''ec'' remain always Parallel to ''MM'', and ''Dd'' or ''ed'' always Parallel to ''LL''; and that the Area of every Parallelogram be equal to every other, one Side being increas'd in the same Proportion wherein the other is diminish'd. By this means the Point ''D'' or ''e'' will describe a Curve-Line within the Angle comprehended by the first Lines;

{{Lorentzbox|Text=This corresponds to squeeze mappings ({{equationNote|9a}}).}}

==={{anchor|Reynaud}} Reynaud (1819) – Hyperbola mapping  ===
[[w:Antoine André Louis Reynaud]] algebraically expressed squeeze mappings by writing:&lt;ref group=M&gt;Reynaud (1819), p. 247&lt;/ref&gt;
:&quot;The system of equations &lt;math&gt;(2)\ x=\frac{y'}{\alpha},\ y=\alpha x'&lt;/math&gt; determines all points of the curve &lt;math&gt;S&lt;/math&gt;, because &lt;math&gt;x'&lt;/math&gt; and &lt;math&gt;y'&lt;/math&gt; being given numbers, each arbitrary value of &lt;math&gt;\alpha&lt;/math&gt; gives a point &lt;math&gt;x,y&lt;/math&gt; of this curve. The elimination of the indeterminate &lt;math&gt;\alpha&lt;/math&gt; between equations (2) will therefore lead to the equation &lt;math&gt;xy=x'y'&lt;/math&gt; of the curve in question. This curve is therefore a hyperbola related to its asymptotes &lt;math&gt;xX,yY&lt;/math&gt;.&quot;

{{Lorentzbox|Text=This is equivalent to (improper) Lorentz transformation ({{equationNote|9a}}-2).}}

==={{anchor|Klein}} Klein (1871) – Projective conic section===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Klein|label 1=History of Lorentz transformations in general § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Klein|label 1=History of Lorentz transformations via Möbius transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}

Elaborating on the [[w:Cayley–Klein metric]], [[w:Felix Klein]] (1871) defined a [[w:Quadric#Projective quadric|w:projective conic]] in order to discuss motions such as rotation and translation in the non-Euclidean plane:&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\
\hline \begin{align}x_{1} &amp; =\alpha_{1}y_{1}\\
x_{2} &amp; =\alpha_{2}y_{2}\\
x_{3} &amp; =\alpha_{3}y_{3}
\end{align}
\\
\left(\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)\\
\hline \frac{x_{1}x_{2}}{x_{3}^{2}}=\text{invariant}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=When the conic section is a hyperbola this is equivalent to squeeze mapping ({{equationNote|9c}}). This becomes ({{equationNote|9a}}) using &lt;math&gt;\left[u,v\right]=\left[\tfrac{x_{1}}{x_{3}},\tfrac{x_{2}}{x_{3}}\right],\ \left[k,\tfrac{1}{k}\right]=\left[\tfrac{\alpha_{1}}{\alpha_{3}},\tfrac{\alpha_{2}}{a_{3}}\right]&lt;/math&gt;.}}

==={{anchor|Laisant1}} Laisant (1874) – Elliptic polar coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Laisant|label 1=History of Lorentz transformations via hyperbolic functions § Laisant}}

[[w:Charles-Ange Laisant]] extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation&lt;ref group=M&gt;Laisant (1874a), pp. 73–76&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=ax,\ y'=\frac{y}{a}\\
x'y'=xy
\end{matrix}&lt;/math&gt;

He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}).}}

==={{anchor|Lie2}} Lie (1879-84) – Transforming pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Lie3|label 1=History of Lorentz transformations in general § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Lie|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}

[[w:Sophus Lie]] (1879/80) derived an operation from [[w:Pierre Ossian Bonnet]]'s (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other.&lt;ref group=M&gt;Lie (1879/80), Collected papers, vol. 3, p. 389&lt;/ref&gt; Lie gave explicit formulas for this operation in two papers published in 1881: If &lt;math&gt;(s,\sigma)&lt;/math&gt; are asymptotic coordinates of two principal tangent curves and &lt;math&gt;\Theta&lt;/math&gt; their respective angle, and &lt;math&gt;\Theta=f(s,\sigma)&lt;/math&gt; is a solution of the Sine-Gordon equation &lt;math&gt;\tfrac{d^{2}\Theta}{ds\ d\sigma}=K\sin\Theta&lt;/math&gt;, then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:&lt;ref group=M&gt;Lie (1879/81), Collected papers, vol. 3, p. 393&lt;/ref&gt;

:&lt;math&gt;\Theta=f(s,\sigma)\Rightarrow\Theta=f\left(ms,\ \frac{\sigma}{m}\right)&lt;/math&gt;

In (1880/81) he wrote these relations as follows:&lt;ref group=M&gt;Lie (1880/81), Collected papers, vol. 3, pp. 477–478&lt;/ref&gt;

:&lt;math&gt;\vartheta=\Phi(s,S)\Rightarrow\vartheta=\Phi\left(ms,\ \frac{S}{m}\right)&lt;/math&gt;

In (1883/84) he showed that the combination of Lie transform ''O'' with Bianchi transform ''I'' produces [[w:Bäcklund transform]] ''B'' of pseudospherical surfaces:&lt;ref group=M&gt;Lie (1883/84), Collected papers, vol. 3, p. 556&lt;/ref&gt;

:&lt;math&gt;B=OIO^{-1}&lt;/math&gt;

{{Lorentzbox|Text=As shown by [[#Bianchi1|Bianchi (1886)]] and [[#Darboux1|Darboux (1891/94)]], the Lie transform is equivalent to Lorentz transformations ({{equationNote|9a}}) and ({{equationNote|9b}}) in terms of light-cone coordinates ''2s=u+v'' and ''2σ=u-v''. In general, it can be shown that the Sine-Gordon equation is Lorentz invariant.}}

==={{anchor|Gunther1}} Günther (1880/81) – Elliptic polar coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Gunther1|label 1=History of Lorentz transformations via hyperbolic functions § Günther}}

Following [[#Laisant1|Laisant (1874)]], [[w:Siegmund Günther]] demonstrated the relation between circular polar coordinates and elliptic polar coordinates as&lt;ref group=M&gt;Günther (1880/81), pp. 383–385&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=ax,\ y'=\frac{1}{a}y\\
x'y'=xy
\end{matrix}&lt;/math&gt;

showing that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}).}}

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Laguerre|label 1=History of Lorentz transformations via sphere transformations § Laguerre}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Laguerre|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Laguerre}}

A transformation (later known as &quot;Laguerre inversion&quot;) of [[../Lorentz transformation (conformal)|E:oriented lines and spheres]] was given by [[w:Edmond Laguerre]] with ''R'' being the radius and ''D'' the distance of its center to the axis:&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}D^{2}-D^{\prime2}=R^{2}-R^{\prime2}\\
\hline \left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change for ''R'') to a squeeze mapping in terms of Lorentz boost ({{equationNote|9d}}).}}

==={{anchor|Darboux1}} Darboux (1883–1891) ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Darboux|label 1=History of Lorentz transformations via sphere transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Darboux2|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Darboux1|label 1=History of Lorentz transformations via trigonometric functions § Darboux}}
====Transforming pseudospherical surfaces====

[[w:Gaston Darboux]] (1883) followed [[#Lie2|Lie (1879/81)]] by transforming pseudospheres into each other as follows:&lt;ref group=M&gt;Darboux (1883), p. 849&lt;/ref&gt;

:&lt;math&gt;f(x,y)\Rightarrow f\left(\frac{x}{m},\ ym\right)&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|9a}}) by interpreting ''x, y'' as light-cone coordinates.}}

Similar to [[#Bianchi1|Bianchi (1886)]], Darboux (1891/94) showed that the Lie transform gives rise to the following relations:&lt;ref group=M&gt;Darboux (1891/94), pp. 381–382&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; u+v=2\alpha,\ u-v=2\beta;\\
(2)\quad &amp; \omega=\varphi\left(\alpha,\beta\right)\Rightarrow\omega=\varphi\left(\alpha m,\ \frac{\beta}{m}\right)\\
(3)\quad &amp; \omega=\psi(u,v)\Rightarrow\omega=\psi\left(\frac{u+v\sin h}{\cos h},\ \frac{v+u\sin h}{\cos h}\right)
\end{align}
&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost ({{equationNote|9a}}) in terms of light-cone coordinates.}}

===={{anchor|Darboux2}} Laguerre inversion====

Following [[#Laguerre|Laguerre (1882)]], Darboux (1887) formulated the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \left.\begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\right|\begin{align}z'+R' &amp; =\frac{1+k}{1-k}(z-R)\\
z'-R' &amp; =\frac{1-k}{1+k}(z+R)
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change for ''R'') to a squeeze mapping in terms of Lorentz boost ({{equationNote|9d}}) where Darboux's ''k'' corresponds to ''a''.}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) - Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Lipschitz1|label 1=History of Lorentz transformations via hyperbolic functions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(1)\\
\hline\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(2)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equation system (1) represents Lorentz boost or squeeze mapping ({{equationNote|9a}}), and (2) represents Lorentz boost ({{equationNote|9b}}).}}

==={{anchor|Bianchi1}} Bianchi (1886–1894) – Transforming pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Bianchi2|label 1=History of Lorentz transformations via Möbius transformations § Bianchi}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Bianchi1|label 1=History of Lorentz transformations via trigonometric functions § Bianchi}}

[[w:Luigi Bianchi]] (1886) followed [[#Lie2|Lie (1879/80)]] by writing the transformation of pseudospheres into each other, obtaining the result:&lt;ref group=M&gt;Bianchi (1886), eq. 1 can be found on p. 226, eq. (2) on p. 240, eq. (3) on pp. 240–241, and for eq. (4) see the footnote on p. 240.&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; u+v=2\alpha,\ u-v=2\beta;\\
(2)\quad &amp; \Omega\left(\alpha,\beta\right)\Rightarrow\Omega\left(k\alpha,\ \frac{\beta}{k}\right);\\
(3)\quad &amp; \theta(u,v)\Rightarrow\theta\left(\frac{u+v\sin\sigma}{\cos\sigma},\ \frac{u\sin\sigma+v}{\cos\sigma}\right)=\Theta_{\sigma}(u,v);\\
 &amp; \text{Inverse:}\left(\frac{u-v\sin\sigma}{\cos\sigma},\ \frac{-u\sin\sigma+v}{\cos\sigma}\right)\\
(4)\quad &amp; \frac{1}{2}\left(k+\frac{1}{k}\right)=\frac{1}{\cos\sigma},\ \frac{1}{2}\left(k-\frac{1}{k}\right)=\frac{\sin\sigma}{\cos\sigma}
\end{align}
&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost ({{equationNote|9a}}) in terms of light-cone coordinates. Plugging equations (4) into (3) gives Lorentz boost ({{equationNote|9b}}) in terms of Bondi's ''k'' factor.}}

In 1894, Bianchi redefined the variables ''u,v'' as asymptotic coordinates, by which the transformation obtains the form:&lt;ref group=M&gt;Bianchi (1894), pp. 433–434&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega\left(u,v\right)\Rightarrow\omega(u,v);\quad\Omega\left(u,v\right)=\omega\left(ku,\ \frac{v}{k}\right);\\
k=\frac{1+\sin\sigma}{\cos\sigma}\Rightarrow\Omega\left(u,v\right)=\omega\left(\frac{1+\sin\sigma}{\cos\sigma}u,\ \frac{1-\sin\sigma}{\cos\sigma}v\right)
\end{matrix}&lt;/math&gt;.

{{Lorentzbox|Text=This is consistent with one of the choices in ({{equationNote|9e}}) where Bianchi's angle σ corresponds to θ.}}

==={{anchor|Lindemann}} Lindemann (1890/91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Lindemann|label 1=History of Lorentz transformations via hyperbolic functions § Lindemann}}

[[w:Ferdinand von Lindemann]] employed the Cayley absolute related to surfaces of second degree and its transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
\\
\hline \Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\\
ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to squeeze mapping ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of &amp;eta; with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

==={{anchor|Haskell}} Haskell (1895) – Hyperbola mapping  ===

[[w:Mellen W. Haskell]] applied the linear transformation 
:&lt;math&gt;\alpha'=k\alpha,\ \beta'=k^{-1}\beta&lt;/math&gt; 
in order to transform a hyperbola into itself.&lt;ref group=M&gt;Haskell (1895), p. 159&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}).}}
==={{anchor|Smith}} Smith (1900) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Smith|label 1=History of Lorentz transformations via sphere transformations § Smith}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Smith|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Smith}}

[[w:Percey F. Smith]] followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\
\hline \kappa=\frac{R'-R}{p'-p}\\
p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|9d}}).}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Elliott|label 1=History of Lorentz transformations via hyperbolic functions § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The second line is equivalent to squeeze mapping or Lorentz boost ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of &amp;eta;.}}

=== {{anchor|Eisenhart}} Eisenhart (1905) – Transforming pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Eisenhart|label 1=History of Lorentz transformations via trigonometric functions § Eisenhart}}

[[w:Luther Pfahler Eisenhart]] followed [[#Lie2|Lie (1879/81)]], [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Darboux1|Darboux (1891/94)]] in transforming pseudospherical surfaces:&lt;ref group=M&gt;Eisenhart (1905), p. 126&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; \alpha=\frac{u+v}{2},\ \beta=\frac{u-v}{2}\\
(2)\quad &amp; \omega\left(\alpha,\beta\right)\Rightarrow\omega\left(m\alpha,\ \frac{\beta}{m}\right)\\
(3)\quad &amp; \omega(u,v)\Rightarrow\omega(\alpha+\beta,\ \alpha-\beta)\Rightarrow\omega\left(\alpha m+\frac{\beta}{m},\ \alpha m-\frac{\beta}{m}\right)\\
 &amp; \Rightarrow\omega\left[\frac{\left(m^{2}+1\right)u+\left(m^{2}-1\right)v}{2m},\ \frac{\left(m^{2}-1\right)u+\left(m^{2}+1\right)v}{2m}\right]\\
(4)\quad &amp; m=\frac{1-\cos\sigma}{\sin\sigma}\Rightarrow\omega\left(\frac{u-v\cos\sigma}{\sin\sigma},\ \frac{v-u\cos\sigma}{\sin\sigma}\right)
\end{align}&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost ({{equationNote|9a}}) in terms of light-cone coordinates. Transformation (3) is equivalent to Lorentz boost ({{equationNote|9b}}) in terms of Bondi's ''k'' factor. Eisenhart's angle σ corresponds to ϑ in ({{equationNote|9e}}).}}

=== {{anchor|Herglotz}} Herglotz (1909/10) – Special relativity ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Herglotz1|label 1=History of Lorentz transformations via hyperbolic functions § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Herglotz1|label 1=History of Lorentz transformations via Möbius transformations § Herglotz}}

In relation to special relativity, [[w:Gustav Herglotz]] (1909/10) defined the Lorentz boost as follows:&lt;ref group=M&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of η.}}

==={{anchor|Born}} Born (1921) – Special relativity===
In the second edition of “Einstein's theory of relativity”, [[w:Max Born]] (1921) discussed the relation of the Lorentz transformation and the hyperbola:&lt;ref group=M&gt;Born (1921), pp. 179-180&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'-ct'=\frac{1+\beta}{\alpha}\left(x-ct\right)\\
x'+ct'=\frac{1-\beta}{\alpha}\left(x+ct\right)\\
\left[\alpha=\sqrt{1-\beta^{2}}\right]\\
\hline \eta=x-ct,\ \xi=x+ct\\
\xi\eta=(x-ct)(x+ct)=x^{2}-c^{2}t^{2}\\
\eta=\frac{1}{\xi}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of β.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|apo}}
*{{#section:History of Topics in Special Relativity/mathsource|apo2}}
*{{#section:History of Topics in Special Relativity/mathsource|bia86lez}} 
*{{#section:History of Topics in Special Relativity/mathsource|bia94diff}} 
*{{#section:History of Topics in Special Relativity/relsource|bornrel2}}
*{{#section:History of Topics in Special Relativity/mathsource|dar83cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar94cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eis0586lez}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|hask}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74a}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie79a}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie79b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie80}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie83}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lip86}} 
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|reyn}}
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}
*{{#section:History of Topics in Special Relativity/mathsource|spei}}
*{{#section:History of Topics in Special Relativity/mathsource|whis}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L10}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
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      <text bytes="42859" sha1="mcr7zi11muj1jl7tplwi4c6n684q5tf" xml:space="preserve">{{../Lorentz transformation (header)}}
==Lorentz transformation via squeeze mappings==
[[File:Hyperbolic sector squeeze mapping.svg|250px|right|thumb|A squeeze mapping relates blue and green parallelograms.]]
As already indicated in [[../Lorentz transformation (hyperbolic)#math_3c|E:'''(3c)''']] in exponential form or [[../Lorentz transformation (Möbius)#math_6f|E:'''(6f)''']] in terms of Cayley–Klein parameter, Lorentz boosts in terms of hyperbolic rotations can be expressed as [[w:squeeze mapping]]s. Using [[w:hyperbola#Hyperbola with equation y = A/x|w:asymptotic coordinates of a hyperbola]] (''u,v''), in relativity also known as [[w:light-cone coordinates]], they have the general form (some authors alternatively add a factor of 2 or &lt;math&gt;\sqrt{2}&lt;/math&gt;):&lt;ref name=terng&gt;Terng &amp; Uhlenbeck (2000), p. 21&lt;/ref&gt;

{{NumBlk|:|&lt;math&gt;\begin{matrix} &amp; \begin{array}{c}
u=x_{0}-x_{1},\ v=x_{0}+x_{1}\\
u'=x_{0}^{\prime}-x_{1}^{\prime},\ v'=x_{0}^{\prime}+x_{1}^{\prime}
\end{array}\\
\hline (1) &amp; (u',v')=\left(ku,\ \frac{1}{k}v\right)\\
(2) &amp; (u',v')=\left(\frac{1}{k}v,\ ku\right)\\
\hline  &amp; u'v'=uv
\end{matrix}&lt;/math&gt;|{{equationRef|9a}}}}

with arbitrary ''k''. This geometrically corresponds to the transformation of one parallelogram to other ones of same area, whose sides touch a hyperbola and both asymptotes. While equation system (1) corresponds to proper Lorentz boosts, equation system (2) produces improper ones. For instance, solving (1) for &lt;math&gt;x'_0, x'_1&lt;/math&gt; gives:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =\frac{1}{2}\left(k+\frac{1}{k}\right)x_{0}-\frac{1}{2}\left(k-\frac{1}{k}\right)x_{1} &amp;  &amp; =\frac{x_{0}\left(k^{2}+1\right)-x_{1}\left(k^{2}-1\right)}{2k}\\
x_{1}^{\prime} &amp; =-\frac{1}{2}\left(k-\frac{1}{k}\right)x_{0}+\frac{1}{2}\left(k+\frac{1}{k}\right)x_{1} &amp;  &amp; =\frac{-x_{0}\left(k^{2}-1\right)+x_{1}\left(k^{2}+1\right)}{2k}\\
\\
x_{0} &amp; =\frac{1}{2}\left(k+\frac{1}{k}\right)x_{0}^{\prime}+\frac{1}{2}\left(k-\frac{1}{k}\right)x_{1}^{\prime} &amp;  &amp; =\frac{x_{0}^{\prime}\left(k^{2}+1\right)+x_{1}^{\prime}\left(k^{2}-1\right)}{2k}\\
x_{1} &amp; =\frac{1}{2}\left(k-\frac{1}{k}\right)x_{0}^{\prime}+\frac{1}{2}\left(k+\frac{1}{k}\right)x_{1}^{\prime} &amp;  &amp; =\frac{x_{0}^{\prime}\left(k^{2}-1\right)+x_{1}^{\prime}\left(k^{2}+1\right)}{2k}
\end{align}
\end{matrix}&lt;/math&gt;|{{equationRef|9b}}}}

The geometrical foundation of squeeze mapping ({{equationNote|9a}}) was known for a long time since [[#Apo|Apollonius (BC)]] and was used to generate hyperbolas by [[#spei|Speidell (1688) and Whiston (1710)]]. Equation ({{equationNote|9a}}-1) was implicitly used by [[#mercator|Mercator (1668)]] and explicitly by [[#Laisant1|Laisant (1874)]] and [[#Gunther1|Günther (1880/81)]] in relation to elliptic trigonometry, or by [[#Lie2|Lie (1879-81)]], [[#Bianchi1|Bianchi (1886, 1894)]], [[#Darboux1|Darboux (1891/94)]], [[#Eisenhart|Eisenhart (1905)]] as [[w:squeeze mapping#Lie transform|Lie transform]]&lt;ref name=terng /&gt; of [[w:pseudospherical surface]]s in terms of the [[w:Sine-Gordon equation]], or by [[#Lipschitz1|Lipschitz (1885/86)]] in transformation theory. Equation ({{equationNote|9a}}-2) was given by [[#Reynaud|Reynaud (1819)]].

From that, different forms of Lorentz transformation were derived: ({{equationNote|9b}}) by [[#Lipschitz1|Lipschitz (1885/86)]], [[#Bianchi1|Bianchi (1886, 1894)]], [[#Eisenhart|Eisenhart (1905)]], trigonometric Lorentz boost [[../Lorentz transformation (trigonometric)#math_8a|E:'''(8a)''']] by [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Darboux1|Darboux (1891/94)]], and trigonometric Lorentz boost [[../Lorentz transformation (trigonometric)#math_8b|E:'''(8b)''']] by [[#Eisenhart|Eisenhart (1905)]]. Lorentz boost ({{equationNote|9b}}) was rediscovered in the framework of special relativity by [[w:Hermann Bondi]] (1964)&lt;ref&gt;Bondi (1964), p. 118&lt;/ref&gt; in terms of [[w:Bondi k-calculus]], by which ''k'' can be physically interpreted as Doppler factor. Since ({{equationNote|9b}}) is equivalent to [[../Lorentz transformation (Möbius)#math_6f|E:'''(6f)''']] in terms of Cayley–Klein parameter by setting &lt;math&gt;k=\alpha^2&lt;/math&gt;, it can be interpreted as the 1+1 dimensional special case of Lorentz Transformation [[../Lorentz transformation (Möbius)#math_6e|E:'''(6e)''']] stated by [[../Lorentz transformation (Möbius)#Gauss3|Gauss around 1800]] (posthumously published 1863), [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]], [[../Lorentz transformation (Möbius)#Bianchi2|E:Bianchi (1888)]], [[../Lorentz transformation (Möbius)#Fricke|E:Fricke (1891)]] and [[../Lorentz transformation (Möbius)#Woods|E:Woods (1895)]].

Rewriting ({{equationNote|9a}}) in terms of [[w:homogeneous coordinates]] signifies squeeze mappings of the unit hyperbola in terms of a [[w:Quadric#Projective quadric|w:projective conic]]:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\left[u,v\right]=\left[\frac{y_{1}}{y_{3}},\frac{y_{2}}{y_{3}}\right]\quad\left(uv=1\quad\Rightarrow\quad y_{1}y_{2}-y_{3}^{2}=0\right)\\
\left[k,\frac{1}{k}\right]=\left[\frac{\alpha_{1}}{\alpha_{3}},\frac{\alpha_{2}}{\alpha_{3}}\right]\quad\left(k\frac{1}{k}=1\quad\Rightarrow\quad\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)\\
\hline y'_{1}=\alpha_{1}y_{1}\\
y'_{2}=\alpha_{2}y_{2}\\
y'_{3}=\alpha_{3}y_{3}\\
\hline y_{1}y_{2}-y_{3}^{2}=y'_{1}y'_{2}-y_{3}^{\prime2}=0\\
\hline uv=\frac{y_{1}y_{2}}{y_{3}^{2}}=u'v'=\frac{y'_{1}y'_{2}}{y_{3}^{\prime2}}
\end{matrix}&lt;/math&gt;|{{equationRef|9c}}}}

Such transformations were given by [[#Klein|Klein (1871)]] to express motions in non-Euclidean space.

Furthermore, variables ''u, v'' in ({{equationNote|9a}}) can be rearranged to produce another form of squeeze mapping, resulting in Lorentz transformation [[../Lorentz transformation (Cayley-Hermite)#math_5b|E:'''(5b)''']] in terms of Cayley-Hermite parameter:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{matrix}\begin{matrix}u=x_{0}-x_{1}\\
v=x_{0}+x_{1}\\
u'=x_{0}^{\prime}-x_{1}^{\prime}\\
v'=x_{0}^{\prime}+x_{1}^{\prime}
\end{matrix}\Rightarrow\begin{matrix}u_{1}=x_{0}+x_{0}^{\prime}\\
v_{1}=x_{0}-x_{0}^{\prime}\\
u_{2}=x_{1}-x_{1}^{\prime}\\
v_{2}=x_{1}+x_{1}^{\prime}
\end{matrix}\\
\hline (u_{2},v_{2})=\left(au_{1},\ \frac{1}{a}v_{1}\right)\Rightarrow u_{2}v_{2}=u_{1}v_{1}\\
(u',v')=\left(\frac{1+a}{1-a}u,\ \frac{1-a}{1+a}v\right)\Rightarrow u'v'=uv
\end{matrix}\Rightarrow\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime2}+x_{1}^{\prime2}\\
\hline \begin{align}x_{0}^{\prime} &amp; =x_{0}\frac{1+a^{2}}{1-a^{2}}-x_{1}\frac{2a}{1-a^{2}} &amp;  &amp; =\frac{x_{0}\left(1+a^{2}\right)-x_{1}2a}{1-a^{2}}\\
x_{1}^{\prime} &amp; =-x_{0}\frac{2a}{1-a^{2}}+x_{1}\frac{1+a^{2}}{1-a^{2}} &amp;  &amp; =\frac{-x_{0}2a+x_{1}\left(1+a^{2}\right)}{1-a^{2}}\\
\\
x_{0} &amp; =x_{0}^{\prime}\frac{1+a^{2}}{1-a^{2}}+x_{1}^{\prime}\frac{2a}{1-a^{2}} &amp;  &amp; =\frac{x_{0}^{\prime}\left(1+a^{2}\right)+x_{1}^{\prime}2a}{1-a^{2}}\\
x_{1} &amp; =x_{0}^{\prime}\frac{2a}{1-a^{2}}+x_{1}^{\prime}\frac{1+a^{2}}{1-a^{2}} &amp;  &amp; =\frac{x_{0}^{\prime}2a+x_{1}^{\prime}\left(1+a^{2}\right)}{1-a^{2}}
\end{align}
\end{matrix}\,&lt;/math&gt;|{{equationRef|9d}}}}

These Lorentz transformations were given (up to a sign change) by [[#Laguerre|Laguerre (1882)]], [[#Darboux2|Darboux (1887)]], [[#Smith|Smith (1900)]] in relation to Laguerre geometry.

On the basis of factors ''k'' or ''a'', all previous Lorentz boosts [[../Lorentz transformation (hyperbolic)#math_3b|E:'''(3b)''']], [[../Lorentz transformation (velocity)#math_4a|E:'''(4a)''']], [[../Lorentz transformation (trigonometric)#math_8a|E:'''(8a)''']], [[../Lorentz transformation (trigonometric)#math_8b|E:'''(8b)''']], can be expressed as squeeze mappings as well:

{{NumBlk|:|&lt;math&gt;\scriptstyle\begin{array}{r|c|c|c|c|c|c}
 &amp; (9a) &amp; (9d) &amp; (3b) &amp; (4a) &amp; (8a) &amp; (8b)\\
\hline \frac{u'}{u}=\frac{x_{0}^{\prime}-x_{1}^{\prime}}{x_{0}-x_{1}}= &amp; k &amp; \frac{1+a}{1-a} &amp; e^{\eta} &amp; \sqrt{\tfrac{1+\beta}{1-\beta}} &amp; \frac{1+\sin\theta}{\cos\theta} &amp; \frac{1+\cos\vartheta}{\sin\vartheta}=\cot\frac{\vartheta}{2}\\
\hline \frac{u_{2}}{u_{1}}=\frac{x_{1}-x_{1}^{\prime}}{x_{0}+x_{0}^{\prime}}= &amp; \frac{k-1}{k+1} &amp; a &amp; \tanh\frac{\eta}{2} &amp; \frac{\gamma-1}{\beta\gamma} &amp; \frac{1-\cos\theta}{\sin\theta}=\tan\frac{\theta}{2} &amp; \frac{1-\sin\vartheta}{\cos\vartheta}\\
\hline  &amp; \frac{k^{2}-1}{k^{2}+1} &amp; \frac{2a}{1+a^{2}} &amp; \tanh\eta &amp; \beta &amp; \sin\theta &amp; \cos\vartheta\\
\hline  &amp; \frac{k^{2}+1}{2k} &amp; \frac{1+a^{2}}{1-a^{2}} &amp; \cosh\eta &amp; \gamma &amp; \sec\theta &amp; \csc\vartheta\\
\hline  &amp; \frac{k^{2}-1}{2k} &amp; \frac{2a}{1-a^{2}} &amp; \sinh\eta &amp; \beta\gamma &amp; \tan\theta &amp; \cot\vartheta
\end{array}&lt;/math&gt;|{{equationRef|9e}}}}

Squeeze mappings in terms of &lt;math&gt;\theta&lt;/math&gt; were used by [[#Darboux1|Darboux (1891/94)]] and [[#Bianchi1|Bianchi (1894)]], in terms of &lt;math&gt;\eta&lt;/math&gt; implicitly by [[#mercator|Mercator (1668)]] and explicitly by [[#Lindemann|Lindemann (1891)]], [[#Elliott|Elliott (1903)]], [[#Herglotz|Herglotz (1909/10)]], in terms of &lt;math&gt;\vartheta&lt;/math&gt; by [[#Eisenhart|Eisenhart (1905)]], in terms of &lt;math&gt;\beta&lt;/math&gt; by [[#Born|Born (1921)]], [[w:Bondi k-calculus|w:Milne (1935) and Bondi (1964)]].

==Historical notation==
==={{anchor|Apo}} Apollonius (BC) – Hyperbola mapping ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Apo|label 1=History of Lorentz transformations in general § Apollonius}}
[[File:Apollonius-Halley-XII.png|thumb|&lt;small&gt;Halley's (1710) illustration of Apollonius prop. XII, showing ΔΕ·ΔΖ=HK·ΘΗ.&lt;/small&gt;]]
[[File:Apollonius-Halley-XIII.png|thumb|&lt;small&gt;Halley's (1710) illustration of Apollonius prop. XIII, showing ΓH·HΘ=ΛK·KΔ (also equal to AΛ·ΛK and EM·MΞ), whereas AE·EZ is smaller.&lt;/small&gt;]]
[[w:Apollonius of Perga]] (c. 240–190 BC, and maybe other Greek geometers such as [[w:Menaechmus]] even earlier) defined the following proposition Nr. XII in his second book on conic sections, which was translated into Latin several times by Giovanni Battista Memmo (1537), [[w:Federico Commandino]] (1566), [[w:Isaac Barrow]] (1675), and in particular by [[w:Edmond Halley]] (1710), with the Halley translation reading as follows:&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XII of book II on p. 114; Latin: &quot;Si ab aliquo puncto eorum, qua sunt in sectione, ad asymptotos duæ rectæ lineæ in quibuslibet angulis ducantur, &amp; ab alio quovis puncto in sectione sumpto ducantur aliæ rectæ his ipsis parallelæ : rectangulum sub parallelis contentum æquale erit contento sub rectis ipsis quibus ductæ fuerant parallelae.&lt;br&gt;
Sit hyperbola, cujus asymptoti AB, BΓ sumatur in sectione aliquod punctum Δ, atque ab eo ad ΑΒ, ΒΓ, ducantur ΔΕ, ΔΓ; sumatur autem &amp; alterum punctum H in sectione, per quod ducantur HΘ, HK ipsis ΔΕ, ΔΖ parallelæ: dico rectangulum EΔZ rectangulo ΘHK æquale esse.&lt;br&gt;
Jungatur enim ΔH, &amp; ad A, Γ producatur. itaque quoniam rectangulum AΔΓ aequatur rectangulo AHΓ; erit ut AH ad AΔ ita ΔΓ ad ΓΗ. sed ut AH ad ΑΔ ita ΗΘ ad ΕΔ, &amp; ut ΔΓ ad ΓΗ ita ΔZ ad ΗΚ; quare ut ΘΗ ad ΔΕ ita ΔZ ad ΗK: rectangulum igitur ΕΔZ rectangulo ΘHK est æquale.&quot;&lt;/ref&gt;

:Let there be a hyperbola whose asymptotes are AB, BΓ, and let some point Δ be taken in that section, from which ΔΕ, ΔΓ are drawn to ΑΒ, ΒΓ; and let another point H be taken in that section, through which HΘ, HK are drawn parallel to ΔΕ, ΔΖ: I say that the rectangle EΔZ is equal to the rectangle ΘHK.
:Let ΔH be joined, and A is connected to Γ. Therefore, since the rectangle AΔΓ is equal to the rectangle AHΓ, it follows that AH is to AΔ as ΔΓ is to ΓΗ. But AH is to ΑΔ as ΗΘ is to ΕΔ, and ΔΓ is to ΓΗ as ΔZ is to ΗΚ; wherefore as ΘΗ is to ΔΕ, so ΔZ is to ΗK: therefore the rectangle EΔZ is equal to the rectangle ΘHK. 

A modernized translation was given by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 34 (Apollonius, Book II, Prop. 12).&lt;/ref&gt;

:If ''Q, q'' be any two points on a hyperbola, and parallel straight lines ''QH, qh'' be drawn to meet one asymptote at any angle, and ''QK, qk'' (also parallel to one another) meet the other asymptote at any angle, then ''HQ·QK = hq·qk''. Let Qq meet the asymptotes in R,r. We have RQ.Qr=Rq.qr; therefore RQ:Rq=qr:Qr. But RQ:Rq=HQ:hq, and qr:Qr=qk:QK; therefore HQ:hq=qk:QK, or HQ.QK=hq.qk.

In the next proposition XIII, Apollonius showed that if a line is drawn parallel to the asymptotes, within the space between asymptotes and hyperbola, it must meet the hyperbola exactly once. In his demonstration, Apollonius used the previous proposition XII when comparing the area of several parallelograms whose sides are drawn parallel to the asymptotes.&lt;ref group=M&gt;Apollonius/Halley (1710), Prop. XIII of book II on p. 114-115&lt;/ref&gt;&lt;ref group=M&gt;Apollonius/Heath (1896), Proposition 35 (Apollonius, Book II, Prop. 13).&lt;/ref&gt;

{{Lorentzbox|Text=The ratios given by Apollonius:
:&lt;math&gt;\frac{\Delta\Gamma}{\Gamma H}=\frac{\Delta Z}{HK},\quad\frac{\Theta H}{\Delta E}=\frac{\Delta Z}{HK}&lt;/math&gt;
represent an equation system that can be solved for ΔΕ, ΔΖ, resulting in the squeeze mapping:
:&lt;math&gt;\Delta E=\frac{\Gamma H}{\Delta\Gamma}\Theta H,\quad\Delta Z=\frac{\Delta\Gamma}{\Gamma H}HK&lt;/math&gt;
producing &lt;math&gt;\Delta E\cdot\Delta Z=HK\cdot\Theta H&lt;/math&gt; in line with Apollonius result that rect. EΔZ is equal to rect. ΘHK.
In case ΔΕ, ΔΖ, HK, ΘΗ are all drawn parallel to the respective asymptotes, it follows u'=ΔΕ, v'=ΔΖ, u=HK, v=ΘΗ, k=&lt;math&gt;\tfrac{\Gamma H}{\Delta\Gamma}&lt;/math&gt; and therefore Apollonius result becomes equivalent to Lorentz boost ({{equationNote|9a}}), signifying squeezed parallelograms located between the asymptotes and the hyperbola.
In general, the identity &lt;math&gt;\Delta E\cdot\Delta Z=HK\cdot\Theta H&lt;/math&gt; demonstrates the invariance of the area of all parallelograms that are constructed in line with the proposition XII, thereby representing all points of a hyperbola defined by ''HK·ΘΗ = const''. That is, the invariant area ''HK·ΘΗ = const.'' together with ''const''=1 gives ''HK''=1/''ΘΗ'', which implies that ''ΘΗ'' is inverse proportional to ''HK''. Thus when ''HK'' is increased into ''k·HK'' using some factor ''k'', it follows that ''ΘΗ'' must be proportionally diminished into ''ΘΗ/k'' in order to preserve  invariance of area.}}

==={{anchor|mercator}} Mercator (1668) – hyperbolic relations ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#mercator|label 1=History of Lorentz transformations via hyperbolic functions § Mercator}}
[[File:Mercator-hyperbola-XIV.png|thumb|&lt;small&gt;Mercator's (1668) illustration of AH·FH=AI·BI.&lt;/small&gt;]]

While deriving the [[w:Mercator series]], [[w:Nicholas Mercator]] (1668) demonstrated Apollonius' proposition on a rectangular hyperbola algebraically as follows:&lt;ref group=M&gt;Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}AD=1+a,\ DF=\sqrt{2a+aa}\\ AH=\frac{1+a+\sqrt{2a+aa}}{\sqrt{2}},\ FH=\frac{1+a-\sqrt{2a+aa}}{\sqrt{2}}\\ AI=BI=\frac{1}{\sqrt{2}}\\ 1+a=c,\ \sqrt{2a+aa}=d,\ 1=cc-dd\\ AH*FH=\frac{cc-dd}{\sqrt{2}*\sqrt{2}}=\frac{1}{2}\\ AI*BI=\frac{1}{2}\\ \hline AH*FH=AI*BI\\ AH.AI::BI.FH \end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=It can be seen that Mercator's relations ''c'' and ''d'' implicitly correspond to hyperbolic functions ''cosh'' and ''sinh'' (which were explicitly introduced by [[../Lorentz transformation (hyperbolic)#Riccati|E:Riccati (1757)]] much later). In particular, his result AH·FH=AI·BI and AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or &lt;math&gt;\tfrac{AH}{AI}=\tfrac{BI}{FH}&lt;/math&gt; in modern notation, corresponds to squeeze mapping or Lorentz boost ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of &amp;eta; because:
:&lt;math&gt;\frac{AH}{AI}=\frac{BI}{FH}=1+a+\sqrt{2a+a^{2}}=c+d=\cosh\eta+\sinh\eta=e^{\eta}=k&lt;/math&gt;
or solved for AH and FH:
:&lt;math&gt;AH=e^{\eta} AI&lt;/math&gt; and &lt;math&gt;FH=e^{-\eta} BI&lt;/math&gt;.}}

==={{anchor|spei}} Speidell (1688), Whiston (1710) – Hyperbola generation ===
[[File:Whiston-hyperbola.png|thumb|&lt;small&gt;Whiston's (1710) illustration of generating a hyperbola by parallelograms of equal area.&lt;/small&gt;]]
The case of squeezing a given square or parallelogram as a means to ''generate'' hyperbolas was discussed by [[w:Euclid Speidell]] (1688):&lt;ref group=M&gt;Speidell (1688), pp. 4-5&lt;/ref&gt;
:[..] from a Square and an infinite company of Oblongs on a Superficies, each Equal to that Square, how a Curve is begotten which shall have the same properties and affections of an Hyperbola inscribed within a Right Angled Cone
:[..] There is a Square ''ABCD'', whose Side or Root is 10, let ''DB'' be prolonged in ''infinitum'', and continually divided equally by the Root, or ''DB'', and those Equal Divisions numbered by 10, 20, 30, 40, 50, 60, 70, &amp;c. in ''infinitum'': Upon these Numbers let Perpendiculars be erected, which call Ordinates, and each of those Perpendiculars of that length, that Perpendiculars let fall from the aforesaid Perpendiculars to the Side or Base ''CD'' (which call Complement Ordinates) the Oblongs made of the Ordinate Perpendiculars, and Complement Ordinate Perpendiculars may be ever Equal to the Square ''AD'', which is easily done thus, for it is &lt;math&gt;\tfrac{100}{20},\tfrac{100}{30},\tfrac{100}{40},\tfrac{100}{50}&lt;/math&gt; &amp;c. produces the Length of the Ordinate Perpendiculars
:[..] all the Oblongs made of the Ordinates, and Complement Ordinates are each of them equal to the Square ''AD'', which is here 100
:[..] the like Demonstration serves for all the Oblongs or Parallelograms standing upon the Base ''CD'', by the Tips or Angular Points of those Parallelograms, or from the Ends of all the Ordinates standing upon 20, 30, 40, 50, 60, 70, in ''infinitum'', draw the Curve Line from ''A'' towards ''E'', so shall you describe the Curve ''AEFGS'' [..].

{{Lorentzbox|Text=This corresponds to squeeze mappings ({{equationNote|9a}}) with ''u=v''=10 and ''k''=1,2,3,4,5,6,7,..., thus ''u'v'=uv''=100.}}

In similar terms, [[w:William Whiston]] (1710/16) wrote:&lt;ref group=M&gt;Whiston (1710/16). In the English version (1716) see pp. 16-17. In the original Latin version (1710) see pp. 16-18&lt;/ref&gt;
:But it is to be acknowledg'd, that many Properties of an Hyperbola are better known from another manner of generating the Figure; which Way is this: Let ''LL'' and ''MM'' be infinite Right Lines intersecting each other in any Angle whatever in the Point ''C'': From any Point whatever, as ''D'' or ''e'', let ''Dc, Dd,'' be drawn parallel to the first Lines, or (''ec, ed''), which with the Lines first drawn make the Parallelograms as ''DcCd'', or ''ecCd''; Now conceive two sides of the Parallelogram as ''Dc, Dd,'' or ''ec, ed'', to be so mov'd this way and that way, that they always keep the same Parallelism, and that at the same time the Area's always remain equal: That is to say, that ''Dc'' and ''ec'' remain always Parallel to ''MM'', and ''Dd'' or ''ed'' always Parallel to ''LL''; and that the Area of every Parallelogram be equal to every other, one Side being increas'd in the same Proportion wherein the other is diminish'd. By this means the Point ''D'' or ''e'' will describe a Curve-Line within the Angle comprehended by the first Lines;

{{Lorentzbox|Text=This corresponds to squeeze mappings ({{equationNote|9a}}).}}

==={{anchor|Reynaud}} Reynaud (1819) – Hyperbola mapping  ===
[[w:Antoine André Louis Reynaud]] algebraically expressed squeeze mappings by writing:&lt;ref group=M&gt;Reynaud (1819), p. 247&lt;/ref&gt;
:&quot;The system of equations &lt;math&gt;(2)\ x=\frac{y'}{\alpha},\ y=\alpha x'&lt;/math&gt; determines all points of the curve &lt;math&gt;S&lt;/math&gt;, because &lt;math&gt;x'&lt;/math&gt; and &lt;math&gt;y'&lt;/math&gt; being given numbers, each arbitrary value of &lt;math&gt;\alpha&lt;/math&gt; gives a point &lt;math&gt;x,y&lt;/math&gt; of this curve. The elimination of the indeterminate &lt;math&gt;\alpha&lt;/math&gt; between equations (2) will therefore lead to the equation &lt;math&gt;xy=x'y'&lt;/math&gt; of the curve in question. This curve is therefore a hyperbola related to its asymptotes &lt;math&gt;xX,yY&lt;/math&gt;.&quot;

{{Lorentzbox|Text=This is equivalent to (improper) Lorentz transformation ({{equationNote|9a}}-2).}}

==={{anchor|Klein}} Klein (1871) – Projective conic section===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Klein|label 1=History of Lorentz transformations in general § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Klein|label 1=History of Lorentz transformations via Möbius transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Klein3|label 1=History of Lorentz transformations via sphere transformations § Klein}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Noether|label 1=History of Lorentz transformations via Quaternions § Klein}}

Elaborating on the [[w:Cayley–Klein metric]], [[w:Felix Klein]] (1871) defined a [[w:Quadric#Projective quadric|w:projective conic]] in order to discuss motions such as rotation and translation in the non-Euclidean plane:&lt;ref group=M&gt;Klein (1871), pp. 601–602&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{1}x_{2}-x_{3}^{2}=0\\
\hline \begin{align}x_{1} &amp; =\alpha_{1}y_{1}\\
x_{2} &amp; =\alpha_{2}y_{2}\\
x_{3} &amp; =\alpha_{3}y_{3}
\end{align}
\\
\left(\alpha_{1}\alpha_{2}-\alpha_{3}^{2}=0\right)\\
\hline \frac{x_{1}x_{2}}{x_{3}^{2}}=\text{invariant}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=When the conic section is a hyperbola this is equivalent to squeeze mapping ({{equationNote|9c}}). This becomes ({{equationNote|9a}}) using &lt;math&gt;\left[u,v\right]=\left[\tfrac{x_{1}}{x_{3}},\tfrac{x_{2}}{x_{3}}\right],\ \left[k,\tfrac{1}{k}\right]=\left[\tfrac{\alpha_{1}}{\alpha_{3}},\tfrac{\alpha_{2}}{a_{3}}\right]&lt;/math&gt;.}}

==={{anchor|Laisant1}} Laisant (1874) – Elliptic polar coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Laisant|label 1=History of Lorentz transformations via hyperbolic functions § Laisant}}

[[w:Charles-Ange Laisant]] extended circular trigonometry to elliptic trigonometry. In his model, polar coordinates x, y of circular trigonometry are related to polar coordinates x', y' of elliptic trigonometry by the relation&lt;ref group=M&gt;Laisant (1874a), pp. 73–76&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=ax,\ y'=\frac{y}{a}\\
x'y'=xy
\end{matrix}&lt;/math&gt;

He noticed the geometrical implication that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}).}}

==={{anchor|Lie2}} Lie (1879-84) – Transforming pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (general)#Lie3|label 1=History of Lorentz transformations in general § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (imaginary)#Lie|label 1=History of Lorentz transformations via imaginary orthogonal transformations § Lie}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Lie|label 1=History of Lorentz transformations via sphere transformations § Lie}}

[[w:Sophus Lie]] (1879/80) derived an operation from [[w:Pierre Ossian Bonnet]]'s (1867) investigations on surfaces of constant curvatures, by which pseudospherical surfaces can be transformed into each other.&lt;ref group=M&gt;Lie (1879/80), Collected papers, vol. 3, p. 389&lt;/ref&gt; Lie gave explicit formulas for this operation in two papers published in 1881: If &lt;math&gt;(s,\sigma)&lt;/math&gt; are asymptotic coordinates of two principal tangent curves and &lt;math&gt;\Theta&lt;/math&gt; their respective angle, and &lt;math&gt;\Theta=f(s,\sigma)&lt;/math&gt; is a solution of the Sine-Gordon equation &lt;math&gt;\tfrac{d^{2}\Theta}{ds\ d\sigma}=K\sin\Theta&lt;/math&gt;, then the following operation (now called Lie transform) is also a solution from which infinitely many new surfaces of same curvature can be derived:&lt;ref group=M&gt;Lie (1879/81), Collected papers, vol. 3, p. 393&lt;/ref&gt;

:&lt;math&gt;\Theta=f(s,\sigma)\Rightarrow\Theta=f\left(ms,\ \frac{\sigma}{m}\right)&lt;/math&gt;

In (1880/81) he wrote these relations as follows:&lt;ref group=M&gt;Lie (1880/81), Collected papers, vol. 3, pp. 477–478&lt;/ref&gt;

:&lt;math&gt;\vartheta=\Phi(s,S)\Rightarrow\vartheta=\Phi\left(ms,\ \frac{S}{m}\right)&lt;/math&gt;

In (1883/84) he showed that the combination of Lie transform ''O'' with Bianchi transform ''I'' produces [[w:Bäcklund transform]] ''B'' of pseudospherical surfaces:&lt;ref group=M&gt;Lie (1883/84), Collected papers, vol. 3, p. 556&lt;/ref&gt;

:&lt;math&gt;B=OIO^{-1}&lt;/math&gt;

{{Lorentzbox|Text=As shown by [[#Bianchi1|Bianchi (1886)]] and [[#Darboux1|Darboux (1891/94)]], the Lie transform is equivalent to Lorentz transformations ({{equationNote|9a}}) and ({{equationNote|9b}}) in terms of light-cone coordinates ''2s=u+v'' and ''2σ=u-v''. In general, it can be shown that the Sine-Gordon equation is Lorentz invariant.}}

==={{anchor|Gunther1}} Günther (1880/81) – Elliptic polar coordinates ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Gunther1|label 1=History of Lorentz transformations via hyperbolic functions § Günther}}

Following [[#Laisant1|Laisant (1874)]], [[w:Siegmund Günther]] demonstrated the relation between circular polar coordinates and elliptic polar coordinates as&lt;ref group=M&gt;Günther (1880/81), pp. 383–385&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'=ax,\ y'=\frac{1}{a}y\\
x'y'=xy
\end{matrix}&lt;/math&gt;

showing that any elliptic polar system of coordinates obtained by this formula is located on the same equilateral hyperbola having its asymptotes as axes.

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}).}}

==={{anchor|Laguerre}} Laguerre (1882) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Laguerre|label 1=History of Lorentz transformations via sphere transformations § Laguerre}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Laguerre|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Laguerre}}

A transformation (later known as &quot;Laguerre inversion&quot;) of [[../Lorentz transformation (conformal)|E:oriented lines and spheres]] was given by [[w:Edmond Laguerre]] with ''R'' being the radius and ''D'' the distance of its center to the axis:&lt;ref group=M name=laguerre&gt;Laguerre (1882), pp. 550–551.&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}D^{2}-D^{\prime2}=R^{2}-R^{\prime2}\\
\hline \left.\begin{align}D' &amp; =\frac{D\left(1+\alpha^{2}\right)-2\alpha R}{1-\alpha^{2}}\\
R' &amp; =\frac{2\alpha D-R\left(1+\alpha^{2}\right)}{1-\alpha^{2}}
\end{align}
\right|\begin{align}D-D' &amp; =\alpha(R-R')\\
D+D' &amp; =\frac{1}{\alpha}(R+R')
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change for ''R'') to a squeeze mapping in terms of Lorentz boost ({{equationNote|9d}}).}}

==={{anchor|Darboux1}} Darboux (1883–1891) ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Darboux|label 1=History of Lorentz transformations via sphere transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Darboux2|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Darboux}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Darboux1|label 1=History of Lorentz transformations via trigonometric functions § Darboux}}
====Transforming pseudospherical surfaces====

[[w:Gaston Darboux]] (1883) followed [[#Lie2|Lie (1879/81)]] by transforming pseudospheres into each other as follows:&lt;ref group=M&gt;Darboux (1883), p. 849&lt;/ref&gt;

:&lt;math&gt;f(x,y)\Rightarrow f\left(\frac{x}{m},\ ym\right)&lt;/math&gt;

{{Lorentzbox|Text=This becomes Lorentz boost ({{equationNote|9a}}) by interpreting ''x, y'' as light-cone coordinates.}}

Similar to [[#Bianchi1|Bianchi (1886)]], Darboux (1891/94) showed that the Lie transform gives rise to the following relations:&lt;ref group=M&gt;Darboux (1891/94), pp. 381–382&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; u+v=2\alpha,\ u-v=2\beta;\\
(2)\quad &amp; \omega=\varphi\left(\alpha,\beta\right)\Rightarrow\omega=\varphi\left(\alpha m,\ \frac{\beta}{m}\right)\\
(3)\quad &amp; \omega=\psi(u,v)\Rightarrow\omega=\psi\left(\frac{u+v\sin h}{\cos h},\ \frac{v+u\sin h}{\cos h}\right)
\end{align}
&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost ({{equationNote|9a}}) in terms of light-cone coordinates.}}

===={{anchor|Darboux2}} Laguerre inversion====

Following [[#Laguerre|Laguerre (1882)]], Darboux (1887) formulated the Laguerre inversions in four dimensions using coordinates ''x,y,z,R'':&lt;ref group=M name=darboux&gt;Darboux (1887)&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x^{\prime2}+y^{\prime2}+z^{\prime2}-R^{\prime2}=x^{2}+y^{2}+z^{2}-R^{2}\\
\hline \left.\begin{align}x' &amp; =x, &amp; z' &amp; =\frac{1+k^{2}}{1-k^{2}}z-\frac{2kR}{1-k^{2}},\\
y' &amp; =y, &amp; R' &amp; =\frac{2kz}{1-k^{2}}-\frac{1+k^{2}}{1-k^{2}}R,
\end{align}
\right|\begin{align}z'+R' &amp; =\frac{1+k}{1-k}(z-R)\\
z'-R' &amp; =\frac{1-k}{1+k}(z+R)
\end{align}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change for ''R'') to a squeeze mapping in terms of Lorentz boost ({{equationNote|9d}}) where Darboux's ''k'' corresponds to ''a''.}}

==={{anchor|Lipschitz1}} Lipschitz (1885/86) - Quadratic forms ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Lipschitz1|label 1=History of Lorentz transformations via hyperbolic functions § Lipschitz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Quaternions)#Lipschitz2|label 1=History of Lorentz transformations via Quaternions § Lipschitz}}

[[w:Rudolf Lipschitz]] (1885/86) discussed transformations leaving invariant the sum of squares
:&lt;math&gt;x_{1}^{2}+x_{2}^{2}\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}&lt;/math&gt;
which he rewrote as 
:&lt;math&gt;x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots+x_{n}^{2}-y_{n}^{2}=0&lt;/math&gt;. 
This led to the problem of finding transformations leaving invariant the pairs &lt;math&gt;x_{a}^{2}-y_{a}^{2}&lt;/math&gt; (where ''a=1...n'') for which he gave the following solution:&lt;ref group=M&gt;Lipschitz (1886), pp. 90–92&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x_{a}^{2}-y_{a}^{2}=\mathfrak{x}_{a}^{2}-\mathfrak{y}_{a}^{2}\\
\hline \begin{align}x_{a}-y_{a} &amp; =\left(\mathfrak{x}_{a}-\mathfrak{y}_{a}\right)r_{a}\\
x_{a}+y_{a} &amp; =\left(\mathfrak{x}_{a}+\mathfrak{y}_{a}\right)\frac{1}{r_{a}}
\end{align}
\quad(1)\\
\hline\begin{align}2\mathfrak{x}_{a} &amp; =\left(r_{a}+\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}-\frac{1}{r_{a}}\right)y_{a}\\
2\mathfrak{y}_{a} &amp; =\left(r_{a}-\frac{1}{r_{a}}\right)x_{a}+\left(r_{a}+\frac{1}{r_{a}}\right)y_{a}
\end{align}
\quad(2)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=Equation system (1) represents Lorentz boost or squeeze mapping ({{equationNote|9a}}), and (2) represents Lorentz boost ({{equationNote|9b}}).}}

==={{anchor|Bianchi1}} Bianchi (1886–1894) – Transforming pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Bianchi2|label 1=History of Lorentz transformations via Möbius transformations § Bianchi}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Bianchi1|label 1=History of Lorentz transformations via trigonometric functions § Bianchi}}

[[w:Luigi Bianchi]] (1886) followed [[#Lie2|Lie (1879/80)]] by writing the transformation of pseudospheres into each other, obtaining the result:&lt;ref group=M&gt;Bianchi (1886), eq. 1 can be found on p. 226, eq. (2) on p. 240, eq. (3) on pp. 240–241, and for eq. (4) see the footnote on p. 240.&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; u+v=2\alpha,\ u-v=2\beta;\\
(2)\quad &amp; \Omega\left(\alpha,\beta\right)\Rightarrow\Omega\left(k\alpha,\ \frac{\beta}{k}\right);\\
(3)\quad &amp; \theta(u,v)\Rightarrow\theta\left(\frac{u+v\sin\sigma}{\cos\sigma},\ \frac{u\sin\sigma+v}{\cos\sigma}\right)=\Theta_{\sigma}(u,v);\\
 &amp; \text{Inverse:}\left(\frac{u-v\sin\sigma}{\cos\sigma},\ \frac{-u\sin\sigma+v}{\cos\sigma}\right)\\
(4)\quad &amp; \frac{1}{2}\left(k+\frac{1}{k}\right)=\frac{1}{\cos\sigma},\ \frac{1}{2}\left(k-\frac{1}{k}\right)=\frac{\sin\sigma}{\cos\sigma}
\end{align}
&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost ({{equationNote|9a}}) in terms of light-cone coordinates. Plugging equations (4) into (3) gives Lorentz boost ({{equationNote|9b}}) in terms of Bondi's ''k'' factor.}}

In 1894, Bianchi redefined the variables ''u,v'' as asymptotic coordinates, by which the transformation obtains the form:&lt;ref group=M&gt;Bianchi (1894), pp. 433–434&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\Omega\left(u,v\right)\Rightarrow\omega(u,v);\quad\Omega\left(u,v\right)=\omega\left(ku,\ \frac{v}{k}\right);\\
k=\frac{1+\sin\sigma}{\cos\sigma}\Rightarrow\Omega\left(u,v\right)=\omega\left(\frac{1+\sin\sigma}{\cos\sigma}u,\ \frac{1-\sin\sigma}{\cos\sigma}v\right)
\end{matrix}&lt;/math&gt;.

{{Lorentzbox|Text=This is consistent with one of the choices in ({{equationNote|9e}}) where Bianchi's angle σ corresponds to θ.}}

==={{anchor|Lindemann}} Lindemann (1890/91) – Weierstrass coordinates and Cayley absolute===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Lindemann|label 1=History of Lorentz transformations via hyperbolic functions § Lindemann}}

[[w:Ferdinand von Lindemann]] employed the Cayley absolute related to surfaces of second degree and its transformation&lt;ref group=M&gt;Lindemann &amp; Clebsch (1890/91), pp. 361–362&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\
X_{1}X_{4}+X_{2}X_{3}=\Xi_{1}\Xi_{4}+\Xi_{2}\Xi_{3}\\
\hline \begin{align}X_{1} &amp; =\left(\lambda+\lambda_{1}\right)U_{4} &amp; \Xi_{1} &amp; =\left(\lambda-\lambda_{1}\right)U_{4} &amp; X_{1} &amp; =\frac{\lambda+\lambda_{1}}{\lambda-\lambda_{1}}\Xi_{1}\\
X_{2} &amp; =\left(\lambda+\lambda_{3}\right)U_{4} &amp; \Xi_{2} &amp; =\left(\lambda-\lambda_{3}\right)U_{4} &amp; X_{2} &amp; =\frac{\lambda+\lambda_{3}}{\lambda-\lambda_{3}}\Xi_{2}\\
X_{3} &amp; =\left(\lambda-\lambda_{3}\right)U_{2} &amp; \Xi_{3} &amp; =\left(\lambda+\lambda_{3}\right)U_{2} &amp; X_{3} &amp; =\frac{\lambda-\lambda_{3}}{\lambda+\lambda_{3}}\Xi_{3}\\
X_{4} &amp; =\left(\lambda-\lambda_{1}\right)U_{1} &amp; \Xi_{4} &amp; =\left(\lambda+\lambda_{1}\right)U_{1} &amp; X_{4} &amp; =\frac{\lambda-\lambda_{1}}{\lambda+\lambda_{1}}\Xi_{4}
\end{align}
\end{matrix}&lt;/math&gt;

into which he put&lt;ref group=M name=linde&gt;Lindemann &amp; Clebsch (1890/91), p. 496&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}\begin{align}X_{1} &amp; =x_{1}+2kx_{4}, &amp; X_{2} &amp; =x_{2}+ix_{3}, &amp; \lambda+\lambda_{1} &amp; =\left(\lambda-\lambda_{1}\right)e^{a},\\
X_{4} &amp; =x_{1}-2kx_{4}, &amp; X_{3} &amp; =x_{2}-ix_{3}, &amp; \lambda+\lambda_{3} &amp; =\left(\lambda-\lambda_{3}\right)e^{\alpha i},
\end{align}
\\
\hline \Omega_{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\\
ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to squeeze mapping ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of &amp;eta; with &lt;math&gt;e^{\alpha i}=1&lt;/math&gt; and ''2k=1'' .}}

==={{anchor|Haskell}} Haskell (1895) – Hyperbola mapping  ===

[[w:Mellen W. Haskell]] applied the linear transformation 
:&lt;math&gt;\alpha'=k\alpha,\ \beta'=k^{-1}\beta&lt;/math&gt; 
in order to transform a hyperbola into itself.&lt;ref group=M&gt;Haskell (1895), p. 159&lt;/ref&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}).}}
==={{anchor|Smith}} Smith (1900) – Laguerre inversion===
{{See also|History of Topics in Special Relativity/Lorentz transformation (conformal)#Smith|label 1=History of Lorentz transformations via sphere transformations § Smith}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Cayley-Hermite)#Smith|label 1=History of Lorentz transformations via Cayley-Hermite transformations § Smith}}

[[w:Percey F. Smith]] followed [[#Laguerre|Laguerre (1882)]] and [[#Darboux2|Darboux (1887)]] and defined the Laguerre inversion as follows:&lt;ref group=M&gt;Smith (1900), p. 159&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}p^{\prime2}-p^{2}=R^{\prime2}-R^{2}\\
\hline \kappa=\frac{R'-R}{p'-p}\\
p'=\frac{\kappa^{2}+1}{\kappa^{2}-1}p-\frac{2\kappa}{\kappa^{2}-1}R,\quad R'=\frac{2\kappa}{\kappa^{2}-1}p-\frac{\kappa^{2}+1}{\kappa^{2}-1}R
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent (up to a sign change) to Lorentz transformation ({{equationNote|9d}}).}}

==={{anchor|Elliott}} Elliott (1903) – Invariant theory ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Elliott|label 1=History of Lorentz transformations via hyperbolic functions § Elliott}}

[[w:Edwin Bailey Elliott]] (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:&lt;ref group=M&gt;Elliott (1903), p. 109&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x=X\cosh\phi+Y\sinh\phi,\quad y=X\sinh\phi+Y\cosh\phi\\
\hline X+Y=e^{-\phi}(x+y),\quad X-Y=e^{\phi}(x-y)
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=The second line is equivalent to squeeze mapping or Lorentz boost ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of &amp;eta;.}}

=== {{anchor|Eisenhart}} Eisenhart (1905) – Transforming pseudospherical surfaces===
{{See also|History of Topics in Special Relativity/Lorentz transformation (trigonometric)#Eisenhart|label 1=History of Lorentz transformations via trigonometric functions § Eisenhart}}

[[w:Luther Pfahler Eisenhart]] followed [[#Lie2|Lie (1879/81)]], [[#Bianchi1|Bianchi (1886, 1894)]] and [[#Darboux1|Darboux (1891/94)]] in transforming pseudospherical surfaces:&lt;ref group=M&gt;Eisenhart (1905), p. 126&lt;/ref&gt;

:&lt;math&gt;\begin{align}(1)\quad &amp; \alpha=\frac{u+v}{2},\ \beta=\frac{u-v}{2}\\
(2)\quad &amp; \omega\left(\alpha,\beta\right)\Rightarrow\omega\left(m\alpha,\ \frac{\beta}{m}\right)\\
(3)\quad &amp; \omega(u,v)\Rightarrow\omega(\alpha+\beta,\ \alpha-\beta)\Rightarrow\omega\left(\alpha m+\frac{\beta}{m},\ \alpha m-\frac{\beta}{m}\right)\\
 &amp; \Rightarrow\omega\left[\frac{\left(m^{2}+1\right)u+\left(m^{2}-1\right)v}{2m},\ \frac{\left(m^{2}-1\right)u+\left(m^{2}+1\right)v}{2m}\right]\\
(4)\quad &amp; m=\frac{1-\cos\sigma}{\sin\sigma}\Rightarrow\omega\left(\frac{u-v\cos\sigma}{\sin\sigma},\ \frac{v-u\cos\sigma}{\sin\sigma}\right)
\end{align}&lt;/math&gt;.

{{Lorentzbox|Text=Equations (1) together with transformation (2) gives Lorentz boost ({{equationNote|9a}}) in terms of light-cone coordinates. Transformation (3) is equivalent to Lorentz boost ({{equationNote|9b}}) in terms of Bondi's ''k'' factor. Eisenhart's angle σ corresponds to ϑ in ({{equationNote|9e}}).}}

=== {{anchor|Herglotz}} Herglotz (1909/10) – Special relativity ===
{{See also|History of Topics in Special Relativity/Lorentz transformation (hyperbolic)#Herglotz1|label 1=History of Lorentz transformations via hyperbolic functions § Herglotz}}
{{See also|History of Topics in Special Relativity/Lorentz transformation (Möbius)#Herglotz1|label 1=History of Lorentz transformations via Möbius transformations § Herglotz}}

In relation to special relativity, [[w:Gustav Herglotz]] (1909/10) defined the Lorentz boost as follows:&lt;ref group=M&gt;Herglotz (1909/10), pp. 404-408&lt;/ref&gt;

:&lt;math&gt;\begin{align}x &amp; =x', &amp; t-z &amp; =(t'-z')e^{\vartheta}\\
y &amp; =y', &amp; t+z &amp; =(t'+z')e^{-\vartheta}
\end{align}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of η.}}

==={{anchor|Born}} Born (1921) – Special relativity===
In the second edition of “Einstein's theory of relativity”, [[w:Max Born]] (1921) discussed the relation of the Lorentz transformation and the hyperbola:&lt;ref group=M&gt;Born (1921), pp. 179-180&lt;/ref&gt;

:&lt;math&gt;\begin{matrix}x'-ct'=\frac{1+\beta}{\alpha}\left(x-ct\right)\\
x'+ct'=\frac{1-\beta}{\alpha}\left(x+ct\right)\\
\left[\alpha=\sqrt{1-\beta^{2}}\right]\\
\hline \eta=x-ct,\ \xi=x+ct\\
\xi\eta=(x-ct)(x+ct)=x^{2}-c^{2}t^{2}\\
\eta=\frac{1}{\xi}
\end{matrix}&lt;/math&gt;

{{Lorentzbox|Text=This is equivalent to Lorentz transformation ({{equationNote|9a}}) as well as ({{equationNote|9e}}) in terms of β.}}

==References==
===Historical mathematical sources===
{{reflist|3|group=M}}
*{{#section:History of Topics in Special Relativity/mathsource|apo}}
*{{#section:History of Topics in Special Relativity/mathsource|apo2}}
*{{#section:History of Topics in Special Relativity/mathsource|bia86lez}} 
*{{#section:History of Topics in Special Relativity/mathsource|bia94diff}} 
*{{#section:History of Topics in Special Relativity/relsource|bornrel2}}
*{{#section:History of Topics in Special Relativity/mathsource|dar83cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar87cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|dar94cou}} 
*{{#section:History of Topics in Special Relativity/mathsource|eis0586lez}} 
*{{#section:History of Topics in Special Relativity/mathsource|eli03}} 
*{{#section:History of Topics in Special Relativity/mathsource|guen80}} 
*{{#section:History of Topics in Special Relativity/mathsource|hask}}
*{{#section:History of Topics in Special Relativity/relsource|herg10}}
*{{#section:History of Topics in Special Relativity/mathsource|klei71}} 
*{{#section:History of Topics in Special Relativity/mathsource|lagu82}} 
*{{#section:History of Topics in Special Relativity/mathsource|lais74a}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie79a}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie79b}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie80}} 
*{{#section:History of Topics in Special Relativity/mathsource|lie83}} 
*{{#section:History of Topics in Special Relativity/mathsource|lind90}} 
*{{#section:History of Topics in Special Relativity/mathsource|lip86}} 
*{{#section:History of Topics in Special Relativity/mathsource|merc}}
*{{#section:History of Topics in Special Relativity/mathsource|reyn}}
*{{#section:History of Topics in Special Relativity/mathsource|smi00}}
*{{#section:History of Topics in Special Relativity/mathsource|spei}}
*{{#section:History of Topics in Special Relativity/mathsource|whis}}

===Secondary sources===
{{reflist|3}}
{{#section:History of Topics in Special Relativity/secsource|L10}}

[[Category:Lorentz transformation]]
[[Category:History of special relativity]]</text>
      <sha1>mcr7zi11muj1jl7tplwi4c6n684q5tf</sha1>
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      <text bytes="46213" sha1="ffhniozrw55mxydi67lwcmje4518qsu" xml:space="preserve">'''Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs at the Planck scale'''

An orbital simulation program is described that emulates both gravitational and atomic orbitals as the sum of individual particle-particle orbital pair rotations. The simulation is dimensionless, the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]], however it can translate to the  [[w:Planck_units |Planck units]] for comparison with real world orbits &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=XXXXXXXXXXXXXXXXXXXXXXXXXXX |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.11496.93445/17}}&lt;/ref&gt;.

[[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point (unit of mass) accordingly, we can 'design' the required orbits. No other parameters are used. The 26 points orbit each other resulting in 325 point-point orbitals.]]
For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''&lt;sub&gt;P&lt;/sub&gt; (for example, a 1kg satellite would be divided into 1kg/''m''&lt;sub&gt;P&lt;/sub&gt; = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs . 

Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... 

The base simulation requires only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). 

For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' is added in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although only the mass state of the electron is mapped during transition, periodically the spiral angles converge to give an integer orbital radius, the transition steps between these radius can then be used to solve the transition frequency. And so although mapping a gravitational orbit on a 2-D plane, a radial quantization (as a function of pi and so of geometrical origin) emerges, (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). In this context it is thus not necessary to develop a separate `quantum' theory of gravity.


=== Theory ===
In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is an oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model.

If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10&lt;sup&gt;23&lt;/sup&gt; units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10&lt;sup&gt;23&lt;/sup&gt; electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity becomes a mass to mass interaction. 

Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map particle to particle orbitals for which both particles are in the wave-state, the H atom is a well-researched particle-to-particle orbital pair (electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the [[v:Quantum_gravity_(Planck)#Photon_orbital_model |photon-orbital model]], in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). 

It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically cancels into integers which correspond with the orbital energy levels where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this quantization has a geometrical origin.

Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period '''t''' measured between these integer radius can be used to solve the transition frequencies '''f''' via the formula &lt;math&gt;f/c = t \lambda_H/(n_f^2-n_i^2)&lt;/math&gt;. 
 
In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The gravitational orbits that we observe are the time averaging sum of the underlying multiple gravitational orbitals. 


=== N-body orbitals ===
[[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]]

The simulation universe is a 4-axis hypersphere expanding in increments &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}&lt;/ref&gt; with 3-axis (the [[v:Black-hole_(Planck)  |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'.

The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m&lt;sub&gt;P&lt;/sub&gt;]], [[w:Planck length |Planck length l&lt;sub&gt;p&lt;/sub&gt;]], [[w:Planck time |Planck time t&lt;sub&gt;p&lt;/sub&gt;]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''&lt;sub&gt;p&lt;/sub&gt;/''t''&lt;sub&gt;p&lt;/sub&gt;) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. 

Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). 

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}}&lt;/math&gt; 

As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation.

Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt; is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence.


[[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]]
==== 2 body orbits ('''x, y''' plane) ====
For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. 

The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units

:&lt;math&gt;r_s = \frac{2 l_p M}{m_P}&lt;/math&gt;

As the simulation itself is dimensionless, we can remove the dimensioned length component &lt;math&gt;2 l_p&lt;/math&gt;, and as each point is analogous to 1 unit of Planck mass &lt;math&gt;m_P&lt;/math&gt;, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius &lt;math&gt;r_s&lt;/math&gt; = number of central points = total points - 1 (the orbiting point).


After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. 

Key:

1. &lt;math&gt;r_s&lt;/math&gt; = '''i'''; number of center mass points (the orbited object). 

2. '''j&lt;sub&gt;max&lt;/sub&gt;''' = radius to mass co-efficient.

3. '''j''' = number of points, including virtual (for simple 2 body orbits with only 1 orbiting point, '''j''' = '''i''' + 1 ).

4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 

5. '''r&lt;sub&gt;α&lt;/sub&gt;''' = a radius constant, here r&lt;sub&gt;α&lt;/sub&gt; = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals.

:&lt;math&gt;r_{orbital} = {r_{\alpha}}^2 \;*\; r_{wavelength} &lt;/math&gt;


==== Orbital formulas (2-D plane)====
Outer = orbiting point, inner = orbited center

:&lt;math&gt;r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2&lt;/math&gt;, orbital radius

:&lt;math&gt;r_{barycenter} = \frac{r_{outer}}{j}&lt;/math&gt;, barycenter

:&lt;math&gt;v_{outer} = \frac{i}{j_{max} r_{\alpha}} &lt;/math&gt;, orbiting point velocity  

:&lt;math&gt;v_{inner} = \frac{1}{j_{max} r_{\alpha}}&lt;/math&gt;, orbited point(s) velocity  

:&lt;math&gt;t_{outer} = \frac{2 \pi r_{outer}}{v_{outer}} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 &lt;/math&gt;, orbiting point period

:&lt;math&gt;l_{outer} = 2 \pi (r_{outer} - r_{barycenter})&lt;/math&gt;, distance travelled 


Simulation data:

:period &lt;math&gt;t_{sim}&lt;/math&gt;

:length &lt;math&gt;l_{sim}&lt;/math&gt;

:radius &lt;math&gt;r_{sim} = \frac{l_{sim}}{2 \pi}&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = \frac{l_{sim}}{t_{sim}}&lt;/math&gt; 

:barycenter &lt;math&gt;b_{sim} = \frac{x_{max} + x_{min}}{2}&lt;/math&gt;


For example; 8 mass points (28 orbitals) divided into ''j'' = 8 (total points), ''i'' = ''j'' - 1 (7 center mass points). After 1 complete orbit, actual period '''t''' and distance travelled '''l''' are noted and compared with the above formulas. 

1) ''j''&lt;sub&gt;max&lt;/sub&gt; = i+1 = 8 

:period &lt;math&gt;t = 74465.0516,\; t_{outer} = 74471.6125&lt;/math&gt; 

:length &lt;math&gt;l = l_{sim} = 3935.7664,\; l_{outer} = 3936.1032&lt;/math&gt; 

:radius &lt;math&gt;r_{sim} = 626.3951&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = 1/18.920137&lt;/math&gt;

:barycenter &lt;math&gt;b_{sim} = 89.5241,\; r_{barycenter} = 89.4929&lt;/math&gt;


2) ''j''&lt;sub&gt;max&lt;/sub&gt; = 32*i+1 = 225 

:period &lt;math&gt;t = 1656793370.3483,\; t_{outer} = 1656793381.3051&lt;/math&gt; 

:length &lt;math&gt;l = l_{sim} = 3113519.1259,\; l_{outer} = 3113519.1385&lt;/math&gt; 

:radius &lt;math&gt;r_{sim} = 495531.959&lt;/math&gt;

:velocity &lt;math&gt;v_{sim} = 1/532.128856&lt;/math&gt;

:barycenter &lt;math&gt;b_{sim} = 70790.283, \;r_{barycenter} = 70790.280&lt;/math&gt;


3) Moon orbit. 

From the [[w:standard gravitational parameter |standard gravitational parameters]], the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising ''i'' = 81 points, ''j'' = i + 1.

:&lt;math&gt;\frac{3.986004418\;x10^{14}}{4.9048695\;x10^{12}} = 81.2663&lt;/math&gt;

:&lt;math&gt;r_{earth-moon}&lt;/math&gt; = 384400km

:&lt;math&gt;M_{earth}&lt;/math&gt; = 0.597378 10&lt;sup&gt;25&lt;/sup&gt;kg


Solving &lt;math&gt;j_{max}&lt;/math&gt;

:&lt;math&gt;r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2 = \frac{2 r_{earth-moon} m_P}{M_{earth} l_p}&lt;/math&gt;

:&lt;math&gt;j_{max} = 1440443&lt;/math&gt;


Gives

:&lt;math&gt;t_{outer} =  4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 (\frac{l_p}{c}) = 0.8643\; 10^{-26}&lt;/math&gt;s

:&lt;math&gt;t_{outer} \frac{M_{earth}} {m_P } = 2371844&lt;/math&gt;s (27.452 days)

:&lt;math&gt;v_{Moon} = (c) \frac{i}{j_{max}{r_{\alpha}}} = 1018.3m/s&lt;/math&gt;

:&lt;math&gt;v_{Earth} = (c) \frac{1}{j_{max} r_{\alpha}}  = 12.57m/s&lt;/math&gt;

:&lt;math&gt;r_{barycenter} = \frac{r_{earth-moon}}{j} = 4688km&lt;/math&gt;


==== Gravitational coupling constant ====
In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α&lt;sub&gt;G&lt;/sub&gt;'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio;

:&lt;math&gt;\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}&lt;/math&gt;

For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α&lt;sub&gt;G&lt;/sub&gt; then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] &lt;ref&gt;Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}&lt;/ref&gt;). 

:&lt;math&gt;{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}&lt;/math&gt;

As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α&lt;sub&gt;G&lt;/sub&gt; is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10&lt;sup&gt;42&lt;/sup&gt; units of Planck time (&lt;math&gt;t_p = 10^{-42}s&lt;/math&gt;), this occurs about once every 3 minutes.

:&lt;math&gt;\frac{{\alpha_G}^{-1}}{t_p}&lt;/math&gt;

Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak.  

If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f&lt;sub&gt;e&lt;/sub&gt;, then the point will require 10&lt;sup&gt;23&lt;/sup&gt; electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time).

:&lt;math&gt;f_e = \frac{m_P}{m_e} = 10^{23}&lt;/math&gt;

For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have &lt;math&gt;1kg/m_P = 45940509&lt;/math&gt; particles in the point-state. The earth (A) will have &lt;math&gt;5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}&lt;/math&gt; particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite;

:&lt;math&gt;N_{orbitals} =  (\frac{m_A}{m_P})(\frac{m_B}{m_P})  = 0.1261\; x10^{41}&lt;/math&gt;
 

Examples:

:&lt;math&gt;i = \frac{M_{earth}}{m_P} = 0.27444  \;x10^{33}&lt;/math&gt; (earth as the center mass)

:&lt;math&gt;i 2 l_p = 0.00887&lt;/math&gt; (earth Schwarzschild radius)

:&lt;math&gt;s = \frac{1kg}{m_P} = 45940509&lt;/math&gt; (1kg orbiting satellite)

:&lt;math&gt;j = N_{orbitals} = i*s = 0.1261  \;x10^{41}&lt;/math&gt;


1) 1kg satellite at earth surface orbit 

:&lt;math&gt;r_{o} = 6371000 km&lt;/math&gt; (earth surface)

:&lt;math&gt;j_{max} = \frac{j}{r_a}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 2289.41&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 7909.7924&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374&lt;/math&gt; s 


2) 1kg satellite at a synchronous orbit radius

:&lt;math&gt;r_o = 42164.17 km&lt;/math&gt; 

:&lt;math&gt;j_{max} =  \frac{j}{r_a} \sqrt{\frac{r_{o}}{i l_p}}  = 0.74256\;x10^{44}&lt;/math&gt;

:&lt;math&gt;n_g = \frac{j_{max}}{j} = 5889.674&lt;/math&gt;

:&lt;math&gt;r = r_{\alpha}^2 n_g^2 i l_p = r_{o} &lt;/math&gt;

:&lt;math&gt;v = \frac{c}{n_g r_{\alpha}} = 3074.66&lt;/math&gt; m/s

:&lt;math&gt;t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165&lt;/math&gt; s 


3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). 

:&lt;math&gt;E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J&lt;/math&gt; (energy per orbital)

:&lt;math&gt;N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}&lt;/math&gt; (number of orbitals)

:&lt;math&gt;E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg&lt;/math&gt;


4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs.

:&lt;math&gt;N_{sun} = \frac{M_{sun}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{planet} = \frac{M_{planet}}{m_P} &lt;/math&gt;

:&lt;math&gt;N_{orbitals} = N_{sun}N_{planet} &lt;/math&gt;

:&lt;math&gt;n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} &lt;/math&gt;

:&lt;math&gt;L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} &lt;/math&gt;

The orbital angular momentum of the planets;

 mercury = .9153 x10&lt;sup&gt;39&lt;/sup&gt;  
 venus    = .1844 x10&lt;sup&gt;41&lt;/sup&gt;  
 earth    = .2662 x10&lt;sup&gt;41&lt;/sup&gt; 
 mars     = .3530 x10&lt;sup&gt;40&lt;/sup&gt; 
 jupiter   = .1929 x10&lt;sup&gt;44&lt;/sup&gt;   
 pluto   = .365 x10&lt;sup&gt;39&lt;/sup&gt;   


Orbital angular momentum combined with orbit velocity cancels ''n&lt;sub&gt;g&lt;/sub&gt;'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly.

:&lt;math&gt;L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} &lt;/math&gt;


[[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2&lt;sup&gt;32&lt;/sup&gt; steps they have clumped to form 1 large mass and 2 orbiting masses.]]
==== Freely moving points ====
The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted.   

In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2&lt;sup&gt;32&lt;/sup&gt; steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. 


[[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]]
==== Orbital trajectory (circular vs. straight) ====
Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. 

In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit.

The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). 

The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other.


==== Precession ====
semi-minor axis: &lt;math&gt;b = \alpha l^2 \lambda_A&lt;/math&gt;

semi-major axis: &lt;math&gt;a = \alpha n^2 \lambda_A&lt;/math&gt;

radius of curvature :&lt;math&gt;L = \frac{b^2}{a} = \frac{a l^4 \lambda_A}{n^2}&lt;/math&gt;

:&lt;math&gt;\frac{3 \lambda_A}{2 L} = \frac{3 n^2}{2 \alpha l^4}&lt;/math&gt;


arc secs per 100 years (drift):

:&lt;math&gt;T_{earth}&lt;/math&gt; = 365.25 days 

drift = &lt;math&gt;\frac{3 n^2}{2 \alpha l^4} 1296000 \frac{100 T_{earth}}{T_{planet}}&lt;/math&gt; 

 Mercury (eccentricity = 0.205630)
 T = 87.9691 days
 a = 57909050 km (''n'' = 378.2734) 
 b = 56671523 km (''l'' = 374.2096)
 drift = 42.98

 Venus (eccentricity = 0.006772) 
 T = 224.701 days
 a = 108208000 km (''n'' = 517.085) 
 b = 108205519 km (''l'' = 517.079)
 drift = 8.6247

 Earth (eccentricity = 0.0167)
 T = 365.25 days
 a = 149598000 km (''n'' = 607.989) 
 b = 149577138 km (''l'' = 607.946)
 drift = 3.8388

 Mars (eccentricity = 0.0934)
 T = 686.980 days
 a = 227939366 km (''n'' = 750.485) 
 b = 226942967 km (''l'' = 748.843)
 drift = 1.351


[[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]]
==== Hyper-sphere orbit ====
{{main|Relativity (Planck)}}

Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates &lt;ref&gt;Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}&lt;/ref&gt;. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. 

While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B&lt;sup&gt;1&lt;/sup&gt; to B&lt;sup&gt;11&lt;/sup&gt;) around the ''A'' time-line (vertical expansion) axis ('''t&lt;sub&gt;d&lt;/sub&gt;''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. 


For object '''B'''

:&lt;math&gt;t_d = \sqrt{t^2 - {t_0}^2} = t \sqrt{1 - v_{outer}^2}&lt;/math&gt; 

For object '''A'''

:&lt;math&gt;t_d = t \sqrt{1 - v_{inner}^2}&lt;/math&gt;


==== Planck force ====

:&lt;math&gt;F_p = \frac{m_P c^2}{l_p}&lt;/math&gt;

:&lt;math&gt;M_a = \frac{m_P \lambda_a}{2 l_p} ,\;m_b = \frac{m_P \lambda_b}{2 l_p}&lt;/math&gt;

:&lt;math&gt;F_g = \frac{M_a m_b G}{R^2} = \frac{\lambda_a \lambda_b F_p}{4 R_g^2}  = \frac{\lambda_a \lambda_b F_p}{4 \alpha^2 n^4 (\lambda_a + \lambda_b)^2} &lt;/math&gt;

a) &lt;math&gt;M_a = m_b&lt;/math&gt;

:&lt;math&gt;F_g = \frac{F_p}{{(4 \alpha n^2)}^2} &lt;/math&gt;

b) &lt;math&gt;M_a &gt;&gt; m_b&lt;/math&gt;

:&lt;math&gt;F_g = \frac{\lambda_b F_p}{{(2 \alpha n^2)}^2 \lambda_a} = \frac{m_b c^2}{2 \alpha^2 n^4 \lambda_a} = m_b a_g&lt;/math&gt;


=== Atomic orbitals ===
[[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]]
In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times. This means that we can treat the atomic orbital as a simple 2-body orbit with the electron as the orbiting point. Although this approach can only map the electron point-state (and so offers no direct information regarding the electron as a wave), during electron transition between ''n''-shell orbitals, we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]], this is significant because periodically the spiral angle components converge reducing to integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r'').

As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin. &lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;.


==== Simulation ====
The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational and atomic orbits is only in the angle of rotation &lt;math&gt;\beta&lt;/math&gt;' which has an additional &lt;math&gt;r_{\alpha}&lt;/math&gt; term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately.

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}&lt;/math&gt; 


[[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]]
===== Rydberg atom =====
For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength), at the ''n'' = 1 orbital, 1 complete rotation becomes (on a 2-D plane);

:&lt;math&gt;t_{ref} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha \lambda_{atom}&lt;/math&gt; 


Adding a relativistic term

:&lt;math&gt;t_{rel} = 2\pi 4\alpha^2 \sqrt{1 - \frac{1}{4\alpha^2}}&lt;/math&gt;

&lt;math&gt;1t_{rel}&lt;/math&gt; = 471961.214...

&lt;math&gt;4t_{rel}&lt;/math&gt; = 1887844.85912...

&lt;math&gt;9t_{rel}&lt;/math&gt; = 4247650.93303...

&lt;math&gt;16t_{rel}&lt;/math&gt; = 7551379.43650...


===== H atom =====
Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]).

H(1s-2s) =  2466 061 413 187.035 kHz &lt;ref&gt;http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf&lt;/ref&gt;  

H(1s-3s) =  2922 743 278 665.79 kHz &lt;ref&gt;https://pubmed.ncbi.nlm.nih.gov/33243883/&lt;/ref&gt;  

H(1s-4s) =  3082 581 563 822.63 kHz &lt;ref&gt;https://codata.org/&lt;/ref&gt;   

H(1s-∞s) =  3288 086 857 127.60 kHz &lt;ref&gt;https://codata.org/ (109678.77174307cm-1)&lt;/ref&gt; (''n'' = ∞)


R = 10973731.568157 &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt; ([[w:Rydberg constant |Rydberg constant]])

α =137.035999177 (inverse fine structure constant &lt;ref&gt;https://codata.org/ (mean)&lt;/ref&gt;


The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. 

&lt;math&gt;r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}&lt;/math&gt;


Dividing (dimensioned) wavelength (&lt;math&gt;r_{wavelength}&lt;/math&gt;) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The &lt;math&gt;(n^2 - 1)&lt;/math&gt; term gives the number of orbital wavelengths in the transition phase;

:&lt;math&gt;h_{(1s-ns)} = (n^2 - 1) \frac{\lambda_H }{H(1s-ns)}&lt;/math&gt;

&lt;math&gt;h_{(1s-2s)}&lt;/math&gt; = 1887839.82626...

&lt;math&gt;h_{(1s-3s)}&lt;/math&gt; = 4247634.04874...

&lt;math&gt;h_{(1s-4s)}&lt;/math&gt; = 7551347.55306...


===== Simulation atom =====
The following example simulates an electron transition, the electron begins at radius &lt;math&gt;r = r_{orbital}&lt;/math&gt; and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model). 

The period &lt;math&gt;t_{sim}&lt;/math&gt; and length &lt;math&gt;l_{sim}&lt;/math&gt; are measured at integer &lt;math&gt;n^2 r&lt;/math&gt; (''n'' = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...) (the angles converge to give integer values at these radius), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period &lt;math&gt;t_{sim}&lt;/math&gt; can then be used to calculate the transition frequencies.


In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''&lt;sub&gt;sim&lt;/sub&gt; = period and ''l''&lt;sub&gt;sim&lt;/sub&gt; = distance travelled by the electron (&lt;math&gt;l_{orbital} = l_{sim}&lt;/math&gt; at ''n''=1), the radius coefficient ''r''&lt;sub&gt;n&lt;/sub&gt; = radius divided by &lt;math&gt;r_{orbital}&lt;/math&gt;. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the ''x'', ''y'' co-ordinates for the electron, nucleus center and barycenter.
[[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]]


:&lt;math&gt;j_{atom} = 250&lt;/math&gt; (atomic mass)

:&lt;math&gt;i_{nucleus} = j_{atom} -1 = 249&lt;/math&gt; (relative nucleus mass)

:&lt;math&gt;r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2&lt;/math&gt; = 2.0160965

:&lt;math&gt;r_{orbital} = 2 \alpha \;*\; r_{wavelength} &lt;/math&gt; (radius) = 552.5556

:&lt;math&gt;t_n = \frac{t_{sim}}{r_{wavelength}}&lt;/math&gt; 

:&lt;math&gt;l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}&lt;/math&gt; 

:&lt;math&gt;r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}&lt;/math&gt; 

:&lt;math&gt;r_n = \frac{r_b}{r_{orbital}}&lt;/math&gt; 


{| class=&quot;wikitable&quot;
|+table 1. Electron transition (mass = 250; ''n''=1 to ''n''=5)
! ''r''&lt;sub&gt;n&lt;/sub&gt;
! ''t''&lt;sub&gt;sim&lt;/sub&gt;
! ''l''&lt;sub&gt;n&lt;/sub&gt;
! angle
! ''x'', ''y'' (electron) 
! ''x'', ''y'' (nucleus) 
! ''x'', ''y'' (barycenter) 
|-1
| 1
| 471957.072
| 0.9999897
| 360°
| 550.334,  0.0036
| -2.2102,  -0.00002 
| -0.00004,  -0.00001
|-
| 4
| 1887867.293
| 2.000012
| 359.952489°
| 2202.8558,  0.0001
| -7.9565,  -1.9475
| 0.8868,  -1.9397
|-
| 9 
| 4247689.502 
| 4.000014
| 119.92712° 
| -2473.180,  4296.283 
| 13.558,  -10.325  
| 3.611,  6.901
|-
| 16 
| 7551439.538 
| 6.000014 
| 179.91669°
| -8815.254,  12.818 
| 25.636,  13.303
| -9.728,  13.301
|-
| 25 
| 11799118.905 
| 8.000014  
| 215.9122°
| -11158.64,  -8081.13
| 16.580,  39.083
| -28.118,  6.602
|}


Comparison of the spiral angle with different mass: (64, 128, 250, 500, Rydberg) suggests nucleus mass and shape influences the results, however note the spiral only measures transition on a 2-D plane. For the proton:electron mass ratio; ''m'' = 1836.15267...

{| class=&quot;wikitable&quot;
|+ table 2. Spiral angle at &lt;math&gt;r_n&lt;/math&gt; = 4, 9 
! mass 
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 4 
! ''r''&lt;sub&gt;n&lt;/sub&gt; = 9
|-
| ''m'' = 64
| 359.80318°
| 119.70323°
|-
| ''m'' = 128
| 359.903935°
| 119.854148°
|-
| ''m'' = 250
| 359.952489°
| 119.92711°
|-
| ''m'' = 500
| 359.977062°
|  
|-
| Rydberg
| 360°
| 360+120°
|}


[[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]]
==== Theory ====
=====Bohr orbital =====
The H atom has 1 proton and 1 electron orbiting the proton, the electron can be found at fixed radius (the [[w:Bohr radius |Bohr radius]]) from the proton (nucleus), these radius represent different energy levels (orbitals) at which the electron may be found orbiting the proton and so are described as quantum levels. Electron transition (to higher energy levels) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to lower energy levels.

The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron is further from the nucleus). Each ''n'' ([[w:electron shell|electron shell]]) can accommodate up to ''n''&lt;sup&gt;2&lt;/sup&gt; electrons (1, 4, 9, 16, 25...), and accounting for two states of spin, 2''n''&lt;sup&gt;2&lt;/sup&gt;. As these orbitals are fixed according to integer ''n'', the atom can be said to be quantized.


The Bohr radius for each ''n'' level uses the fine structure constant alpha (α = 137.036) whereby;

&lt;math&gt;r_{orbital} = 2\alpha n^2 (\lambda_p + \lambda_e)&lt;/math&gt;


Such that at ''n'' = 1, the start radius alpha component ''r'' = 2α. We can map the electron orbit around the orbital as a series of steps. The steps are defined according to the rotation angle β;

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;


[[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]]

This gives a length travelled per (integer) step as the inverse of the radius (omitting the wavelength component to reduce computation);

:&lt;math&gt;l_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt; 

:&lt;math&gt;v_{orbital} = \frac{1}{2\alpha n}&lt;/math&gt; 


The number of steps (orbital period) for 1 orbit of the electron then becomes 

:&lt;math&gt;t_{orbital} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha n^3&lt;/math&gt;


===== Photon orbital model =====
The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the Photon-orbital model&lt;ref&gt;Macleod, Malcolm J.; {{Cite journal |title=XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX |journal=RG |date=Feb 2011 | doi=10.13140/RG.2.2.23106.71367/9}}&lt;/ref&gt;, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that &lt;math&gt;orbital \;radius + photon = zero&lt;/math&gt; (cancel).

The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path.


The photon is actually 2 photons as per the Rydberg formula (denoted initial and final).

:&lt;math&gt;\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}&lt;/math&gt;

:&lt;math&gt;\lambda_{photon} = (+\lambda_i) - (+\lambda_f)&lt;/math&gt;


The wavelength of the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon corresponds to the wavelength of the orbital radius. The (+&lt;math&gt;\lambda_i&lt;/math&gt;) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-&lt;math&gt;\lambda_f&lt;/math&gt;), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+&lt;math&gt;\lambda_i&lt;/math&gt;) photon wavelength, the orbital radius does not change as the 2 photons cancel each other;

:&lt;math&gt;r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)&lt;/math&gt;


However, the (&lt;math&gt;\lambda_f&lt;/math&gt;) has the longer wavelength, and so after the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon has been absorbed, and for the remaining duration of this (&lt;math&gt;\lambda_f&lt;/math&gt;) photon wavelength, the orbital radius will be extended until the (&lt;math&gt;\lambda_f&lt;/math&gt;) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial &lt;math&gt;n_i = 1&lt;/math&gt; orbital to a final &lt;math&gt;n_f = 2&lt;/math&gt; orbital, first the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed (&lt;math&gt;\lambda_i + \lambda_{orbital} = zero&lt;/math&gt; which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (&lt;math&gt;\lambda_f&lt;/math&gt;) photon continues until it too is absorbed (the '''transition phase''').

:&lt;math&gt;t_{ref} \sim 2\pi 4\alpha^2 &lt;/math&gt;

:&lt;math&gt;\lambda_i = 1t_{ref}&lt;/math&gt;

:&lt;math&gt;\lambda_f = 4t_{ref}&lt;/math&gt; (''n'' = 2)


After the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon is absorbed, the (&lt;math&gt;\lambda_f&lt;/math&gt;) photon still has &lt;math&gt;\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}&lt;/math&gt; steps remaining until it too is absorbed.


[[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]]
This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of &lt;math&gt;r_{incr}&lt;/math&gt; (both photon and orbital radius may be considered as constructs from multiple units of this geometry);

:&lt;math&gt;r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}&lt;/math&gt;


In summary; the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step

&lt;math&gt;r = r_{orbital}&lt;/math&gt;

WHILE (&lt;math&gt;\lambda_i &gt; 0&lt;/math&gt;) 

:&lt;math&gt;r = r + r_{incr}&lt;/math&gt; 
://&lt;math&gt;\lambda_i&lt;/math&gt; photon


Conversely, because of its minus term, the (&lt;math&gt;\lambda_i&lt;/math&gt;) photon will simultaneously extend the orbital radius accordingly;

WHILE (&lt;math&gt;r &lt; 4 r_{orbital}&lt;/math&gt;) 

:&lt;math&gt;r = r - r_{incr}&lt;/math&gt; 
://&lt;math&gt;\lambda_f&lt;/math&gt; photon


The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]]

:&lt;math&gt;t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}&lt;/math&gt; 

The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius;

:&lt;math&gt;t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}&lt;/math&gt; 


===== Alpha spiral =====
[[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]]
A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a''').

In its simplest form, a [[w:fine structure constant|fine structure constant]] spiral (or alpha spiral) is a specific hyperbolic spiral that appears in [[w:Atomic electron transition|electron transitions]] between [[w:atomic orbital|atomic orbitals]] in a [[w:Hydrogen atom|Hydrogen atom]].


It can be represented in Cartesian coordinates by

:&lt;math&gt;x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 &lt; \varphi &lt; 4\pi&lt;/math&gt;


This spiral has only 2 revolutions approaching 720° as the radius approaches infinity. If we set start radius '''r''' = 1, then at given angles radius '''r''' will have integer values (the angle components cancel).

:&lt;math&gt;\varphi = (2)\pi, \; r = 4&lt;/math&gt; (360°)
:&lt;math&gt;\varphi = (4/3)\pi,\; r = 9&lt;/math&gt; (240°)
:&lt;math&gt;\varphi = (1)\pi, \;  r = 16&lt;/math&gt; (180°)
:&lt;math&gt;\varphi = (4/5)\pi, \;  r = 25&lt;/math&gt; (144°)
:&lt;math&gt;\varphi = (2/3)\pi, \;  r = 36&lt;/math&gt; (120°)


For a Rydberg atom, starting the simulation with &lt;math&gt;\varphi = 0, \;r = 2\alpha&lt;/math&gt; (''n''=1), such that for each step during transition;

:&lt;math&gt;\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}&lt;/math&gt;

:&lt;math&gt;\varphi = \varphi + \beta&lt;/math&gt;


As &lt;math&gt;\beta&lt;/math&gt; is proportional to the radius, as the radius increases the value of &lt;math&gt;\beta&lt;/math&gt; will reduce correspondingly (likewise reducing the orbital velocity). 

{{see|Fine-structure_constant_(spiral)}}


Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and the &lt;math&gt;n_f^2&lt;/math&gt; at each step.

:&lt;math&gt;r = 2 \alpha + \frac{t}{2\pi 2\alpha}&lt;/math&gt; (number of increments ''t'' of &lt;math&gt;r_{incr}&lt;/math&gt;)

:&lt;math&gt;n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}&lt;/math&gt; (&lt;math&gt;n_f^2&lt;/math&gt; as a function of ''t'') 

:&lt;math&gt;\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}&lt;/math&gt; (&lt;math&gt;\varphi&lt;/math&gt; at any &lt;math&gt;n_f^2&lt;/math&gt;) 


We can then re-write (&lt;math&gt;n_f&lt;/math&gt; is only an integer at prescribed spiral angles);

:&lt;math&gt;\beta = \frac{1}{4\alpha^2 n_f^3}&lt;/math&gt;


== External links ==
* [[v:Fine-structure_constant_(spiral) | Fine structure constant hyperbolic spiral]]
* [[v:Physical_constant_(anomaly) | Physical constant anomalies]]
* [[v:Planck_units_(geometrical) | Planck units as geometrical objects]] 
* [[v:electron_(mathematical) | The mathematical electron]]
* [[v:Relativity_(Planck) | Programming relativity at the Planck scale]]
* [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck level]]
* [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]]
* [[v:God_(programmer) | The Programmer God]]
* [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] 
* [https://theprogrammergod.com/ The Programmer God, are we in a computer simulation? - eBook] 


==References==
{{Reflist}}


[[Category:Physics| ]]
[[Category:Philosophy of science| ]]</text>
      <sha1>ffhniozrw55mxydi67lwcmje4518qsu</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
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    <revision>
      <id>2692614</id>
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      <timestamp>2024-12-19T14:22:54Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
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      <text bytes="9878" sha1="mu848vila6p27o4up6sihe50tw82oy1" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

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

=== Handling Series of Data ===

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

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

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241214.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

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

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

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


=== Class Notes ===

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


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

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

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

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


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

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>mu848vila6p27o4up6sihe50tw82oy1</sha1>
    </revision>
    <revision>
      <id>2692619</id>
      <parentid>2692614</parentid>
      <timestamp>2024-12-19T14:26:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2692619</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="buuq80sgvrvu8hm5ug3ol8f49w9grzz" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

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

=== Handling Series of Data ===

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

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

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241216.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

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

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

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


=== Class Notes ===

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


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

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

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

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


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

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>buuq80sgvrvu8hm5ug3ol8f49w9grzz</sha1>
    </revision>
    <revision>
      <id>2692621</id>
      <parentid>2692619</parentid>
      <timestamp>2024-12-19T14:27:26Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2692621</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="hcee70jsimkrpsv39ex3ke40boffypl" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

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

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

=== Handling Series of Data ===

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

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

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20241217.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

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

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

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


=== Class Notes ===

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


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

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

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

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


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

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>hcee70jsimkrpsv39ex3ke40boffypl</sha1>
    </revision>
  </page>
  <page>
    <title>Workings of gcc and ld in plain view</title>
    <ns>0</ns>
    <id>285384</id>
    <revision>
      <id>2692749</id>
      <parentid>2692537</parentid>
      <timestamp>2024-12-20T01:04:13Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Linking Libraries */</comment>
      <origin>2692749</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3430" sha1="jfzd57s4ksmzw0ccaz4kdwlbim934o4" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241217.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])

* Shared Libraries ([[Media:LIB.2A.Shared.20241220.pdf |pdf]])

==== Directories and Symbolic Links ====

* Shared Library Names ([[Media:MNG.1A.Names.20241218.pdf |pdf]])
* Managing Shared Libraries ([[Media:MNG.2A.Manage.20241218.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>jfzd57s4ksmzw0ccaz4kdwlbim934o4</sha1>
    </revision>
    <revision>
      <id>2692756</id>
      <parentid>2692749</parentid>
      <timestamp>2024-12-20T03:46:13Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Workings of the GNU Linker for IA-32 */</comment>
      <origin>2692756</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3547" sha1="e5cd0vslnrtxj4x7s0rdvk0271xmahm" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241217.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241220.pdf |pdf]])

==== Dynamic Linking - Directories and Symbolic Links ====

* Shared Library Names ([[Media:DIR.1A.Names.20241218.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241218.pdf |pdf]])

==== Dynamic Loading - API Functions ====

* DL API ([[Media:API.1A.Functions.20241220.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>e5cd0vslnrtxj4x7s0rdvk0271xmahm</sha1>
    </revision>
    <revision>
      <id>2692760</id>
      <parentid>2692756</parentid>
      <timestamp>2024-12-20T06:01:06Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Dynamic Linking - Directories and Symbolic Links */</comment>
      <origin>2692760</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3547" sha1="lp1gw5x157n9xj0rttyk9ov32en2o95" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241217.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241220.pdf |pdf]])

==== Dynamic Linking - Directories and Symbolic Links ====

* Shared Library Names ([[Media:DIR.1A.Names.20241219.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241219.pdf |pdf]])

==== Dynamic Loading - API Functions ====

* DL API ([[Media:API.1A.Functions.20241220.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>lp1gw5x157n9xj0rttyk9ov32en2o95</sha1>
    </revision>
    <revision>
      <id>2692761</id>
      <parentid>2692760</parentid>
      <timestamp>2024-12-20T06:01:28Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Linking Libraries */</comment>
      <origin>2692761</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3549" sha1="djc83fkr0aa7wo9kt6ro1w17puvig89" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241217.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241220-1.pdf |pdf]])

==== Dynamic Linking - Directories and Symbolic Links ====

* Shared Library Names ([[Media:DIR.1A.Names.20241219.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241219.pdf |pdf]])

==== Dynamic Loading - API Functions ====

* DL API ([[Media:API.1A.Functions.20241220.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>djc83fkr0aa7wo9kt6ro1w17puvig89</sha1>
    </revision>
    <revision>
      <id>2692764</id>
      <parentid>2692761</parentid>
      <timestamp>2024-12-20T06:04:02Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Dynamic Linking - Directories and Symbolic Links */</comment>
      <origin>2692764</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3549" sha1="jugac7mcr9gde7tzokh3miw4q1gf7g5" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241217.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241220-1.pdf |pdf]])

==== Dynamic Linking - Directories and Symbolic Links ====

* Shared Library Names ([[Media:DIR.1A.Names.20241220.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241219.pdf |pdf]])

==== Dynamic Loading - API Functions ====

* DL API ([[Media:API.1A.Functions.20241220.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>jugac7mcr9gde7tzokh3miw4q1gf7g5</sha1>
    </revision>
    <revision>
      <id>2692767</id>
      <parentid>2692764</parentid>
      <timestamp>2024-12-20T06:06:26Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Dynamic Linking - Directories and Symbolic Links */</comment>
      <origin>2692767</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3549" sha1="j3osj3qcqzmrz423stsnc4ptn4b77if" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241217.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241220-1.pdf |pdf]])

==== Dynamic Linking - Directories and Symbolic Links ====

* Shared Library Names ([[Media:DIR.1A.Names.20241220.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241220.pdf |pdf]])

==== Dynamic Loading - API Functions ====

* DL API ([[Media:API.1A.Functions.20241220.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
      <sha1>j3osj3qcqzmrz423stsnc4ptn4b77if</sha1>
    </revision>
  </page>
  <page>
    <title>User:Indexcard88</title>
    <ns>2</ns>
    <id>296072</id>
    <revision>
      <id>2692726</id>
      <parentid>2673193</parentid>
      <timestamp>2024-12-19T22:24:55Z</timestamp>
      <contributor>
        <username>Indexcard88</username>
        <id>118020</id>
      </contributor>
      <origin>2692726</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="249" sha1="dw3fpunq6t6hx1814fcwmacvl9l2x0o" xml:space="preserve">https://drive.google.com/drive/folders/1m3oXP9EtHLMZfzXEJBJKUkNb2l6pUUcf?usp=sharing (Private)

https://javerikr.blogspot.com/ (Private)

https://pastebin.com/u/javerikr


[[Wikipedia:User:Indexcard88]]

[[Metawiki:User:Indexcard88]]

[[/Mysteries]]</text>
      <sha1>dw3fpunq6t6hx1814fcwmacvl9l2x0o</sha1>
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  </page>
  <page>
    <title>Talk:WikiJournal Preprints/Practical applications of moisture sorption models for predicting the drying characteristics and shelf-life of malted and/or fermented FARO 44 rice plus soybean-based complementary foods</title>
    <ns>1</ns>
    <id>298615</id>
    <revision>
      <id>2692777</id>
      <parentid>2636754</parentid>
      <timestamp>2024-12-20T09:45:02Z</timestamp>
      <contributor>
        <username>Charles Bristone</username>
        <id>2968113</id>
      </contributor>
      <comment>/* Peer review 1 */ Reply</comment>
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      <text bytes="6961" sha1="l3dudg9t5hswa78wbbfok0iqacb9zvv" xml:space="preserve">{{#section-h:{{ARTICLEPAGENAMEE}}}}

== Plagiarism check ==

* {{Pass}} Report from [https://copyvios.toolforge.org/?lang=en&amp;project=wikiversity&amp;action=search&amp;turnitin=1&amp;title=WikiJournal_Preprints/Practical_applications_of_moisture_sorption_models_for_predicting_the_drying_characteristics_and_shelf-life_of_malted_and/or_fermented_FARO_44_rice_plus_soybean-based_complementary_foods WMF copyvios tool]: 0% plagiarism detected. [[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;]] 13:46, 11 August 2023 (UTC)

== Peer review 1 ==

{{review
 |credentials= Retired professor with interest and contribution to the specific field of moisture sorptions in foods and f moisture sorpotion 
 |date       = 10 July 2024
 |text       = 
The paper reports the experimental results of a set of moisture sorption isotherms determinations of several rice/soybean mixtures at several temperatures and their fitting with the BET and GAB model equations. It has several serious shortcomings and is unacceptable for publication in WikiJournal of Science for both lack of general interest and poor technical quality. Specific comments:
# The authors, like many in their field, confuse curve-fitting for scientific prediction.
# Comparison of the statistical fit a 3-parameters model (GAB) and 2-parameters model (BET) is a futile exercise.
# There is no information on whether or how representative the experimental data are.
# The very existence a water monolayer, the essence of the two models, has been questioned in the literature and hence the validity of the thermodynamic analysis is highly questionable too.
# Most of the data are presented with a ridiculous number of insignificant digits.
Recommendation: Reject.
}} [[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;]] 04:16, 10 July 2024 (UTC)

:The expiration date and best-by date on every food label is an estimated time. Establishing such a unique time duration requires many scientific approaches such as computer simulation study, actual storage conditions, accelerated shelf-life study etc. Food dehydration processes for shelf-life stability or rehydration of food which favour deterioration is a serious concern to researchers, food manufacturers and consumers. Every year, the world economy is being affected by the spoilage of foods during harvest, transportation or storage. Global food shortages due to spoilage are on the increase.
:1.    Authors comments: Many model equations such as BET and GAB are not linear. Many researchers may like to choose the easy way of estimating the model parameters by linearization of the non-linear equation. Others may choose the direct method or the non-linear regression procedures. The model may then be fitted into the experimental data either directly or through regression (curve-fitting). The goal of linear or non-linear regression is to fit a model to data. However, the goal of curve-fitting will depend on what the researcher intends to achieve. If the goal is to obtain the best-fit values of the parameters, then the best-fit values obtained can be used to interpret or predict the context of the model.
: 
:2.    Authors comments: Statistical fit is a guide to evaluate the best-fit model into data. GAB is an extension of BET model and this makes it relevant for comparison. 
: 
:3.    Authors comment: This statement is not clear.
: 
:4.    Authors comments: First of all, the concept of the monolayer is useful because of its relationship with many aspects of the physical and chemical deterioration in dehydrated products. In addition, the monolayer calculation was reported as an effective method for estimating the amount of bound water to specific polar sites in dehydrated foods.
:All mathematical models have limitations. For BET model, it is limited to 0.45 or 0.5 water activity. For GAB model, it is limited up to 0.94 water activity and requires at least five experimental data points. For a goodness of fits of the GAB isotherms, it is up to 0.90 water activity. Example, in our elementary mathematics, we have seen the limitations of factorization, completing the square, quadratic formula or the use of graphical method for finding the roots of a quadratic equation. For these reasons, it is important to be aware of the limitations of any mathematical model and not to try the use of the model outside of its useful range. Every mathematical model has its concepts. Problems arise if the reader does not understand those model concepts. The validation of the model is not that it is “true” but it generates good testable hypotheses relevant to solving important problems. A mathematical model is an equation that describes a physical, chemical or biological state or process. Earlier, it was for this reason we mentioned the use of BET-region in the abstract and also methods for predicting the self-life.
:A report by Rahman (1995) on the heat of binding of water at monolayer moisture content estimated by different methods indicates that the calorimetric and isosteric methods did not differ much. Therefore, the isosteric method based on the monolayer moisture is an alternative thermodynamic method of calculating the heat of sorption.  Notwithstanding, these models (GAB and BET) have been used successfully for food characterization, classification, optimization, process design and control etc.
: 
:5.    Authors comments: These products were formulated using a material balance equation to meet 16% protein as the target in each as recommended by the regulatory agencies. It was reported that the composition of food plays a vital role in the sorption phenomenon. The trends observed in this study correspond to their composition. Our recent research (Bristone et al., 2021; Bristone et al., 2024) on these same products exhibited similar trends.
: 
:Motulsky. H. J. (2020). GraphPad Curve Fitting Guide GraphPad Software Inc. www.graphpad.com
: 
:Bristone, C., Eke, M. O., Ikya, J. K., Ariahu, C. C. (2024) Influence of Malting and/or Fermentation on Proximate Composition of FARO 44 Rice Plus Soybean Based Complementary Foods. Dutse Journal of Pure and Applied Sciences (DUJOPAS). 10 (2a).1 – 15. https://dx.doi.org/10.4314/dujopas.v10i2a.1
: 
:Bristone, C., Ariahu, C. C., Ikya, J. K., Eke, M. O. (2021). Microbiological, essential dietary minerals and amino acids composition of malted and /or fermented FARO 44 rice plus soybean based complementary foods. Annals Clinical Nutrition. 4 (2): 1022. 1-12.
:Rahman, S. (1995): Food properties handbook. CRC Press LLC, 2000 N, W. Corporate Blvd., Boca Raton, Florida, pp. 1 - 86. [[User:Charles Bristone|Charles Bristone]] ([[User talk:Charles Bristone|discuss]] • [[Special:Contributions/Charles Bristone|contribs]]) 09:45, 20 December 2024 (UTC)</text>
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  </page>
  <page>
    <title>Linear and noble Boolean functions</title>
    <ns>0</ns>
    <id>302536</id>
    <revision>
      <id>2692744</id>
      <parentid>2642667</parentid>
      <timestamp>2024-12-19T23:50:27Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>/* linear to patron (noble) */</comment>
      <origin>2692744</origin>
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      <text bytes="10059" sha1="k3tz6icl1jgxx5h4dwyxva6nh700ir1" xml:space="preserve">{{Boolf header}}
&lt;templatestyles src=&quot;Collapsible with classes/style.css&quot; /&gt;

{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center; &quot;
! arity
! &lt;math&gt;n&lt;/math&gt;
! 1
! 2
! 3
! 4 
! 5
|-
! linear
| &lt;math&gt;2^{n+1}&lt;/math&gt;
| 4
| 8
| 16
| 32
| 64
|-
! noble
| &lt;math&gt;2^{2^{n-1}}&lt;/math&gt;
| 2
| 4
| 16
| 256 
| 65536
|}

Among the truth tables for a given arity, the [[Linear Boolean functions|linears]] and the [[Noble Boolean functions|nobles]] are important subsets.

Each linear can be assigned a [[Noble Boolean functions#patrons|patron]], which is noble. Each noble can be assigned a prefect, which is linear.

For arity 3 they form a bijection. For higher arities the nobles outnumber the linears &lt;small&gt;(i.e. the patrons of the linears are a subset of the nobles)&lt;/small&gt;.


==overview==

{| class=&quot;collapsible-with-classes collapsible collapsed wide followed&quot;
!colspan=&quot;2&quot;| 2-ary
|-
|
{{multiple image
 | align = center | total_width = 500
 | image1 = 2T linears with quadrants.svg
 | image2 = 2Z linears with quadrants.svg
 | footer = linears &amp;nbsp; &lt;small&gt;(truth tables and Zhegalkin indices)&lt;/small&gt;
}}
| [[File:2-ary nobles in matrix.svg|thumb|center|245px|nobles]]
|}

{| class=&quot;collapsible-with-classes collapsible open wide followed&quot;
!colspan=&quot;2&quot;| 3-ary
|-
|
{{multiple image
 | align = center | total_width = 700
 | image1 = 3T linears with quadrants.svg
 | image2 = 3Z linears with quadrants.svg
 | footer = linears &amp;nbsp; &lt;small&gt;(truth tables and Zhegalkin indices)&lt;/small&gt;
}}
| [[File:3-ary nobles in matrix.svg|thumb|center|345px|nobles]]
|}

{{Collapsible START|4-ary|collapsed wide}}
{{Collapsible START|linears &amp;nbsp; &lt;small&gt;(truth tables)&lt;/small&gt;|collapsed wide light followed}}
[[File:4T linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|linears &amp;nbsp; &lt;small&gt;(Zhegalkin indices)&lt;/small&gt;|collapsed wide light followed}}
[[File:4Z linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|nobles|collapsed wide light}}
[[File:4-ary nobles in matrix.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible END}}

==3-ary==

{| class=&quot;collapsible-with-classes collapsible collapsed wide light gap-below&quot;
!colspan=&quot;3&quot; style=&quot;background-color: #f0f0f0; color: gray;&quot; | nobles in tesseract
|-
| [[File:3-ary nobles in tesseract.svg|500px]]
| [[File:2T principality; faction size 1; king index 0.svg|thumb|center|280px|dihedral symmetry]]
| [[File:2T principality; faction size 3; king index 2.svg|thumb|center|280px|mirror symmetry]]
|-
|}

{| class=&quot;collapsible-with-classes collapsible open wide followed&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | quadrants
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, quadrant 0.svg|thumb|right|300px| {{colorbox|#e30000}} &amp;nbsp; 0 &amp;nbsp; &lt;small&gt;(even, evil)&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, quadrant 3.svg|thumb|left|300px| {{colorbox|#00cc00}} &amp;nbsp; 3 &amp;nbsp; &lt;small&gt;(odd, odious)&lt;/small&gt;]]
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, quadrant 2.svg|thumb|right|300px| {{colorbox|#ffb000}} &amp;nbsp; 2 &amp;nbsp; &lt;small&gt;(even, odious)&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, quadrant 1.svg|thumb|left|300px| {{colorbox|#0055ff}} &amp;nbsp; 1 &amp;nbsp; &lt;small&gt;(odd, evil)&lt;/small&gt;]]
|}

{| class=&quot;collapsible-with-classes collapsible collapsed wide followed&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | halves
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, even.svg|thumb|right|300px|even&lt;br&gt;&lt;small&gt;Walsh functions&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, odd.svg|thumb|left|300px|odd&lt;br&gt;&lt;small&gt;complements of Walsh functions&lt;/small&gt;]]
|}

{| class=&quot;collapsible-with-classes collapsible collapsed wide&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | all
|-
| [[File:3-ary noble to linear.svg|300px|center]]
|}


==4-ary==

===linear to patron (noble)===

There are 32 linears, which form 10 factions. &lt;small&gt;(Even and odd faction for each Walsh weight 0...4.)&lt;/small&gt; Their patrons are 32 nobles, which also form 10 factions &lt;small&gt;(among the 44 noble factions)&lt;/small&gt;.

Their [[Boolf-term#junior|juniors]] are the 3-ary Boolean functions with consul 0.

{{Collapsible START|list|collapsed wide followed}}
The faction is determined by Walsh weight and parity (or quadrant), and represented by variants of the quadrant colors.&lt;br&gt;
&lt;small&gt;A good sort order is first by quadrant, and [[w:Help:Sortable tables#Secondary key|then]] by Walsh weight.&lt;/small&gt;
{{Linear to patron 4-ary list}}
{{Collapsible START|sequences|collapsed wide light}}
{{Linear to patron 4-ary sequences}}
{{Collapsible END}}
{{Collapsible END}}

{{Collapsible START|details for each faction|collapsed wide followed}}
The numbers next to the images correspond to the truth tables.&lt;br&gt;
The linears are in the left columns, their patrons in the middle, and their [[Zhegalkin twins|twins]] on the right. &amp;nbsp;
&lt;small&gt;(So the [[Zhegalkin matrix|Zhegalkin indices]] of the linear functions are in the columns on the right.)&lt;/small&gt;
{{Linear to patron 4-ary factions}}
{{Collapsible END}}

{{Collapsible START|3-ary noble indices &amp;nbsp; &lt;small&gt;(juniors)&lt;/small&gt;|collapsed wide}}
{| style=&quot;text-align: center; width: 100%;&quot;
|colspan=&quot;2&quot; style=&quot;text-align: left;&quot;|
The factions are shown in ten different colors, which are variants of the quadrant colors. &amp;nbsp;
&lt;small&gt;(There are three shades of red and green, and two shades of blue and yellow.)&lt;/small&gt;
|-
| [[File:3T linear to patron.svg|thumb|center|500px|truth tables]]
| [[File:3Z linear to patron.svg|thumb|center|500px|Zhegalkin indices]]
|}
{{Collapsible END}}

The following images show that pairs of complementary factions are in the same principality.

{{Collapsible START|quadrants 0 and 3|collapsed wide followed}}
{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center;&quot;
|- style=&quot;font-size: 65%;&quot;
! king
| 0
| 3808
| 26752
|-
! king index
| 0
| 14
| 104
|}
{{colorbox|#e30000}} &amp;nbsp; The patrons of the 8 even Walsh functions &lt;small&gt;(quadrant 0)&lt;/small&gt; are the red entries in these 3 principalities.&lt;br&gt;
{{colorbox|#00cc00}} &amp;nbsp; The patrons of their complements &lt;small&gt;(quadrant 3)&lt;/small&gt; are the green entries.
{| class=&quot;collapsible-with-classes collapsible collapsed light wide followed&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| truth tables
|-
| [[File:3T principality; faction size 1; king index 0.svg|400px]]
| [[File:3T principality; faction size 6; king index 14.svg|400px]]
| [[File:3T principality; faction size 1; king index 104.svg|400px]]
|}
{| class=&quot;collapsible-with-classes collapsible collapsed light wide&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| Zhegalkin indices
|-
| [[File:3Z principality; faction size 1; king index 0.svg|400px]]
| [[File:3Z principality; faction size 6; king index 14.svg|400px]]
| [[File:3Z principality; faction size 1; king index 104.svg|400px]]
|}
{{Collapsible END}}

{{Collapsible START|quadrants 1 and 2|collapsed wide}}
{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center;&quot;
|- style=&quot;font-size: 65%;&quot;
! king
| 5760
| 10920
|-
! king index
| 22
| 42
|}
{{colorbox|#ffb000}} &amp;nbsp; The patrons of the 8 odious Walsh functions &lt;small&gt;(quadrant 2)&lt;/small&gt; are the yellow entries in these 2 principalities.&lt;br&gt;
{{colorbox|#0055ff}} &amp;nbsp; The patrons of their complements &lt;small&gt;(quadrant 1)&lt;/small&gt; are the blue entries.
{| class=&quot;collapsible-with-classes collapsible collapsed light wide followed&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| truth tables
|-
| [[File:3T principality; faction size 4; king index 22.svg|400px]]
| [[File:3T principality; faction size 4; king index 42.svg|400px]]
|}
{| class=&quot;collapsible-with-classes collapsible collapsed light wide&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| Zhegalkin indices
|-
| [[File:3Z principality; faction size 4; king index 22.svg|400px]]
| [[File:3Z principality; faction size 4; king index 42.svg|400px]]
|}
{{Collapsible END}}

===noble to prefect (linear)===

There are 256 nobles. A prefect is one of the 32 linears. Every linear is the prefect of 8 nobles.

{{Collapsible START|list &amp;nbsp; &lt;small&gt;(32 × 8 matrix)&lt;/small&gt;|collapsed wide followed}}
This table is an extension of the one shown above (from linear to patron). The leftmost noble column is equal to that of patrons.&lt;br&gt;
But here the assignment goes in the other direction. Each of the nobles on the right has the prefect on the left. &lt;small&gt;E.g. the nobles 3870 and 2600 have prefect Ж 17.&lt;/small&gt;

The nobles are represented by the integer values of their truth tables, which are also their Zhegalkin indices.&lt;br&gt;
The small numbers to their right are the juniors 0...255.&lt;br&gt;
&lt;small&gt;They correspond to 3-ary Boolean functions, so among them are also linears and nobles. The linears are highlighted in bold, and the nobles with a box.&lt;/small&gt;

The background colors of the nobles denote the [[Noble Boolean functions#quadrants|noble quadrants]].&lt;br&gt;
The tiny red numbers are the king indices. Together with the noble quadrants they denote the noble factions. &amp;nbsp; &lt;small opacity: .5;&gt;(The eleven king indices are 0, 2, 6, 8, 14, 22, 26, 42, 44, 104, 110.)&lt;/small&gt;

{{Noble to prefect 4-ary}}
{{Collapsible END}}

{{Collapsible START|3-ary subprefects &amp;nbsp; &lt;small&gt;(16 × 16 matrices)&lt;/small&gt;|collapsed wide}}
The right side of the following table shows the same information as the table above.&lt;br&gt;
Entries of one color in one matrix are the eight juniors in a row of the table above. &amp;nbsp; &lt;small&gt;(Linears are highlighted in bold, and the nobles with a crown.)&lt;/small&gt;

The left side of the table shows the prefects of all 3-ary Boolean functions. &lt;small&gt;(And the great prefects as sort keys.)&lt;/small&gt;&lt;br&gt;
It can be seen, that the {{w|Image (mathematics)|preimages}} of the 4-ary prefects are a refinement of the 3-ary prefects. This motivates the term ''subprefect''.

{{Noble to prefect 4-ary matrices}}

&lt;small&gt;A more detailed version of this table can be seen [[commons:Category:3-ary Boolean functions in octeract matrix; great subprefects|here]].&lt;/small&gt;
{{Collapsible END}}


[[Category:Linear and noble Boolean functions]]</text>
      <sha1>k3tz6icl1jgxx5h4dwyxva6nh700ir1</sha1>
    </revision>
    <revision>
      <id>2692748</id>
      <parentid>2692744</parentid>
      <timestamp>2024-12-20T00:14:26Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>/* linear to patron (noble) */</comment>
      <origin>2692748</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10182" sha1="hb4q1mh6rwrgd588tgsfmk8sgnd7ene" xml:space="preserve">{{Boolf header}}
&lt;templatestyles src=&quot;Collapsible with classes/style.css&quot; /&gt;

{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center; &quot;
! arity
! &lt;math&gt;n&lt;/math&gt;
! 1
! 2
! 3
! 4 
! 5
|-
! linear
| &lt;math&gt;2^{n+1}&lt;/math&gt;
| 4
| 8
| 16
| 32
| 64
|-
! noble
| &lt;math&gt;2^{2^{n-1}}&lt;/math&gt;
| 2
| 4
| 16
| 256 
| 65536
|}

Among the truth tables for a given arity, the [[Linear Boolean functions|linears]] and the [[Noble Boolean functions|nobles]] are important subsets.

Each linear can be assigned a [[Noble Boolean functions#patrons|patron]], which is noble. Each noble can be assigned a prefect, which is linear.

For arity 3 they form a bijection. For higher arities the nobles outnumber the linears &lt;small&gt;(i.e. the patrons of the linears are a subset of the nobles)&lt;/small&gt;.


==overview==

{| class=&quot;collapsible-with-classes collapsible collapsed wide followed&quot;
!colspan=&quot;2&quot;| 2-ary
|-
|
{{multiple image
 | align = center | total_width = 500
 | image1 = 2T linears with quadrants.svg
 | image2 = 2Z linears with quadrants.svg
 | footer = linears &amp;nbsp; &lt;small&gt;(truth tables and Zhegalkin indices)&lt;/small&gt;
}}
| [[File:2-ary nobles in matrix.svg|thumb|center|245px|nobles]]
|}

{| class=&quot;collapsible-with-classes collapsible open wide followed&quot;
!colspan=&quot;2&quot;| 3-ary
|-
|
{{multiple image
 | align = center | total_width = 700
 | image1 = 3T linears with quadrants.svg
 | image2 = 3Z linears with quadrants.svg
 | footer = linears &amp;nbsp; &lt;small&gt;(truth tables and Zhegalkin indices)&lt;/small&gt;
}}
| [[File:3-ary nobles in matrix.svg|thumb|center|345px|nobles]]
|}

{{Collapsible START|4-ary|collapsed wide}}
{{Collapsible START|linears &amp;nbsp; &lt;small&gt;(truth tables)&lt;/small&gt;|collapsed wide light followed}}
[[File:4T linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|linears &amp;nbsp; &lt;small&gt;(Zhegalkin indices)&lt;/small&gt;|collapsed wide light followed}}
[[File:4Z linears with quadrants.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible START|nobles|collapsed wide light}}
[[File:4-ary nobles in matrix.svg|center|1040px]]
{{Collapsible END}}
{{Collapsible END}}

==3-ary==

{| class=&quot;collapsible-with-classes collapsible collapsed wide light gap-below&quot;
!colspan=&quot;3&quot; style=&quot;background-color: #f0f0f0; color: gray;&quot; | nobles in tesseract
|-
| [[File:3-ary nobles in tesseract.svg|500px]]
| [[File:2T principality; faction size 1; king index 0.svg|thumb|center|280px|dihedral symmetry]]
| [[File:2T principality; faction size 3; king index 2.svg|thumb|center|280px|mirror symmetry]]
|-
|}

{| class=&quot;collapsible-with-classes collapsible open wide followed&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | quadrants
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, quadrant 0.svg|thumb|right|300px| {{colorbox|#e30000}} &amp;nbsp; 0 &amp;nbsp; &lt;small&gt;(even, evil)&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, quadrant 3.svg|thumb|left|300px| {{colorbox|#00cc00}} &amp;nbsp; 3 &amp;nbsp; &lt;small&gt;(odd, odious)&lt;/small&gt;]]
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, quadrant 2.svg|thumb|right|300px| {{colorbox|#ffb000}} &amp;nbsp; 2 &amp;nbsp; &lt;small&gt;(even, odious)&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, quadrant 1.svg|thumb|left|300px| {{colorbox|#0055ff}} &amp;nbsp; 1 &amp;nbsp; &lt;small&gt;(odd, evil)&lt;/small&gt;]]
|}

{| class=&quot;collapsible-with-classes collapsible collapsed wide followed&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | halves
|-
|style=&quot;padding-right: 15px;&quot;| [[File:3-ary linear to noble, even.svg|thumb|right|300px|even&lt;br&gt;&lt;small&gt;Walsh functions&lt;/small&gt;]]
|style=&quot;padding-left: 15px;&quot;| [[File:3-ary linear to noble, odd.svg|thumb|left|300px|odd&lt;br&gt;&lt;small&gt;complements of Walsh functions&lt;/small&gt;]]
|}

{| class=&quot;collapsible-with-classes collapsible collapsed wide&quot;
!colspan=&quot;2&quot; bgcolor=&quot;#ddd&quot; | all
|-
| [[File:3-ary noble to linear.svg|300px|center]]
|}


==4-ary==

===linear to patron (noble)===

There are 32 linears, which form 10 factions. &lt;small&gt;(Even and odd faction for each Walsh weight 0...4.)&lt;/small&gt; Their patrons are 32 nobles, which also form 10 factions &lt;small&gt;(among the 44 noble factions)&lt;/small&gt;.

Their [[Boolf-term#junior|juniors]] are the 3-ary Boolean functions with consul 0. &amp;nbsp; &lt;small style=&quot;opacity: .5;&quot;&gt;(This is not the case for arities 2 and 3, but probably for all arities ≥ 4.)&lt;/small&gt;

{{Collapsible START|list|collapsed wide followed}}
The faction is determined by Walsh weight and parity (or quadrant), and represented by variants of the quadrant colors.&lt;br&gt;
&lt;small&gt;A good sort order is first by quadrant, and [[w:Help:Sortable tables#Secondary key|then]] by Walsh weight.&lt;/small&gt;
{{Linear to patron 4-ary list}}
{{Collapsible START|sequences|collapsed wide light}}
{{Linear to patron 4-ary sequences}}
{{Collapsible END}}
{{Collapsible END}}

{{Collapsible START|details for each faction|collapsed wide followed}}
The numbers next to the images correspond to the truth tables.&lt;br&gt;
The linears are in the left columns, their patrons in the middle, and their [[Zhegalkin twins|twins]] on the right. &amp;nbsp;
&lt;small&gt;(So the [[Zhegalkin matrix|Zhegalkin indices]] of the linear functions are in the columns on the right.)&lt;/small&gt;
{{Linear to patron 4-ary factions}}
{{Collapsible END}}

{{Collapsible START|3-ary noble indices &amp;nbsp; &lt;small&gt;(juniors)&lt;/small&gt;|collapsed wide}}
{| style=&quot;text-align: center; width: 100%;&quot;
|colspan=&quot;2&quot; style=&quot;text-align: left;&quot;|
The factions are shown in ten different colors, which are variants of the quadrant colors. &amp;nbsp;
&lt;small&gt;(There are three shades of red and green, and two shades of blue and yellow.)&lt;/small&gt;
|-
| [[File:3T linear to patron.svg|thumb|center|500px|truth tables]]
| [[File:3Z linear to patron.svg|thumb|center|500px|Zhegalkin indices]]
|}
{{Collapsible END}}

The following images show that pairs of complementary factions are in the same principality.

{{Collapsible START|quadrants 0 and 3|collapsed wide followed}}
{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center;&quot;
|- style=&quot;font-size: 65%;&quot;
! king
| 0
| 3808
| 26752
|-
! king index
| 0
| 14
| 104
|}
{{colorbox|#e30000}} &amp;nbsp; The patrons of the 8 even Walsh functions &lt;small&gt;(quadrant 0)&lt;/small&gt; are the red entries in these 3 principalities.&lt;br&gt;
{{colorbox|#00cc00}} &amp;nbsp; The patrons of their complements &lt;small&gt;(quadrant 3)&lt;/small&gt; are the green entries.
{| class=&quot;collapsible-with-classes collapsible collapsed light wide followed&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| truth tables
|-
| [[File:3T principality; faction size 1; king index 0.svg|400px]]
| [[File:3T principality; faction size 6; king index 14.svg|400px]]
| [[File:3T principality; faction size 1; king index 104.svg|400px]]
|}
{| class=&quot;collapsible-with-classes collapsible collapsed light wide&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| Zhegalkin indices
|-
| [[File:3Z principality; faction size 1; king index 0.svg|400px]]
| [[File:3Z principality; faction size 6; king index 14.svg|400px]]
| [[File:3Z principality; faction size 1; king index 104.svg|400px]]
|}
{{Collapsible END}}

{{Collapsible START|quadrants 1 and 2|collapsed wide}}
{| class=&quot;wikitable&quot; style=&quot;float: right; text-align: center;&quot;
|- style=&quot;font-size: 65%;&quot;
! king
| 5760
| 10920
|-
! king index
| 22
| 42
|}
{{colorbox|#ffb000}} &amp;nbsp; The patrons of the 8 odious Walsh functions &lt;small&gt;(quadrant 2)&lt;/small&gt; are the yellow entries in these 2 principalities.&lt;br&gt;
{{colorbox|#0055ff}} &amp;nbsp; The patrons of their complements &lt;small&gt;(quadrant 1)&lt;/small&gt; are the blue entries.
{| class=&quot;collapsible-with-classes collapsible collapsed light wide followed&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| truth tables
|-
| [[File:3T principality; faction size 4; king index 22.svg|400px]]
| [[File:3T principality; faction size 4; king index 42.svg|400px]]
|}
{| class=&quot;collapsible-with-classes collapsible collapsed light wide&quot; style=&quot;text-align: center;&quot;
!colspan=&quot;3&quot;| Zhegalkin indices
|-
| [[File:3Z principality; faction size 4; king index 22.svg|400px]]
| [[File:3Z principality; faction size 4; king index 42.svg|400px]]
|}
{{Collapsible END}}

===noble to prefect (linear)===

There are 256 nobles. A prefect is one of the 32 linears. Every linear is the prefect of 8 nobles.

{{Collapsible START|list &amp;nbsp; &lt;small&gt;(32 × 8 matrix)&lt;/small&gt;|collapsed wide followed}}
This table is an extension of the one shown above (from linear to patron). The leftmost noble column is equal to that of patrons.&lt;br&gt;
But here the assignment goes in the other direction. Each of the nobles on the right has the prefect on the left. &lt;small&gt;E.g. the nobles 3870 and 2600 have prefect Ж 17.&lt;/small&gt;

The nobles are represented by the integer values of their truth tables, which are also their Zhegalkin indices.&lt;br&gt;
The small numbers to their right are the juniors 0...255.&lt;br&gt;
&lt;small&gt;They correspond to 3-ary Boolean functions, so among them are also linears and nobles. The linears are highlighted in bold, and the nobles with a box.&lt;/small&gt;

The background colors of the nobles denote the [[Noble Boolean functions#quadrants|noble quadrants]].&lt;br&gt;
The tiny red numbers are the king indices. Together with the noble quadrants they denote the noble factions. &amp;nbsp; &lt;small opacity: .5;&gt;(The eleven king indices are 0, 2, 6, 8, 14, 22, 26, 42, 44, 104, 110.)&lt;/small&gt;

{{Noble to prefect 4-ary}}
{{Collapsible END}}

{{Collapsible START|3-ary subprefects &amp;nbsp; &lt;small&gt;(16 × 16 matrices)&lt;/small&gt;|collapsed wide}}
The right side of the following table shows the same information as the table above.&lt;br&gt;
Entries of one color in one matrix are the eight juniors in a row of the table above. &amp;nbsp; &lt;small&gt;(Linears are highlighted in bold, and the nobles with a crown.)&lt;/small&gt;

The left side of the table shows the prefects of all 3-ary Boolean functions. &lt;small&gt;(And the great prefects as sort keys.)&lt;/small&gt;&lt;br&gt;
It can be seen, that the {{w|Image (mathematics)|preimages}} of the 4-ary prefects are a refinement of the 3-ary prefects. This motivates the term ''subprefect''.

{{Noble to prefect 4-ary matrices}}

&lt;small&gt;A more detailed version of this table can be seen [[commons:Category:3-ary Boolean functions in octeract matrix; great subprefects|here]].&lt;/small&gt;
{{Collapsible END}}


[[Category:Linear and noble Boolean functions]]</text>
      <sha1>hb4q1mh6rwrgd588tgsfmk8sgnd7ene</sha1>
    </revision>
  </page>
  <page>
    <title>Internet Protocol Analysis/Collection</title>
    <ns>0</ns>
    <id>302547</id>
    <revision>
      <id>2692641</id>
      <parentid>2594441</parentid>
      <timestamp>2024-12-19T16:32:50Z</timestamp>
      <contributor>
        <username>Fkhfoukh</username>
        <id>2995234</id>
      </contributor>
      <minor />
      <comment>Learn More About IPTV: For a comprehensive understanding of IPTV, visit this detailed guide on IPTV</comment>
      <origin>2692641</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6051" sha1="mslhnimy8jnaiwerjswalw0x14y9d6f" xml:space="preserve">{{Noprint|{{Notice|To download this collection as a PDF, select '''&lt;code&gt;Download as PDF&lt;/code&gt;''' on the left.}}}}
== Internet Protocol Analysis ==
=== Learning Guide ===

This learning guide supports the Wikiversity course ''Internet Protocol Analysis'', available at http://en.wikiversity.org/wiki/Internet_Protocol_Analysis.

'''Learn More About IPTV''':

For a comprehensive understanding of IPTV, visit this [https://iptvsubscriptionhub.com/https-iptvsubscriptionhub-com-what-is-iptv-comprehensive-guide/ detailed guide on IPTV] which covers all the essential aspects, from technology to benefits.

== Overview ==
{{:Internet Protocol Analysis | Internet Protocol Analysis}}

== Lesson 1 - Introduction ==
{{:Internet Protocol Analysis/Introduction | Introduction}}
{{:Ipconfig/Default | Ipconfig}}
{{:Private_networks | Private Networks}}

== Lesson 2 - Packet Analyzers ==
{{:Internet Protocol Analysis/Packet Analyzers | Packet Analyzers}}
{{:Wireshark/Install | Install Wireshark}}
{{:Wireshark/Capture | Capture Network Traffic}}
{{:Wireshark/Display_filter | Filter Displayed Traffic}}
{{:Wireshark/Capture_filter | Filter Captured Traffic}}

== Lesson 3 - Link Layer ==
{{:Internet Protocol Analysis/Link Layer | Link Layer}}
{{:getmac | Display MAC Addresses Using Getmac}}
{{:Ipconfig/All | Display MAC Addresses Using Ipconfig}}
{{:MAC_address/OUI | Search for a MAC Address OUI}}
{{:Wireshark/Ethernet | Capture and Analyze Ethernet Traffic}}

== Lesson 4 - Address Resolution Protocol (ARP) ==
{{:Internet Protocol Analysis/Address Resolution Protocol | Address Resolution Protocol (ARP)}}
{{:Computer Networks/Management/Utilities/Arp/View | View the ARP Cache}}
{{:Computer Networks/Management/Utilities/Arp/Modify | Modify the ARP Cache}}
{{:Wireshark/Arp | Capture and Analyze Address Resolution Protocol (ARP) Traffic}}

== Lesson 5 - Internet Layer / IPv4 ==
{{:Internet Protocol Analysis/Internet Layer IPv4 | Internet Layer / IPv4}}
{{:Whois/IP_address | Search the Whois Database}}
{{:Wireshark/Arp | Capture and Analyze Address Resolution Protocol (ARP) Traffic}}
{{:Wireshark/IPv4_local | Capture and Analyze Local IPv4 Traffic}}
{{:Wireshark/IPv4_remote | Capture and Analyze Remote IPv4 Traffic}}
{{:Wireshark/IPv4_fragments | Capture and Analyze Fragmented IPv4 Traffic}}

== Lesson 6 - Subnetting ==
{{:Internet Protocol Analysis/Subnetting | Subnetting}}

== Lesson 7 - IPv6 ==
{{:Internet Protocol Analysis/IPv6 | IPv6}}
{{:Netsh/IPv6 | Configure IPv6 Settings}}
{{:Wireshark/IPv6_local | Capture and Analyze Local IPv6 Traffic}}
{{:Wireshark/IPv6_remote | Capture and Analyze Remote IPv6 Traffic}}
{{:Wireshark/IPv6_Teredo | Capture and Analyze IPv6 Teredo Traffic}}
{{:Wireshark/IPv6_6to4 | Capture and Analyze IPv6 6to4 Traffic}}
{{:Wireshark/IPv6_6in4 | Capture and Analyze IPv6 6in4 Traffic}}

== Lesson 8 - Internet Control Message Protocol (ICMP) ==
{{:Internet Protocol Analysis/Internet Control Message Protocol | Internet Control Message Protocol (ICMP)}}
{{:Wireshark/ICMP_Echo | Capture and Analyze ICMP Echo Traffic}}
{{:Wireshark/ICMP_Time_Exceeded | Capture and Analyze ICMP Time Exceeded Traffic}}
{{:Wireshark/ICMP_Trace | Capture and Analyze ICMP tracert/traceroute Traffic}}
{{:Wireshark/ICMPv6_Echo | Capture and Analyze ICMPv6 Echo Traffic}}
{{:Wireshark/ICMPv6_Time_Exceeded | Capture and Analyze ICMPv6 Time Exceeded Traffic}}
{{:Wireshark/ICMPv6_Trace | Capture and Analyze ICMPv6 tracert/traceroute Traffic}}
{{:Ping/MTU | Ping MTU}}

== Lesson 9 - Multicast ==
{{:Internet Protocol Analysis/Multicast | Multicast}}
{{:Wireshark/IPv4_multicast | Capture and Analyze IPv4 Multicast Traffic}}
{{:Wireshark/IPv6_multicast | Capture and Analyze IPv6 Multicast Traffic}}
{{:Wireshark/ICMPv6_NDP | Capture and Analyze ICMPv6 Neighbor Discovery Protocol (NDP) Traffic}}

== Lesson 10 - Transport Layer ==
{{:Internet Protocol Analysis/Transport Layer | Transport Layer}}
{{:Netstat/Statistics | Display Protocol Statistics}}
{{:Netstat/All | Display All Active Connections and Listening Ports}}
{{:Wireshark/UDP | Capture and Analyze User Datagram Protocol (UDP) Traffic}}
{{:Wireshark/TCP | Capture and Analyze Transmission Control Protocol (TCP) Traffic}}

== Lesson 11 - Address Assignment ==
{{:Internet Protocol Analysis/Address Assignment | Address Assignment}}
{{:Link-local_address | View and Test a Link-Local Address}}
{{:Wireshark/DHCP | Capture and Analyze Dynamic Host Configuration Protocol (DHCP) Traffic}}
{{:Wireshark/DHCPv6 | Capture and Analyze DHCPv6 Traffic}}

== Lesson 12 - Name Resolution ==
{{:Internet Protocol Analysis/Name Resolution | Name Resolution}}
{{:Hosts_file/View | View the Hosts File}}
{{:Hosts_file/Edit | Edit the Hosts File}}
{{:Computer Networks/Management/Utilities/Nslookup/Host | Display Host Addresses}}
{{:Computer Networks/Management/Utilities/Nslookup/Type | Display Other Record Types}}
{{:Computer Networks/Management/Utilities/Nslookup/Recurse | Simulate a Recursive Query}}
{{:Wireshark/DNS | Capture and Analyze Domain Name System (DNS) Traffic}}
{{:Wireshark/LLMNR | Capture and Analyze Link Local Multicast Name Resolution (LLNMR) Traffic}}
{{:Nbtstat | Display NetBIOS Over TCP/IP Statistics}}

== Lesson 13 - Application Layer ==
{{:Internet Protocol Analysis/Application Layer | Application Layer}}
{{:Wireshark/HTTP | Capture and Analyze Hypertext Transfer Protocol (HTTP) Traffic}}
{{:Wireshark/HTTPS | Capture and Analyze HTTP Secure (HTTPS) Traffic}}
{{:Wireshark/SMTP | Capture and Analyze Simple Mail Transfer Protocol (SMTP) Traffic}}

== Lesson 14 - Routing Protocols ==
{{:Internet Protocol Analysis/Routing | Routing Protocols}}
{{:Route_command/Print | Display the Local Routing Table}}
{{:Route_command/Modify | Modify the Local Routing Table}}

== Lesson 15 - Network Monitoring ==
{{:Internet Protocol Analysis/Network Monitoring | Network Monitoring}}
{{:SNMP_Service/Install | Install the SNMP Service}}
{{:SNMP_Service/Configure | Configure the SNMP Service}}
{{:SNMP_Service/Test | Test the SNMP Service}}

{{CourseCat}}</text>
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      <comment>Reverted edits by [[Special:Contributions/Fkhfoukh|Fkhfoukh]] ([[User_talk:Fkhfoukh|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]]</comment>
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== Internet Protocol Analysis ==
=== Learning Guide ===

This learning guide supports the Wikiversity course ''Internet Protocol Analysis'', available at http://en.wikiversity.org/wiki/Internet_Protocol_Analysis.

== Overview ==
{{:Internet Protocol Analysis | Internet Protocol Analysis}}

== Lesson 1 - Introduction ==
{{:Internet Protocol Analysis/Introduction | Introduction}}
{{:Ipconfig/Default | Ipconfig}}
{{:Private_networks | Private Networks}}

== Lesson 2 - Packet Analyzers ==
{{:Internet Protocol Analysis/Packet Analyzers | Packet Analyzers}}
{{:Wireshark/Install | Install Wireshark}}
{{:Wireshark/Capture | Capture Network Traffic}}
{{:Wireshark/Display_filter | Filter Displayed Traffic}}
{{:Wireshark/Capture_filter | Filter Captured Traffic}}

== Lesson 3 - Link Layer ==
{{:Internet Protocol Analysis/Link Layer | Link Layer}}
{{:getmac | Display MAC Addresses Using Getmac}}
{{:Ipconfig/All | Display MAC Addresses Using Ipconfig}}
{{:MAC_address/OUI | Search for a MAC Address OUI}}
{{:Wireshark/Ethernet | Capture and Analyze Ethernet Traffic}}

== Lesson 4 - Address Resolution Protocol (ARP) ==
{{:Internet Protocol Analysis/Address Resolution Protocol | Address Resolution Protocol (ARP)}}
{{:Computer Networks/Management/Utilities/Arp/View | View the ARP Cache}}
{{:Computer Networks/Management/Utilities/Arp/Modify | Modify the ARP Cache}}
{{:Wireshark/Arp | Capture and Analyze Address Resolution Protocol (ARP) Traffic}}

== Lesson 5 - Internet Layer / IPv4 ==
{{:Internet Protocol Analysis/Internet Layer IPv4 | Internet Layer / IPv4}}
{{:Whois/IP_address | Search the Whois Database}}
{{:Wireshark/Arp | Capture and Analyze Address Resolution Protocol (ARP) Traffic}}
{{:Wireshark/IPv4_local | Capture and Analyze Local IPv4 Traffic}}
{{:Wireshark/IPv4_remote | Capture and Analyze Remote IPv4 Traffic}}
{{:Wireshark/IPv4_fragments | Capture and Analyze Fragmented IPv4 Traffic}}

== Lesson 6 - Subnetting ==
{{:Internet Protocol Analysis/Subnetting | Subnetting}}

== Lesson 7 - IPv6 ==
{{:Internet Protocol Analysis/IPv6 | IPv6}}
{{:Netsh/IPv6 | Configure IPv6 Settings}}
{{:Wireshark/IPv6_local | Capture and Analyze Local IPv6 Traffic}}
{{:Wireshark/IPv6_remote | Capture and Analyze Remote IPv6 Traffic}}
{{:Wireshark/IPv6_Teredo | Capture and Analyze IPv6 Teredo Traffic}}
{{:Wireshark/IPv6_6to4 | Capture and Analyze IPv6 6to4 Traffic}}
{{:Wireshark/IPv6_6in4 | Capture and Analyze IPv6 6in4 Traffic}}

== Lesson 8 - Internet Control Message Protocol (ICMP) ==
{{:Internet Protocol Analysis/Internet Control Message Protocol | Internet Control Message Protocol (ICMP)}}
{{:Wireshark/ICMP_Echo | Capture and Analyze ICMP Echo Traffic}}
{{:Wireshark/ICMP_Time_Exceeded | Capture and Analyze ICMP Time Exceeded Traffic}}
{{:Wireshark/ICMP_Trace | Capture and Analyze ICMP tracert/traceroute Traffic}}
{{:Wireshark/ICMPv6_Echo | Capture and Analyze ICMPv6 Echo Traffic}}
{{:Wireshark/ICMPv6_Time_Exceeded | Capture and Analyze ICMPv6 Time Exceeded Traffic}}
{{:Wireshark/ICMPv6_Trace | Capture and Analyze ICMPv6 tracert/traceroute Traffic}}
{{:Ping/MTU | Ping MTU}}

== Lesson 9 - Multicast ==
{{:Internet Protocol Analysis/Multicast | Multicast}}
{{:Wireshark/IPv4_multicast | Capture and Analyze IPv4 Multicast Traffic}}
{{:Wireshark/IPv6_multicast | Capture and Analyze IPv6 Multicast Traffic}}
{{:Wireshark/ICMPv6_NDP | Capture and Analyze ICMPv6 Neighbor Discovery Protocol (NDP) Traffic}}

== Lesson 10 - Transport Layer ==
{{:Internet Protocol Analysis/Transport Layer | Transport Layer}}
{{:Netstat/Statistics | Display Protocol Statistics}}
{{:Netstat/All | Display All Active Connections and Listening Ports}}
{{:Wireshark/UDP | Capture and Analyze User Datagram Protocol (UDP) Traffic}}
{{:Wireshark/TCP | Capture and Analyze Transmission Control Protocol (TCP) Traffic}}

== Lesson 11 - Address Assignment ==
{{:Internet Protocol Analysis/Address Assignment | Address Assignment}}
{{:Link-local_address | View and Test a Link-Local Address}}
{{:Wireshark/DHCP | Capture and Analyze Dynamic Host Configuration Protocol (DHCP) Traffic}}
{{:Wireshark/DHCPv6 | Capture and Analyze DHCPv6 Traffic}}

== Lesson 12 - Name Resolution ==
{{:Internet Protocol Analysis/Name Resolution | Name Resolution}}
{{:Hosts_file/View | View the Hosts File}}
{{:Hosts_file/Edit | Edit the Hosts File}}
{{:Computer Networks/Management/Utilities/Nslookup/Host | Display Host Addresses}}
{{:Computer Networks/Management/Utilities/Nslookup/Type | Display Other Record Types}}
{{:Computer Networks/Management/Utilities/Nslookup/Recurse | Simulate a Recursive Query}}
{{:Wireshark/DNS | Capture and Analyze Domain Name System (DNS) Traffic}}
{{:Wireshark/LLMNR | Capture and Analyze Link Local Multicast Name Resolution (LLNMR) Traffic}}
{{:Nbtstat | Display NetBIOS Over TCP/IP Statistics}}

== Lesson 13 - Application Layer ==
{{:Internet Protocol Analysis/Application Layer | Application Layer}}
{{:Wireshark/HTTP | Capture and Analyze Hypertext Transfer Protocol (HTTP) Traffic}}
{{:Wireshark/HTTPS | Capture and Analyze HTTP Secure (HTTPS) Traffic}}
{{:Wireshark/SMTP | Capture and Analyze Simple Mail Transfer Protocol (SMTP) Traffic}}

== Lesson 14 - Routing Protocols ==
{{:Internet Protocol Analysis/Routing | Routing Protocols}}
{{:Route_command/Print | Display the Local Routing Table}}
{{:Route_command/Modify | Modify the Local Routing Table}}

== Lesson 15 - Network Monitoring ==
{{:Internet Protocol Analysis/Network Monitoring | Network Monitoring}}
{{:SNMP_Service/Install | Install the SNMP Service}}
{{:SNMP_Service/Configure | Configure the SNMP Service}}
{{:SNMP_Service/Test | Test the SNMP Service}}

{{CourseCat}}</text>
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|-
|class=&quot;walsh-weight&quot;| 4
|class=&quot;walsh-index&quot;| 15
|class=&quot;parity&quot;| 1
|class=&quot;leader&quot;| 14
|class=&quot;quadrant faction-4-odd&quot;| 3
|class=&quot;tt&quot;| !..! .!!. .!!. !..!
|class=&quot;intval&quot;| 38505
|class=&quot;zhe&quot;| 279
|class=&quot;tt border&quot;| .!!! !!!. !!!. !..!
|class=&quot;intval&quot;| 38782
|class=&quot;junior&quot;| 151
|}&lt;noinclude&gt;
----

styles: {{tl|Linear to patron 4-ary list/style.css}}

compare: {{tl|Linear to patron 4-ary factions}}

[[Category:Linear and noble Boolean functions]]
&lt;/noinclude&gt;</text>
      <sha1>ohdtyurlw4po5cylw9sjz88f37muytk</sha1>
    </revision>
  </page>
  <page>
    <title>African Arthropods/Arthropods on ''Ficus burkei''</title>
    <ns>0</ns>
    <id>306078</id>
    <revision>
      <id>2692598</id>
      <parentid>2692597</parentid>
      <timestamp>2024-12-19T13:05:24Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Ants */</comment>
      <origin>2692598</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5178" sha1="lmf9y5c4czv08z0ny8x3qcatycdbmgl" xml:space="preserve">==Parasitic wasps==
===Agaonidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===Epichrysomallidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===Eurytomidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===Pteromalidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===Encyrtidae===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===Eulophidae===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== Eupelmidae ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===Chalcididae===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps (Aculeata)==

===Bethylidae===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===Pemphredonidae===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

== Ants ==

=== Formicidae ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>lmf9y5c4czv08z0ny8x3qcatycdbmgl</sha1>
    </revision>
    <revision>
      <id>2692601</id>
      <parentid>2692598</parentid>
      <timestamp>2024-12-19T13:26:07Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Stinging wasps (Aculeata) */</comment>
      <origin>2692601</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5411" sha1="4x3rd2iz66zjdz1sbe59seuqeuzkowr" xml:space="preserve">==Parasitic wasps==
===Agaonidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===Epichrysomallidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===Eurytomidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===Pteromalidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===Encyrtidae===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===Eulophidae===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== Eupelmidae ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===Chalcididae===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps (Aculeata)==

===Bethylidae===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===Pemphredonidae===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===Pompilidae===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== Formicidae ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>4x3rd2iz66zjdz1sbe59seuqeuzkowr</sha1>
    </revision>
    <revision>
      <id>2692654</id>
      <parentid>2692601</parentid>
      <timestamp>2024-12-19T20:06:43Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Parasitic wasps */</comment>
      <origin>2692654</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5548" sha1="bsgb06enuhah1z6xdvtvbu8w5qkobwq" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===Encyrtidae===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===Eulophidae===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== Eupelmidae ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===Chalcididae===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps (Aculeata)==

===Bethylidae===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===Pemphredonidae===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===Pompilidae===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== Formicidae ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>bsgb06enuhah1z6xdvtvbu8w5qkobwq</sha1>
    </revision>
    <revision>
      <id>2692656</id>
      <parentid>2692654</parentid>
      <timestamp>2024-12-19T20:11:26Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2692656</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5697" sha1="3pqs31pbh0ys4a66cj8qxzhl0orumq8" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>3pqs31pbh0ys4a66cj8qxzhl0orumq8</sha1>
    </revision>
    <revision>
      <id>2692659</id>
      <parentid>2692656</parentid>
      <timestamp>2024-12-19T20:17:26Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera) */</comment>
      <origin>2692659</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5763" sha1="l25iacqkb3wnlu900ndm3ap7wqynxk8" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>l25iacqkb3wnlu900ndm3ap7wqynxk8</sha1>
    </revision>
    <revision>
      <id>2692671</id>
      <parentid>2692659</parentid>
      <timestamp>2024-12-19T20:28:37Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Beetles */</comment>
      <origin>2692671</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5836" sha1="gll5a0yg9ilj74zz4h63i3tehablltb" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microfreudea cyclica|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Sciobius pullus|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>gll5a0yg9ilj74zz4h63i3tehablltb</sha1>
    </revision>
    <revision>
      <id>2692677</id>
      <parentid>2692671</parentid>
      <timestamp>2024-12-19T20:32:54Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Beetles */</comment>
      <origin>2692677</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5829" sha1="58n6ck54elbnd8xxjbeneqheob39oa3" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Sciobius pullus|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>58n6ck54elbnd8xxjbeneqheob39oa3</sha1>
    </revision>
    <revision>
      <id>2692680</id>
      <parentid>2692677</parentid>
      <timestamp>2024-12-19T20:34:50Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Beetles */</comment>
      <origin>2692680</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5827" sha1="1iya1616beqs9vzrw8839sihgtvp80j" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''Lasioglossum'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>1iya1616beqs9vzrw8839sihgtvp80j</sha1>
    </revision>
    <revision>
      <id>2692687</id>
      <parentid>2692680</parentid>
      <timestamp>2024-12-19T20:41:17Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Halictidae */</comment>
      <origin>2692687</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5846" sha1="g98x2cslrrkobdx2xvz3hnkzqjfzqn8" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|Honey Bee (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>g98x2cslrrkobdx2xvz3hnkzqjfzqn8</sha1>
    </revision>
    <revision>
      <id>2692689</id>
      <parentid>2692687</parentid>
      <timestamp>2024-12-19T20:42:00Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Apidae */</comment>
      <origin>2692689</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5862" sha1="f9yhzjpmytesly61w2i00l7aw0di4ud" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Honey Bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>f9yhzjpmytesly61w2i00l7aw0di4ud</sha1>
    </revision>
    <revision>
      <id>2692693</id>
      <parentid>2692689</parentid>
      <timestamp>2024-12-19T20:43:56Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Apidae */</comment>
      <origin>2692693</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5870" sha1="3o73fklhnv3icaa6f0d8k5opx13919l" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''Lepisiota'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''Lepisiota'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>3o73fklhnv3icaa6f0d8k5opx13919l</sha1>
    </revision>
    <revision>
      <id>2692698</id>
      <parentid>2692693</parentid>
      <timestamp>2024-12-19T20:52:12Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Formicidae */</comment>
      <origin>2692698</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5918" sha1="e9gp9a4afe19cd9xx091l0x2fnc5eh3" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting honeydew from aphids (''Greenidea'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>e9gp9a4afe19cd9xx091l0x2fnc5eh3</sha1>
    </revision>
    <revision>
      <id>2692699</id>
      <parentid>2692698</parentid>
      <timestamp>2024-12-19T20:53:00Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Formicidae */</comment>
      <origin>2692699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5949" sha1="7sjhmoo420lx4dz3qbt9x0msm7i2fw4" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:honeydew|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>7sjhmoo420lx4dz3qbt9x0msm7i2fw4</sha1>
    </revision>
    <revision>
      <id>2692703</id>
      <parentid>2692699</parentid>
      <timestamp>2024-12-19T21:01:10Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Formicidae */</comment>
      <origin>2692703</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5961" sha1="pzybu0vooa657nepkmw35vzbysftlft" xml:space="preserve">==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
      <sha1>pzybu0vooa657nepkmw35vzbysftlft</sha1>
    </revision>
    <revision>
      <id>2692770</id>
      <parentid>2692703</parentid>
      <timestamp>2024-12-20T07:24:24Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2692770</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6468" sha1="3118tyio6d8o95x0kww8hhfasbnitgk" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. 

&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. Figweb: figs and fig wasps of the world. URL: www.figweb.org(Accessed on 20-12-2024).&lt;/ref&gt;

==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>3118tyio6d8o95x0kww8hhfasbnitgk</sha1>
    </revision>
    <revision>
      <id>2692771</id>
      <parentid>2692770</parentid>
      <timestamp>2024-12-20T07:28:08Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2692771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6523" sha1="5t1qrx3rnbm0q1mjktjkskoeauo6min" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. 

&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;

==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>5t1qrx3rnbm0q1mjktjkskoeauo6min</sha1>
    </revision>
    <revision>
      <id>2692772</id>
      <parentid>2692771</parentid>
      <timestamp>2024-12-20T07:42:26Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>refs</comment>
      <origin>2692772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7196" sha1="0q7l4s2xa1vxkp8bgahvtr18m70iqcw" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''Ficus burkei'')&lt;ref name=Figwebburkei2024&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>0q7l4s2xa1vxkp8bgahvtr18m70iqcw</sha1>
    </revision>
    <revision>
      <id>2692773</id>
      <parentid>2692772</parentid>
      <timestamp>2024-12-20T07:44:02Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>link</comment>
      <origin>2692773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7215" sha1="mety4d6pi9pfw11ys3668j83xhf4emu" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]'')&lt;ref name=Figwebburkei2024&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>mety4d6pi9pfw11ys3668j83xhf4emu</sha1>
    </revision>
    <revision>
      <id>2692774</id>
      <parentid>2692773</parentid>
      <timestamp>2024-12-20T08:08:10Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <minor />
      <origin>2692774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7216" sha1="f83hwrahy0r9yys2ixy3uoqpihec51n" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]'').&lt;ref name=Figwebburkei2024&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>f83hwrahy0r9yys2ixy3uoqpihec51n</sha1>
    </revision>
    <revision>
      <id>2692775</id>
      <parentid>2692774</parentid>
      <timestamp>2024-12-20T08:28:14Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2692775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7317" sha1="fy3co81va09x2ms2e88sntgaorqsmmy" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>fy3co81va09x2ms2e88sntgaorqsmmy</sha1>
    </revision>
    <revision>
      <id>2692778</id>
      <parentid>2692775</parentid>
      <timestamp>2024-12-20T10:21:08Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Moths and butterflies */</comment>
      <origin>2692778</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7601" sha1="jr2v4sr81k5kbrqpdejx8kzan0l4vxx" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Mantispidae==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>jr2v4sr81k5kbrqpdejx8kzan0l4vxx</sha1>
    </revision>
    <revision>
      <id>2692779</id>
      <parentid>2692778</parentid>
      <timestamp>2024-12-20T10:22:07Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera) */</comment>
      <origin>2692779</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7604" sha1="m7ju92bnzrbzbg8yez3wjbwfmqn6rz9" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Mantispidae==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>m7ju92bnzrbzbg8yez3wjbwfmqn6rz9</sha1>
    </revision>
    <revision>
      <id>2692780</id>
      <parentid>2692779</parentid>
      <timestamp>2024-12-20T10:23:34Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Apidae */</comment>
      <origin>2692780</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7636" sha1="iug2xutxnjgz8fuqx844newtoa8cnzu" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''; This resin is used as propolis
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Mantispidae==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>iug2xutxnjgz8fuqx844newtoa8cnzu</sha1>
    </revision>
    <revision>
      <id>2692781</id>
      <parentid>2692780</parentid>
      <timestamp>2024-12-20T10:23:54Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Apidae */</comment>
      <origin>2692781</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7637" sha1="9wq7qp7wlvwa3awrvd53ed7vll8p7di" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as propolis.
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Mantispidae==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>9wq7qp7wlvwa3awrvd53ed7vll8p7di</sha1>
    </revision>
    <revision>
      <id>2692782</id>
      <parentid>2692781</parentid>
      <timestamp>2024-12-20T10:24:26Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Apidae */</comment>
      <origin>2692782</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7652" sha1="7gbhlno4hokej7671i9aez2usi6eydd" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as [[w:propolis|propolis]].
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Mantispidae==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>7gbhlno4hokej7671i9aez2usi6eydd</sha1>
    </revision>
    <revision>
      <id>2692783</id>
      <parentid>2692782</parentid>
      <timestamp>2024-12-20T10:28:00Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Mantispidae */</comment>
      <origin>2692783</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7670" sha1="75glvtl09o843yg6xhf9s8lscllf6g6" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as [[w:propolis|propolis]].
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==[[w:Mantispidae|Mantisflies]]==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>75glvtl09o843yg6xhf9s8lscllf6g6</sha1>
    </revision>
    <revision>
      <id>2692784</id>
      <parentid>2692783</parentid>
      <timestamp>2024-12-20T10:50:08Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Mantisflies */</comment>
      <origin>2692784</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7905" sha1="epgtkj1iw8jp6qcj2wzm9har57o6p5x" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as [[w:propolis|propolis]].
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==[[w:Mantispidae|Mantisflies]]==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==[[w:Chrysopidae|Green lacewings]]==
&lt;gallery mode=packed heights=200&gt;
Chrysopidae iNat 41850659.jpg|Green lacewing larva
Chrysopidae iNat 41850677.jpg|Green lacewing larva
Chrysopidae iNat 41852893.jpg|Green lacewing egg
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>epgtkj1iw8jp6qcj2wzm9har57o6p5x</sha1>
    </revision>
    <revision>
      <id>2692785</id>
      <parentid>2692784</parentid>
      <timestamp>2024-12-20T10:53:30Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Parasitic wasps */</comment>
      <origin>2692785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7954" sha1="j5cl65cvhpi5g4egprme3wmo37ehzfn" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps of the [[w:Proctotrupomorpha|Proctotrupomorpha]]==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as [[w:propolis|propolis]].
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==[[w:Mantispidae|Mantisflies]]==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==[[w:Chrysopidae|Green lacewings]]==
&lt;gallery mode=packed heights=200&gt;
Chrysopidae iNat 41850659.jpg|Green lacewing larva
Chrysopidae iNat 41850677.jpg|Green lacewing larva
Chrysopidae iNat 41852893.jpg|Green lacewing egg
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>j5cl65cvhpi5g4egprme3wmo37ehzfn</sha1>
    </revision>
    <revision>
      <id>2692786</id>
      <parentid>2692785</parentid>
      <timestamp>2024-12-20T10:54:59Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <origin>2692786</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8017" sha1="7vf91jivp49qs0brlr4e2jsn2jkr1nr" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps of the [[w:Proctotrupomorpha|Proctotrupomorpha]]==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

==Parasitic wasps of the [[w:Ichneumonoidea|Ichneumonoidea]]==
===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==Bees==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as [[w:propolis|propolis]].
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==[[w:Mantispidae|Mantisflies]]==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==[[w:Chrysopidae|Green lacewings]]==
&lt;gallery mode=packed heights=200&gt;
Chrysopidae iNat 41850659.jpg|Green lacewing larva
Chrysopidae iNat 41850677.jpg|Green lacewing larva
Chrysopidae iNat 41852893.jpg|Green lacewing egg
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
{{BookCat}}</text>
      <sha1>7vf91jivp49qs0brlr4e2jsn2jkr1nr</sha1>
    </revision>
    <revision>
      <id>2692787</id>
      <parentid>2692786</parentid>
      <timestamp>2024-12-20T10:58:35Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Bees */</comment>
      <origin>2692787</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8027" sha1="93r9vlmcb199a75mqrg4p01ntnz911v" xml:space="preserve">The genus ''[[w:Ficus|Ficus]]'' includes the cultivated ''[[w:Fig|Ficus carica]]'' which is native to the [[w:Mediterranean basin|Mediterranean basin]]. The genus, however, includes more than 750 species of fig tree worldwide, including 25 species native to South Africa, and 112 species in the [[w:Afrotropical realm|Afrotropics]]. ''Ficus'' species are fairly well known for their remarkable interaction with fig wasps, but there are a host of other animals that interact with these trees (such as the many species of birds that eat the fruit and the [[w:syconium|syconium]] that encloses the flowers and fruit.&lt;ref name=figweb&gt; van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Figs%20and%20fig%20wasps/index.htm Figweb: Figs and fig wasps of the world]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt; 

This page explores the diversity of arthropods that have been found associated with the common wild fig (''[[w:Ficus burkei|Ficus burkei]]''). This species is found in southern and eastern Africa, from South Africa to southern Kenya and Uganda.&lt;ref name=Figwebburkei2024&gt;van Noort, S. &amp; Rasplus, JY. 2024. [https://www.figweb.org/Ficus/subgenus_urostigma/Section_Galoglychia/Subsection_Chlamydodorae/Ficus_burkei.htm Ficus burkei (Miq.) Miq. 1867 (Common Wild Fig)]. www.figweb.org. Accessed on 20-12-2024.&lt;/ref&gt;


==Parasitic wasps of the [[w:Proctotrupomorpha|Proctotrupomorpha]]==
===[[w:Agaonidae|Agaonidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===[[w:Epichrysomallidae|Epichrysomallidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===[[w:Eurytomidae|Eurytomidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
&lt;/gallery&gt;
===[[w:Pteromalidae|Pteromalidae]] ([[w:Fig wasp|Fig wasps]])===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;

===[[w:Encyrtidae|Encyrtidae]]===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg|''Homalotylus'' sp.
Encyrtidae lateral view with annotations.jpg|Encyrtid wasp, possibly ''Psyllaephagus'' sp.
&lt;/gallery&gt;

===[[w:Eulophidae|Eulophidae]]===
&lt;gallery mode=packed heights=200&gt;
Tetrastichinae iN 226221658.jpg|Subfamily Tetrastichinae 
Entedoninae iN 228280710.jpg|Subfamily Entedoninae
&lt;/gallery&gt;

=== [[w:Eupelmidae|Eupelmidae]] ===
&lt;gallery mode=packed heights=200&gt;
Brasema 2024 06 26 iN 226745239 02.jpg|''Brasema'' sp.
Brasema 2024 06 26 iN 226745239 01.jpg|''Brasema'' sp.
&lt;/gallery&gt;

===[[w:Chalcididae|Chalcididae]]===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg|''Brachymeria'' sp.
&lt;/gallery&gt;

==Parasitic wasps of the [[w:Ichneumonoidea|Ichneumonoidea]]==
===Braconidae===
&lt;gallery mode=packed heights=200&gt;
Brachistinae iN 119243068.jpg|A braconid wasp (Brachistinae) ovipositing into a ''Ficus burkei'' syconium
&lt;/gallery&gt;

==Stinging wasps ([[w:Aculeata|Aculeata]])==

===[[w:Bethylidae|Bethylidae]]===
&lt;gallery mode=packed heights=200&gt;
Bethylinae inaturalist28661558.jpg|Subfamily Bethylinae
&lt;/gallery&gt;


===[[w:Pemphredonidae|Pemphredonidae]]===
&lt;gallery mode=packed heights=200&gt;
Polemistus braunsii iNaturalist 228280708.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
Polemistus braunsiii iNaturalist229894212.jpg|''Polemistus braunsii'' collecting resin (dried ''Ficus'' latex)
&lt;/gallery&gt;

===[[w:Pompilidae|Pompilidae]]===
&lt;gallery mode=packed heights=200&gt;
Pompilidae inaturalist 123577538.jpg|Spider-hunting wasp (probably ''Auplopus'' sp.)
Pompilidae inaturalist 46961473.jpg||Spider-hunting wasp (probably ''Auplopus'' sp.)
&lt;/gallery&gt;

== Ants ==

=== [[w:Ant|Formicidae]] ===
&lt;gallery mode=packed heights=200&gt;
Lepisiota, Elisabethiella stueckenbergi inaturalist 124232407.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Elisabethiella stueckenbergi'')
Lepisiota, Lachaisea inaturalist 122348752.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) carrying a fig wasp (''Lachaisea'' sp.)
Lepisiota, Greenidea inaturalist 122435456.jpg|An ant, (''[[w:Lepisiota|Lepisiota]]'' sp.) harvesting [[w:Honeydew (secretion)|honeydew]] from aphids (''[[w:Greenidea|Greenidea]]'' sp.)
&lt;/gallery&gt;

==[[w:Bee|Bees]]==

===Apidae===
&lt;gallery mode=packed heights=200&gt;
Apis mellifera collecting dried latex 2024 06 26 iN 226725676 03.jpg|[[w:Western honey bee|Honey Bee]] (''Apis mellifera'' ssp. ''scutellata'') collecting dried latex from a damaged stem of ''Ficus burkei''. This resin is used as [[w:propolis|propolis]].
&lt;/gallery&gt;

===Halictidae===
&lt;gallery mode=packed heights=200&gt;
Lasioglossum iN 41852932.jpg|''[[w:Lasioglossum|Lasioglossum]]'' sp.
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''[[w:Harmonia axyridis|Harmonia axyridis]]''
Microfreudea cyclica iNat 226950451.jpg|''[[w:Microweiseini|Microfreudea cyclica]]''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''[[w:Otiorhynchini|Sciobius pullus]]''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Psylloids (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A [[w:Homotomidae|homotomid]] psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''[[w:Greenidea|Greenidea]]'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. ([[w:Triozidae|Triozidae]]) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==[[w:Mantispidae|Mantisflies]]==
&lt;gallery mode=packed heights=200&gt;
Afromantispa iN 44503671 2020 04 28 a.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa iN 44503671.jpg|A mantisfly, ''Afromantispa'' sp.
Afromantispa 2020 05 09 15 38 03 iN 46988873.jpg|A mantisfly, ''Afromantispa'' sp.
&lt;/gallery&gt;

==[[w:Chrysopidae|Green lacewings]]==
&lt;gallery mode=packed heights=200&gt;
Chrysopidae iNat 41850659.jpg|Green lacewing larva
Chrysopidae iNat 41850677.jpg|Green lacewing larva
Chrysopidae iNat 41852893.jpg|Green lacewing egg
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;

==References==
{{reflist}}
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      <text bytes="126464" sha1="nmz4clcl7a11ufwh9ygldrsaxiyaabs" xml:space="preserve">=Event=
On Monday 28 June 1897, Queen Victoria hosted a garden party at Buckingham Palace, inviting between 5,000 and 6,000 people. This party was the final official event of the London Diamond Jubilee celebrations. The Queen released to the press the names of people invited, which means the newspapers could print some or all of this list. The very long article in the London ''Morning Post'', for example, prints what may be the comprehensive list of those invited, although two columns are illegible in some places.

The original newspaper account seems to have been published by the ''Court Circular'', and then the popular newspapers reprinted pieces of that story, many adding contextualizing paragraphs of their own. Some of these later reports are quite long, perhaps 5 or more full columns. Sometimes the newspapers included short descriptions of the women's dresses, suggesting that for the list of people invited, the source was the ''Court Circular'', but the parts of the stories devoted to context, history or fashion might have been written by a reporter present at the event.

==Logistics==
* 28 June 1897, Monday, in the gardens at Buckingham Palace, hosted by Queen Victoria.
* Between 5,000 and 6,000 guests were invited.
* Many visitors from the empire who were in town for the Jubilee celebrations were invited to this garden party.
* The weather was fine, having improved since the day before.
* The garden party was held in the grounds around Buckingham Palace, and the Palace itself was open and available for guests to visit:&lt;blockquote&gt;Great preparations had been made in the splendid grounds adjoining the Royal Palace for the party, the whole scene presenting a fascinating appearance. The beautifully-kept grounds were partially covered with tents and marquees for the convenience of the many guests, and the lovely lake was really in the hands of the Queen’s bargemen, who had charge of the many boats which had been placed on the extensive ornamental waters for the use of guests. There was also plenty of music, several regimental bands being in attendance, while for those who wished to become acquainted with the valuable pictures and works art which are to be found at the Royal residence, all the State and reception rooms of the Palace were thrown open.&lt;ref name=&quot;:2&quot;&gt;“The Queen’s Garden Party. Brilliant Scene at Buckingham Palace.” ''Globe'' 29 June 1897, Tuesday: 6 [of 8], Col. 3a–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18970629/050/0006. Print p. 6.&lt;/ref&gt; (6, Col. 3a)&lt;/blockquote&gt;
*The streets around the entrances to Buckingham Palace were lined with spectators beginning hours before the Queen was to arrive:&lt;blockquote&gt;Although the Garden Party was not timed to commence until after five o’clock, the Mall from Marlborough House to Buckingham Palace was well lined by two o’clock, and an hour afterwards large crowds, for the most part composed of ladies, had taken up their positions. This was also the case along Constitution-hill, where the assembly which had gathered to witness the Queen’s arrival at the Palace from Windsor nad [sic] to a large extent remained. The heat was somewhat oppressive, but the trees along the Mall and the Green Park afforded welcome shelter. Many ladies had evidently come prepared for a long wait, as they had provided themselves with the now familiar camp stool, which is always prominent on these occasions. On the other hand, the police were waging war against the men who frequent such places with stools and forms, and as soon as any of them put in an appearance they were quickly pounced upon by the officers, who at once proceeded to destroy the intended stands before the eyes of the helpless owners. Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.&lt;ref name=&quot;:2&quot; /&gt; (6, Col. 3a)&lt;/blockquote&gt;
==Related Events==
This garden party was the culminating event of the official celebrations for Queen Victoria's Diamond Jubilee, and more specific events led up to it:

# Trip from Windsor to Paddington Station Queen Victoria and a large retinue traveled by train from Windsor to Paddington Station the day before, preceded on an earlier train by &quot;the royal equipages sent from Buckingham Palace for the use of the Queen and her suite,&quot; which were&lt;blockquote&gt;First came the splendied semi-state landau in which the Queen made her now famous journey on June 22d. It was preceded by scarlet-coated outriders, and horsed by four magnificent bays driven by postilions in navy blue and white uniforms. Two similar carriages followed, and these were in turn succeeded by a number of pair-horse clarences for the conveyance of the household and suite, and several breaks and ‘buses for luggage. A captain's escort, furnished by the 2d Life Guards, and commanded by Captain Ellison, clattered along in rear of the carriages, and took up a position opposite the spot where, by prior arrangement, Her Majesty’s saloon was to be brought to a standstill. These magnificent troops, riding their great black horses, and with the sunlight dancing upon their nodding plumes, and reflected by their burnished helmets, cuirasses, and trappings, made a very fine show indeed. The escort did not carry the colour, as it did on the 21st, nor was it accompanied by the regimental trumpets.&lt;ref name=&quot;:0&quot;&gt;&quot;Jubilee Festivities. The Queen Again in London. Interesting Functions. A Visit to Kensington. The Garden Party.&quot; ''North British Daily Mail'' 29 June 1897, Tuesday: 5 [of 8], Col. 3a–7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002683/18970629/083/0005. Print p. 5.&lt;/ref&gt;{{rp|5, Col. 3b}}&lt;/blockquote&gt;
# Reception at Paddington
# Visit to Kensington
# Kensington to Buckingham Palace
# The Garden Party
# Return to Windsor by Way of Paddington

=== Foreign Admirals ===
On 29 June 1897, the day after the garden party, the ''North British Daily Mail'' reports that, after the Queen's garden party, the foreign admirals would return to Spithead for a tour around the dockyard and luncheon:&lt;blockquote&gt;THE FLEET AT SPITHEAD&lt;p&gt;
The fleet at Spithead was again illuminated last night, the railway companies having duplicated the ordinary train service to bring visitors down. The Koenig Wilhelm was to have sailed on Sunday evening, but her departure has been deferred, and last night her officers gave a private dinner party aboard for the anniversary of the Queen’s Coronation. All the commissioned ships in the harbour were dressed at noon. A royal salute was fired. The [Col. 6c–7a] foreign admirals will return from their visit to London on the occasion of the Queen’s garden party to be conducted round the dockyard to-day, and they will be entertained to luncheon.&lt;ref name=&quot;:0&quot; /&gt;{{rp|5, Col. 6c–7a}}&lt;/blockquote&gt;

=== Colonial Premiers ===
The day of the garden party the colonial premiers attended a meeting with Secretary of State for the Colonies, [[Social Victorians/People/Chamberlain|Joseph Chamberlain]]:&lt;blockquote&gt;THE COLONIAL PREMIERS

The whole of the Colonial Premiers went to the Colonial Office yesterday for further conference with Mr Chamberlain, who received them in his private room, attended by Mr F. H. Wilson, legal assistant, Mr Reid and the Hon. T. Cochrane, M.P., assistant private secretaries. The conference lasted hours, and was of a strictly private and confidential character, the matters discussed involving several points of high State policy.

Premiers will be entertained at Warwick Castle by the Earl and Countess of Warwick on July 15th. On the same occasion the Attorney General of Queensland will present a loving cup from Warwick, in Queensland, to the old county town of Warwick, from which it takes its name. He will be accompanied by the Colonial troopers.&lt;ref name=&quot;:0&quot; /&gt;{{rp|5, Col. 7a}}&lt;/blockquote&gt;

For these visitors to London during the Diamond Jubilee, the next major social event was on 15 July, at Warwick Castle, hosted by [[Social Victorians/People/Warwick|Daisy, Countess of Warwick and Francis, 5th Earl of Warwick]], although perhaps some attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]].

== Who Was Present ==
In the absence of a copy of the report about the garden party in the ''Court Circular'', the newspaper account with the fullest list of names is from the ''Morning Post'', although people further down the list can be impossible to identify, and two full columns are damaged (Col. 7 on p. 4 and Col. 1 on p. 5).&lt;ref name=&quot;:1&quot;&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005.&lt;/ref&gt; Whenever possible, then, what is here has been amended with other newspaper reports that have names to help decipher the illegible ones in the ''Morning Post'' account. The names in the Morning Post are grouped, mostly by rank and name.

=== People of Color at This Event ===

One purpose of a closer look at this event is to get a more precise list of names of people of color from the various countries in the empire, who were not recognized and thus not named in newspaper descriptions of other events. For example, the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]] was said to include a number of South Asian dignitaries, but because the Duchess did not release to the newspapers the names of those who were invited, those dignitaries went mostly unnamed in the newspaper reports, if their presence was noted at all. Besides the South Asian guests invited to this garden party, some South Asian visitors to London were spectators as well:&lt;blockquote&gt;Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.&lt;ref name=&quot;:2&quot; /&gt; (6, Col. 3a)&lt;/blockquote&gt;In a section on what people — mostly women — wore, the reporter for the ''Daily News'' said,&lt;blockquote&gt;Suffice to say, the modistes had done their best, and that their achievements excited general admiration. Here and there, however, was an Eastern beauty whose golden lace drapery, loosely enveloping a figure that owed nothing to the corset, challenged comparison, we will not say with what success, with the European model. In the almost entire absence of uniforms or Court dress, the costumes of the East Indian notables lent colour to the assemblage, while their pearls and diamonds, the wealth of Ormuz and of Ind, were not allowed to pass unobserved.&lt;ref name=&quot;:3&quot; /&gt; (5, Col. 6b)&lt;/blockquote&gt;

=== People Invited ===
# Queen Victoria, with escort and attendants
## Captain's Escort of the 2nd Life Guards
## The Duchess of Buccleuch, Mistress of the Robes
## The Dowager Lady Churchill, Lady in Waiting
## The Hon. Harriet Phipps, Woman of the Bedchamber
## Maids of Honour in Waiting
### The Hon. Mary Hughes
### The Hon. Aline Majendie
## the Earl of Kintore, Lord in Waiting
## Captain Drummond, Groom in Waiting
## Equerries in Waiting
### Major-General Sir John M'Neill, V.C.
### Lieutenant Colonel Davidson, M.V O. [sic]
#Grand Duke and Grand Duchess Serge of Russia
#Princess Henry of Battenberg, with attendants
##Miss Minnie Cochrane
##Colonel John Clerk, C.S.I., C.V.O.
#Her Imperial Majesty the Empress Frederic, attended by 
##the Dowager Lady Ampthill
##Lord Harris
##Colonel S. Waller
##Princess Hatzfeldt Trachenberg
##Count Seckendorff
##Baron and Baroness Reischach
#Their Royal Highnesses the Prince and Princess of Wales, with attendants
##Lady Suffield, Lady in Waiting
##Miss Knollys, Woman of the Bedchamber
##Lord Colville of Culross, K.T., G.C.V.O., Chamberlain to the Princess of Wales
##The Earl of Gosford, K.P., Lord in Waiting
##General Sir D. Probyn, G.C.V.O., K.C.B., K.C.S.I., V.C, Comptroller
##Sir Francis Knollys, K.C.M.G., C.B., Groom in Waiting
##Major-General Stanley Clarke, C.M.G., Equerry in Waiting
#Princess Victoria of Wales
#Their Royal Highnesses Prince and Princess Charles of Denmark
#Their Royal Highnesses the Grand Duke and Grand Duchess of Mecklenburg-Strelitz, attended by
##Lady Caroline Cust
##Mr. Hugo Erskine Wemyss
##Count Reventlow Criminil
##Baron von der Wense
#Their Royal Highnesses Prince and Princess Christian, attended by
##Baroness von und zu Egloffstein
##Colonel the Hon. Charles Eliot
#Her Highness Princess Victoria
#His Highness Prince Christian Victor
#His Highness Prince Albert of Schleswig-Holstein
#Her Royal Highness Princess Louise Marchioness of Lorne and the Marquis of Lorne, attended by
##Lady Sophia Macnamara
##[[Social Victorians/People/Arthur Collins|Colonel Arthur Collins]], M.V.O.
#Their Royal Highnesses Prince and Princess Henry of Prussia, attended by
##Admiral of the Fleet Sir Edmund Commerell
##Baron and Baroness Seckendorff
##Count Hahn
##Captain Muller
#Their Royal Highnesses the Duke and Duchess of Saxe-Coburg and Gotha, attended by
##The Hon. Mrs. Monson
##His Excellency Herr von Schön
##Captain the Hon. D. J. Mouson [sic, s/b Monson?], M.V.O.
##Mr. A. D. J. Monson
##Captain von Ruxleben
#Princess Beatrice of Saxe-Coburg and Gotha
#The Hereditary Prince of Saxe-Coburg and Gotha
#Their Royal Highnesses the Duke and Duchess of Connaught and Strathearn, attended by
##Colonel and the Hon. Mrs. A. Egerton
#Her Royal Highness the Duchess of Albany, attended by
##Sir Robert and Lady Collins
##Miss Potts
#Her Royal Highness Princess Frederica of Hanover and Baron von Pawel Raminingen, attended by
##Mr. and Mrs. Charles Wood
#His Royal Highness the Duke of Cambridge, attended by
##Colonel A. C. FitzGeorge, C.B.
#Her Royal Highness the Duchess of Teck and his Highness the Duke of Teck, attended by
##Lady Katherine Coke
##The Hon. A. Nelson Hood
#Her Royal Highness Princess Louise Duchess of Fife and the Duke of Fife
#His Highness the Prince and her Royal Highness Princess Frederic Charles of Hesse, attended by
##The Hon. A. Hay
##Fraulein von Tasmund
##Baron von Kotwitz
#Their Highnesses Prince and Princess Aribert of Anhalt, attended by
##Miss Deverell
##Major Evan Martin
#Her Royal Highness the Hereditary Princess of Saxe-Meiningen and her Serene Highness Princess Feodore of Saxe-Meiningen, attended by
##The Hon. Aubrey FitzClarence
##Miss von Dreskan
##Baron von Roeder
#His Serene Highness the Prince of Schaumburg-Lippe
#Their Highnesses Prince and Princess Edward of Saxe-Weimar
#Her Serene Highness Princess Victor of Hohenlohe
#Countess Gleichen (x2)
#Their Serene Highnesses Prince and Princess Adolphus of Teck
#The Prince Francis and Prince Alexander of Teck
#His Highness Prince Augustus Leopold of Saxe-Coburg
#Their Serene Highnesses Prince and Princess Blucher von Wahlstatt
#Their Serene Highnesses Prince and Princess Joachim Murat
#Their Serene Highnesses [[Social Victorians/People/Pless|Prince and Princess Hans Henry Pless]]
#Prince and Princess Loewenstein
#Their Serene Highnesses the Duke and Duchess of Arenberg
#Prince Victor Duleep Singh
#Prince Frederick Duleep Singh
#Princess Duleep Singh (x2)
#ARGENTINE REPUBLIC — M. Florencio Dominguez and M. Carlos Dominguez
#BADEN — Herr yon Brauer, Mr. Brook Taylor, and Baron Bohlen Halbach
#BAVARIA — His Royal Highness the Prince Rupert, General Sir L. Gardiner, K.C.V.O., C.B., Major Fairholme, Lieutenant-Colonel Emile von le Bret Nucourt, and Captain Othon von Stettin
#BELGIUM — His Serene Highness the Prince Charles de Ligne, Princess de Ligne, Madlle. de Ligne, Mr. C. lnnes Ker, Count de Jonghe d'Ardoye, and the Marquis d’Asshe
#BOLIVIA — M. Caso, Mr. Conway Seymour, M. Pedro Suarez, Madame Suarez, and M. Adolfo Bolivian
#BRAZIL — M. [[Social Victorians/People/Souza Correa|de Souza Correa]] [Corréa?]
#BULGARIA — Their Royal Highnesses the Prince and Princess of Bulgaria, Colonel J. R. Slade, C.B., Madame Petrow Tchomakoff, Count Robert de Bourboulon, Lieutenant-Colonel Marcoff, Major Petrew, Captain Stoïanow, and Mr. Martin Furth
#CENTRAL AMERICA (Greater Republic) — M. Medina and Miss Medina
#CHILI — M. Ramon Subercasseaux and Mr. Raglan Somerset
#CHINA — His Excellency Chang Yen Hoon, Colonel Mark Bell, V.C,. Mr. Liang, Mr. Jui, and Mr. Koo
#COREA — His Excellency Min Young Hwan, Major A. Cavendish, Mr. Min Young Chan, Mr. Min Shangho, and Mr. von Rautenfeld
#COSTA RICA — Senor Don Demetrio Iglesias, Mr. C. Alban Young, Dona Eudoxia Castro, Señorita Maria Iglesias, Don Ricardo Fernandez Guardia, and Dona Christina Castro Keith
#DENMARK — His Royal Highness the Prince Waldemar, Major-General Arthur Ellis, C.S.I., M. Charles Rothe, and Captain Evers
#EGYPT — Prince Mohammed Ali Pasha, Colonel Larking, Tigrane Pasha, Colonel Aziz Bey, Mr. George Smart, Said Zoulfikar Bey
#ECUADOR — M. Navares, Colonel Concha
#FRANCE — General Davoust, Duc d'Auerstadt, Duchesse d'Auerstadt, and Madlle. Davoust, Colonel Brabazon, Colonel Dawson, General Hagron, M. Crozier, Colonel Humbert, and Captain Riviers de Mauny
#GERMANY — His Royal Highness the Prince Albert of Prussia, Prince Regent of Brunswick, Major-General Sir C. du Piat, K.C.B., Colonel Grierson, Lieutenant-General von Plessen, Colonel von Arnim, Captain Fischel, Count von der Schulenberg (Hofmarschall), Major Freiherr von Stein, Dr. Schreibe, Captain von Unzer
#GREECE — M. Rangabi, Mr. R. D. Norton
#GUATEMALA — Dr. Cruz, Madlles. Cruz (2), Señor Estrada
#HAWAIIAN ISLANDS — Mr. S. M. Damon, Captain the Hon. H. Napier, Major Curtis P. Jaukea
#HESSE — Their Royal Highnesses the Grand Duke and Grand Duchess of Hesse, Colonel the Hon. H. Byng, C.B., Baroness de Grancy, Baron Riedesel zu Eisenbach, Baron de Genadius Grancy
#ITALY — Their Royal Highnesses the Crown Prince and Princess, the Earl of Clarendon, Colonel Needham, Countess Giulia Trigona, Lieutenant-General Terzaghi, Major Cavaliere Viganoni, Captain Cavaliere Merli Miglietti, Count Romnaldo Trigona, Cavaliere F. Comotto
#JAPAN — His Imperial Highness the Prince Arisugawa, Mr. R. F. Synge, Captain Beaumont, R.N., Marquis Ito, Mr. S. Saito, Marquis Kido, Captain Funaki, Lieutenant-Colonel Murata, Lieutenant Kato, Mr. Nabeshima
#LIBERIA — Mr. H. Hayman
#LUXEMBURG — His Royal Highness the Hereditary Grand Duke of Luxemburg, Colonel H. D. Browne, Baron Ritter yon Grünstein
#MECKLENBURG-SCHWERIN — His Excellency Herr D. yon Vietinghoff, Mr. Eyre A. Crowe
#MEXICO — Don Antonio Mier y Celis, Mr. Arnold Royle, C.B., Don Francisco R. Gallardo, Don Eustagino dc Escaudon, and Captain Don Ponfirio Diaz
#MONTENEGRO — His Highness the Prince Danilo, Major the Hon. C Harbord, Colonel Djurcovitch, and Captain Pejanovitch
#NETHERLANDS — Count van Lynden, Countess van Lynden, Mr. Horace West, and Count W. de Bylandt
#PARAGUAY — M. E. Machain and Madame Machain
#PERSIA — His Imperial Highness the Prince Amir Khan, General Sir Thomas Gordon, K.C.I.E., C.B., C.S.I.[,] Mr. Harry Churchill, General Karim Khan, Mirza Ahmad Khan, Mirza Ohaness Khan, Mirza Mohamad Ali Khan
#PERU — Senor Canevaro, Duchesse de Zoagli Canevaro, Dr. Don A. N. Puente, Don Alfredo Elster, and Don Carlos von der Heyde
#PORTUGAL — His Royal Highness the Duke of Oporto, Major the Hon. H. C Legge, M.V.O., Colonel Duval Telles, Captain Moreira de Sà, Major d'Albuquerque, and Lieutenant Jose de Melie[?]
#ROME — Right Rev. Monsignore Sambucetti, [[Social Victorians/People/Stonor|Hon. Harry Stonor]], Right Rev. Monsignore Belmont, the Right Rev. Monsignore de Vaz, Marchesi and Marchesa Muccioli, of the Noble Guard
#ROUMANIA — General Pancovici, Colonel G. P. Georgescu
#RUSSIA — Their Imperial Highnesses the Grand Duke Serge and Grand Duchess Feodrowna, the Grand Duke Cyril, Lord Churchill, Lieutenant-Colonel Waters, Countess Olsouffiew, Princess Youssoupoff, Princess Lobanoff de Rostow, General Stépanoff, Colonel Gadon, and Prince Youssoupoff, Colonel Clements, Mr. Alexander Gordon Ross, and Sub-Lieutenant N. Coubé (A.D.C. to Grand Duke Cyril)
#SAXE-COBURG — His Royal Highness the Prince Philip of Saxe-Coburg, Captain Walter Campbell, and Herr von Schön
#SAXE-WEIMAR — His Highness the Prince Hermann of Saxe-Weimar, Mr. Frederick Campbell, and Count Zeppelin
#SAXONY — His Royal Highness the Prince Frederick Augustus, Duke of Saxony, Colonel Howard, Freiherr yon Reitzenstern, First Lieutenant von Metzsch, and Baron von Oppell
#SERVIA — M. Mijatovich and Madame Mijatovich
#SIAM — His Royal Highness the Crown Prince and the Prince Mahit of Siam, Colonel E. H. Sartorius, V.C., Lieutenant-Colonel Rajavallabha, Lieutenant-Colonel C. Vernon Hume, Colonel Indaraty, Surgeon-Major Yarr
#SPAIN — Duke of Sotomayor, Captain the Hon. A. Greville, Señor José Caro, Señor Alfonso Merry del Val, and Señor Benitez al Villar
#SWEDEN AND NORWAY — His Royal Highness the Prince Eugène of Sweden and Norway, Captain G. L. Holford, Count G. Gyldenstolpe, Captain Roeder, Captain Baron Cederstrom
#TURKEY — Munir Pasha, Major Surtees, Brigadier-General Nassir Pasha, Captain Enver Bey, Colonel Gordon Ponsonby
#UNITED STATES — His Excellency the Hon. Whitelaw Reid, Mrs. Whitelaw Reid, Colonel Hallam Parr, Major-General Nelson A. Miles, Mrs. Nelson Miles, Rear-Admiral Joseph N. Miller, Captain M. P. Maus, Mr. Ogden Mills, Mrs. Ogden Mills, Mr. G. Creighton Webb, Mr. Erskine Hewett, Commander W. H. Emory, Lieutenant Philip Andrews, Lieutenant T. S. Rogers
#URUGUAY — Dr. Alberto Nin, Madlle. Nin, Don Alfonso Saenz de Zumaran, Don Luis Posadas, Colonel C. Robido
#WURTEMBURG— His Royal Highness the Duke Albert of Wurtemburg, Colonel C. Swaine, Lieutenant-General von Bilfinger, First Lieutenant Count von Degenfeld- Schonburg; five officers of the Queen's German Regiment: Major C. R. Burn (in attendance), Lieutenant-Colonel von Falkenhayn, Major von Arnim, First Lieutenant Baron von Moeller-Lilienstern, First Lieutenant von Gerlach, Second Lieutenant von Studnitz
#&quot;Native Princes, and gentlemen and ladies accompanying them&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##His Highness the Raja of Kaparthala
##His Highness the Thakur Sahib of Morvi, K.C.I.E.
##His Highness the Thakur Sahib of Gondal, C.I., and her Highness the Maharani of Gondal, C.I.
##Colonel Maharaj Dhiraz
##Sir Pratab Singh, K.C.S.I.
##Thakur Hari Singh[,?]
##Kunwar Dhokal Singh
##Rajah Ajit Singh of Khetri, attended by
##Rajkumar Unmaid Singh of Shahpura, attended by
###Colonel Trevor (in attendance upon the Rajah Ajit Singh of Khetri and the Rajkumar Unmaid Singh of Shahpura)
##Bijey Singh
##Sir Jamaetjee Jejeebhoy, Bart., C.S.I., Miss Jejeebhoy, Mr. Jejeebhoy
##Mr. and Mrs. Powrala
##Major J. G. Turner and Mrs. Turner
##Mr. A. R. Wood and Mrs. Wood
#The &quot;officers of the Imperial Service Troops, with British officers and ladies&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Captain Mir Hashim Ali Khan Hyderabad-Resaldar
##Major Sunayat Singh, Kashmir
##Commandant Abdul Ganny, Gwalior
##Commandant Gooind, Rao Matkar, Indore
##Commandant Mirza Kurim Beg, Bhopal
##Rai Bahadur Dhunpat Rai, Jeypore
##Commandant Nand Singh, Patiala
##Commandant Rai Bahadur Thakur Dip Sing, Bikanir
##Commandant Chatru Singh, Bhartpur
##Resaldar Abdul Majid Khan, Babawalpur
##Commandant Daud Khan, Ulwar
##Commandant Nazir Khan, Rampur
##Risalda-Major Didar Singh, Sindi
##Risaldar-Major Kishan Singh, Nabha
##Risaldar Hara Singh, Karpurthala
##Risaldar Dhan Singhi, Bhavnagar
##Colonel H. Melliss, C.S.I., and Mrs. Melliss
##Major F. H. R. Drummond and Mrs. Drummond
##Captain F. Angelo
##Lieutenant H. Coape-Smith
##Captain G. F. Chenevix-Trench
#The &quot;officers of Native Cavalry Corps with British officers and ladies&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Risaldar-Major Baha-ud-din-Khan
##Sardar Bahadur, A.D.C. to Viceroy
##Risaldar-Major Sayyid Abdul Aziz
##Risaldar-Major Khan Bahadur
##Risaldar-Major Izzat Khan
##Risaldar-Major Hukam Singh
##Risaldar-Major Sher Singh
##Risaldar-Major Husain Khan
##Risaldar-Major Mangal Singh
##Risaldar-Major Kesar Singh
##Risaldar- Major Faiz Khan
##Risaldar-Major Muhammad Umar Khan
##Risaldar-Major Ali Mahomed Khan
##Risaldar-Major Mihrab Ali Khan
##Risaldar Kaddam Khan
##Risaldar Jahanzir Khan
##Risaldar Nadir Khan
##Risaldar Mir Haidar Shah Khan
##Risaldar Makbul Khan
##Risaldar Net Ram
##Ressaidar Gurdatt Singh
##Subadar Muhammed Beg Junadar
##Abdul Karin Khan
##Lieutenant-Colonel J. C. H. Gordon and Mrs. Gordon
##Major A. Phayre and Mrs. Phayre
##Captain C. F. Campbell
##Captain P. Melville, in attendance on his Highness Thakur Sahib of Morvi
##Captain M'Cartney Filgate, in attendance on their Highnesses the Thakur Sahib and Maharani of Gondal
##Mr. Nowroz
##M. Parveez
##Sir M. Mansherjee Bhownaggree, M.P.
##Mr. Percy Armytage and Mrs. Armytage
##Mr. Frank Cook, C.I.E., and Mrs. Frank Cook
#The &quot;commanding officers of Colonial contingents, with the ladies accompanying them&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Colonel the Hon. M. and Mrs. Aylmer, Canada
##Colonel and Mrs. Lassetter, New South Wales
##Major Reay, Victoria
##Colonel Pitt, New Zealand
##Major and Miss King, Queensland
##Lieutenant and Mrs. Phillips, Cape of Good Hope
##Lieutenant-Colonel Rowell, South Australia
##Major Strickland, Western Australia
##Captain Shepstone, Natal
##Major and Miss Reeves, Ceylon
##Mr. Badeley, Hong Kong
##Colonel Walker, C.M.G., and Mrs. Walker, Straits Settlements
##Captain Lucie Smith, Jamaica
##Lieutenant-Colonel E. B. M'lnnis, C.M.G., and Mrs. M'lnnis, British Guiana
##Major Rooks, Trinidad
##Captain Bernard, Malta
##Captain Kershaw, Cyprus
##Captain and Mrs. Middlemist, Gold Coast
##Inspector Hook, Lagos
##Captain Blakeney, Sierra Leone
##Lieutenant Festing, Royal Niger Company
##Captain Flint, British North Borneo Company
##The Hon. M. Gifford, Rhodesian Horse
##The following British officers attached: Lieutenant-Colonel Boulton, Lieutenant-Colonel Prior, Lieutenant-Colonel Tucker, Lieutenant-Colonel Domville, Lieutenant-Colonel Gibson, and Lieutenant-Colonel Tyrwhitt
#The &quot;gentlemen representing the various races in the Island of Ceylon&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)
##Maha Mudaliyar don Solomon Dias Bandaranaihe
##The Hon. Alexander Dealius Sonewiratne
##M. E. Rowland Goonoratne
##M. Charles de Soysa Dessanayaka
##Panabokko Jikiri Banda
##Nugawela Kuia Banda
##Kobbokeduwe Loku Banda
##M. E. S. W. Senathi rajah [sic] and Mrs. Senathi
##M. J. H. de Saram and Miss de Saram
##M. P. Ramanathan
##M. Saunders and Miss Saunders
#The &quot;members of the Corps Diplomatique and other foreigners of distinction&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)
##The Russian Ambassador, Madame de Staal, Madlle. de Staal, Madame de Stoeckl, Princess de San Donato, Madame Yermoloff, Madlle. Yermoloff, the Councillor, three Secretaries, and four Attachés of Embassy
##The German Ambassador, Countess Paul Hatzfeldt-Wildenburg, her Serene Highness Princess Hans Hohenlohe-Oehringen, Baroness yon Eckardtstein, the Councillor, two Secretaries, three Attachés of Embassy, and the Director of the Chancery
##The Austro-Hungarian Ambassador, Countess Deym, Countess Isabella Deym, Countess Clary Aldringen, Baroness Ferstel, the Councillor, two Secretaries, and four Attachés of Embassy
##The French Ambassador, Baroness de Courcel[,] Madlle. de Courcel, Madame Geoffray, the Minister Plenipotentiary, five Secretaries, and three Attachés of Embassy
##The Italian Ambassador, Princess Ruspoli, three Secretaries, and three Attachés of Embassy
##The Spanish Ambassador, Countess de Casa Valencia; Mesdlles. de Alcala Galiano (2), Marquise de Guiria, Donna de Zea Bermudez, Countess de Morella, Donna de Ia Camara y Livermore, three Secretaries, and four Attachés of Embassy
##The Turkish Ambassador, Madame Antbopoulos, the Councillor, and two Secretaries of Embassy
##The United States Ambassador, Mrs. Hay, Miss Hay, Mrs. Henry White, Mrs. Carter, Mrs. Colwell, two Secretaries, one Attaché of Embassy, and the Private Secretary to the Ambassador
##The Argentine Minister, Madame Dominguez, Mesdlles. Dominguez (3), and the Secretary of Legation
##The Persian Minister, and one Secretary of Legation
##The Danish Minister, Madame de Bille, Madame Gosch, and the Secretary of Legation
##The Siamese Minister, Mrs. Verney, Miss Verney, Mrs. Loftus, the Councillor, the Secretary, the Attaché, and the Interpreter to the Legation
##The Liberian Minister
##The Roumanian Minister and the Councillor of the Legation
##The Netherlands Minister, Baroness de Goltstein d'Oldenaller, Baroness Schimmelpenninck van der Oye, and the Councillor of Legation
##The Belgian Minister, the Councillor, and two Secretaries of Legation
##The Mexican Minister, Madame Yturbe, Madame Romero, Madame Farias, Madame Garcia, two Secretaries and three Attachés of Legation
##The Japanese Minister, Madame Kato, two Secretaries, and three Atachés [sic] of Legation
##The Minister for Sweden and Norway, Countess Lewenhaupt, and the Attaché of Legation
##The Chinese Minister, Lady Macartney, the English Secretary, three Secretaries, and four Attachés of Legation
##The Portuguese Minister, Madlle. de Quilinan, three Secretaries, and one Attaché of Legation
##The Swiss Minister, Madame Bourcart, Madame de Salis, the Secretary, and the Attaché of Legation
##The Haytian Chargé d’Affaires
##The Chargé d’Affaires of Greece, Madame Metaxas, and the Attaché
##The Chargé d’Affaires of Chile and Madame Bascunan
##Two Secretaries and one Attaché of the Brazilian Legation
##Count E. van Rosen
##Mr. Hippolyte de Aranjo
##Vice-Admiral Montt
##Mr. Pinto, Mrs. Pinto
##Mr. and Mrs. Scaramanga
##Vicomte de Galard
##Dr. Arnold, and Madlle. von Rappoport
##Mrs. John Meiggs, Miss Meiggs
##Miss Margaret Butler
##Mrs. Henry Morgan
##Hon. Chauncey Depew
##Mr. and Mrs. James Taylor
##Mr. and Mrs. Charles Marshall
##Mr. and Mrs. Edmund Bayliss
##Mrs. Colgate
##Miss Furniss
##Miss Wells
##Miss Harris
##Hon. Levi P. Morton, Mrs. Morton, and the Misses Morton
##The Bishop of Illinois and Mrs. Leonard, Miss Leonard
##The Bishop of Albany and Mrs. Doane
##The Bishop of New York and Mrs. Potter
##the Bishop of Minnesota and Mrs. Whipple
##Mr. and Mrs. Walter Burns
##Mrs. Douglas Grant
##Miss Scott
##Mrs. Grace, Miss Margarita Grace
##Mrs. Wentworth
##Miss van Wart
##M. Valentin de Courcel
##Madame la Marquise de Talleyrand Perigord
##Comte Boson de Perigord
##Vicomte d'Espenilles
##Madame and Madlle. Thierry Delanoue
##Madlle. de la Cherè
##M. Cellerier
##M. and Madame Delawarre
##Madame Evelina Fenzi
##Count A. Zannini
##M. and Madame Jules Cottran
##Chevalier E. Mazzuechi
##Signor A. Tedeschi
##Signor A. Mariotti
##Captain Lucian von Ziegler
##Chevalier Lieutenant von Barry
##Baron Georg Rothschild
##Privy Councillor Count Berchtold
##Baron G. E. Levi, Baroness Levi
##Commander E. Philipson, Mrs. E. Philipson
##The Duke and Duchess of San Germano Calabritto
##The Marquis of San Vito
##Donna Lidia Serramezzana
##Donna Margherita Chigi
##Marchioness Vitelleschi
##Chevalier Elia
##Count de Franqueville
##Count Urbain Chevrau
##M. Marcel Fonquier
##M. Baudon de Mony, Madame Baudon de Mony
##Duchess de Rohan
##Marquis de Lastorgrie, Marchioness de Lastorgrie
##Count de Boisgelin, Countess L. de Boisgelin
##M. Stern, Madame Stern, Madlle. Stern
##Count Charles du Luart
##General de Saucy
##M. E. Seydoux
##Count Jean de Madre
##M. de Monbrison
##Baron de la Chevrelière
##Count de la Villestreux, Countess de la Villestreux
##Count Urbain de Maille, Countess Urbain de Maille
##General Faveret de Kerbrich
##Monsieur de la Haye Jousselin
##Baronne Faveret de Kerbrich
##Colonel Matton
##M. Ferinier Didet
##Madame Ferinier Didet
##Donna Isabella Colonna, Donna Victoria Colonna
##Pom-k-Soh
##Madame Reyntiens
##Marquis de Fuente Hermosa
##Herr Rudolf Swobody
##M. Lauritz Tuxen
##Duchesse de Baiten
##M. de Marcoarti
##Comte de Heeren, Madlle. de Heeren
##Monsieur M. de Mauny Talvande
##Senor Don Nicolas Campero
##Lieutenant Charny
##Lieutenant Sanders
##Madame and Madlle. de Mouni
##Comtesse de Montsoulmin
#&quot;Foreign Admirals and Commanding Officers and Staffs&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 3a / Col. 3b)
##Austrian Admiral Baron von Spaun, Commander von Ziegler, Lieutenant Retter yon Barry, Lieutenant Mitchell, R.N. (attached)
##Danish Admiral H. H. Koch, Captain Waudel, Lieutenant Middelboc, Lieutenant Majendie, R.N. (attached)
##French Admiral C. F. E. De Courthille, Captain Germinet, Commander Poidlone, Lieutenant Perdriel, Sub-Lieutenant de Caqueray, Lieutenant Phillimore, R.N. (attached)
##Italian Admiral C. E. Morin, Commander Count Prasca, Lieutenant Lunghetti, Lieutenant Count Morano, Lieutenant Henderson. R.N. (attached)
##German Admiral his Royal Highness Prince Henry of Prussia, Captain Muller, Lieutenant von Spee, Sub-Lieutenant Wittman, Lieutenant Garforth, R.N. (attached)
##Japanese Admiral H.I.H. Prince Arizugawa, Captain Miura, Commander Tsuda, Lieutenant Stewart, R.N. (attached)
##Netherlands Admiral F. K. Englebrecht, Captain de Groot, Lieutenant Baron von Hardenbrock, Lieutenant Woolcombe, R.N. (attached)
##Norwegian Rear-Admiral von Krogh, Captain Muller, Lieutenant Petersen, Lieutenant Kerr Pearse, R.N. (attached)
##Portuguese Captain Barreto de Vascomellos, Captain de Cartillo, Lieutenant Trye, R.N. (attached)
##Russian Admiral Nicholas Skrydloff, Captain Domojiroff, Lieutenant Stetsenkoff, Lieutenant Twisleton Wykeham Fiennes, R.N. (attached)
##Spanish Admiral Don Segismundo Bermijo y Merelo, Captain Don Antonio Eulate y Fery, Lieutenant Don Juan Romero, Lieutenant Don Antonio Romero, Lieutenant Fair, R.N. (attached)
##Swedish Admiral A. F. H. Klintberg, Captain Ingelman, Commander Flack, Lieutenant Alton, R.N. (attached)
##United States Admiral J. N. Miller, Lieutenaut Richmond (attached)
##Captain de Mar E. Guerra
##Captain R. S. D. Cumins
#The Lord Lieutenant of Ireland and Countess Cadogan
#The Right Hon. the Speaker and Mrs. Gully, Miss Gully, and Miss Shelly Gully
#Cardinal Vaughan
#Right Hon. the Lord Mayor and Lady Mayoress, and Misses Faudel Phillips (2)
#The Gold Stick in Waiting, Silver Stick in Waiting, Silver Stick Adjutant in Waiting
#Officer Commanding 1st Life Guards and five officers
#Officer Commanding 2nd Life Guards and four officers
#Officer Commanding Royal Horse Guards and four officers
#Officer Commanding 2nd Dragoons and three officers
#Field Officer in Brigade Waiting, Adjutant in Brigade Waiting
#Commanding Officer Grenadier Guards
#Commanding Officer Coldstream Guards
#Commanding Officer Scots Guards, a Regimental Adjutant
#Commanding Officer 1st, 2nd, and 3rd Battalions Grenadier Guards and three officers of each Battalion
#Commanding Officer 1st and 2nd Battalions Coldstream Guards and three officers of each Battalion
#Commanding Officer 1st and 2nd Battalions of Scots Guards and three officers of each Battalion
#Commanding Officer Woolwich District and six officers
#Commanding Officer R.H.A. Home District and two officers
#Commanding Officer R.E. and four officers
#Commanding Officer 2nd Battalion Lincolnshire Regiment and three officers
#Commanding Officer Royal Marines (Chatham) and four officers
#Commanding Officer Royal Marines (Portsmouth) and two officers
#Four officers of the Honourable Corps of the Gentlemen at Arms
#Archbishops — Canterbury, York, Armagh, Ontario, Rupertsland
#Dukes and Duchesses
##The Duke and Duchess of [[Social Victorians/People/Argyll|Argyll]]
##The Duke and Duchess of [[Social Victorians/People/Abercorn|Abercorn]]
##The Duchess of De Baileu
##The Duke and Duchess of [[Social Victorians/People/Buccleuch|Buccleuch]]
##The Duchess of [[Social Victorians/People/Cleveland|Cleveland]]
##The Duke and Duchess of [[Social Victorians/People/Devonshire|Devonshire]]
##The Duchess of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Hamilton]]
##The Duke and Duchess of [[Social Victorians/People/Leeds|Leeds]]
##The Duke and Duchess of [[Social Victorians/People/Marlborough|Marlborough]]
##The Duke and Duchess of [[Social Victorians/People/Manchester|Manchester]]
##The Duke and Duchess of [[Social Victorians/People/Montrose|Montrose]]
##The Duke and Duchess of [[Social Victorians/People/Newcastle|Newcastle]]
##The Duke of [[Social Victorians/People/Norfolk|Norfolk]]
##The Duke of [[Social Victorians/People/Northumberland|Northumberland]]
##The Duke and Duchess of [[Social Victorians/People/Portland|Portland]]
##The Duke of [[Social Victorians/People/Richmond and Gordon|Richmond and Gordon]]
##The Duke and Duchess of [[Social Victorians/People/Roxburghe|Roxburghe]]
##The Duke and Duchess of [[Social Victorians/People/Somerset|Somerset]]
##The Duke and Duchess of [[Social Victorians/People/Sutherland|Sutherland]]
##The Duke and Duchess of St. Albans
##The Duke and Duchess of Wellington
##The Duchess of [[Social Victorians/People/Westminster|Westminster]]
#Marquises and Marchionesses
##The Marquis of Abergavenny
##The Marchioness of Ailesbury
##The Marquis and Marchioness of Ailsa
##The Marquis of Anglesey
##The Marquis and Marchioness of [[Social Victorians/People/Breadalbane|Breadalbane]]
##The Marchioness of [[Social Victorians/People/Marlborough#Marchioness of Blandford|Blandford]]
##The Marquis and Marchioness of Bristol
##The Marquis of [[Social Victorians/People/Camden|Camden]]
##The Marquis and Marchioness of Conyngham
##Dowager [Marchioness of] Conyngham
##The Marchioness of Cassar de Sai[n]
##The Marquis and Marchioness of Cholmondeley
##The Marquis of D'Auerstadt
##The Marquis and Marchioness [[Social Victorians/People/Stonor|D'Hautpoul]]
##The Marquis and Marchioness of Downshire
##Dowager [Marchioness of] Downshire
##The Marquis and Marchioness of [[Social Victorians/People/Hamilton Temple Blackwood|Dufferin and Ava]]
##The Marquis and Marchioness of [[Social Victorians/People/Exeter|Exeter]]
##The Marquis and Marchioness of Granby
##The Marchioness of [[Social Victorians/People/Florence Rawdon-Hastings Chetwynd|Hastings]]
##The Marquis and Marchioness of [[Social Victorians/People/Bective|Headfort]]
##The Marquis and Marchioness of Hertford
##The Marquis and Marchioness of Huntly
##The Marquis and Marchioness of [[Social Victorians/People/Abercorn#James Hamilton, Marquess of Hamilton|Hamilton]]
##The Marquis and Marchioness of [[Social Victorians/People/Lansdowne|Lansdowne]]
##The Marquis and Marchioness of Lothian
##Dowager (Marchioness of) [[Social Victorians/People/Londonderry|Londonderry]]
##The Marquis and Marchioness of [[Social Victorians/People/Londonderry|Londonderry]]
##The Marquis and Marchioness of [[Social Victorians/People/Ormonde|Ormonde]]
##The Marchioness of [[Social Victorians/People/Queensberry|Queensberry]]
##The Marquis and Marchioness of [[Social Victorians/People/Ripon|Ripon]]
##The Marquis and Marchioness of [[Social Victorians/People/Salisbury|Salisbury]]
##The Marquis and Marchioness of [[Social Victorians/People/Tweeddale|Tweeddale]]
##Dowager (Marchioness of) [[Social Victorians/People/Tweeddale|Tweeddale]]
##John Stewart-Murray, [[Social Victorians/People/Atholl|Marquess of Tullibardine]]
##Lawrence, [[Social Victorians/People/Zetland|Marquess of Zetland]] and Lilian, [[Social Victorians/People/Zetland|Marchioness of Zetland]]
#Earls and Countesses 
##Countess of Aberdeen and Dowager Countess of Aberdeen
##Earl and Countess of Albemarle and Dowager Countess of Albemarle
##Earl and Countess of Ancaster
##Earl and Countess of Amherst
##Earl of Ava
##Earl and Countess of Antrim
##Earl and Countess of Aylesford
##Earl and Countess of Annesley
##Earl and Countess of Airlie
##Earl and Countess of Arran
##Earl of Aberdeen
##Earl and Countess of Bandon
##Countess of Bantry
##Earl and Countess of Beauchamp
##Earl and Countess of Bathurst and Dowager Countess of Bathurst
##Countess of Bective
##Earl and Countess of Belmore
##Earl of Bradford
##Countess of Bremer
##Earl and Countess of Brownlow
##Earl and Countess of Buckinghamshire
##Earl of Burford
##Earl and Countess of Cairns
##Earl and Countess of Caledon
##Earl of Camperdown
##Earl of Cardigan
##Earl and Countess of Carnarvon and Dowager Countess of Carnarvon
##Earl of Carnwath
##Earl and Countess of Carrington
##Earl and Countess of Carysfort
##Earl and Countess of Castlestuart
##Earl and Countess of Cathcart
##Earl and Countess of Cavan
##Earl and Countess of Chesterfield
##Earl and Countess of Chichester
##Dowager Countess of Clancarty
##Countess of Clanwilliam
##Earl and Countess of Compton
##Countess of Cottenham
##Earl of Courtown
##Earl and Countess of Cowper
##Earl and Countess of Cranbrook
##Earl and Countess of Craven and Dowager Countess of Craven
##Earl and Countess of Crawford
##Earl of Crewe
##Earl and Countess of Cork and Orrery
##Earl and Countess of Coventry
##Countess of Cromartie and Dowager Countess of Cromartie
##Earl and Countess of Dalkeith
##Earl and Countess of Dartmouth
##Earl and Countess of De Grey
##Dowager Countess of De La Warr
##Earl and Countess of Denbigh
##Earl and Countess of Derby
##Earl and Countess of Donoughmore
##Earl and Countess of Drogheda
##Earl of Ducie
##Earl and Countess of Dudley and Dowager Countess of Dudley
##Earl and Countess of Dundonald
##Earl and Countess of Dunmore
##Earl and Countess of Dunraven
##Earl of Durham
##Earl and Countess of Eglinton and Winton
##Earl of Eldon
##Earl and Countess of Ellesinere
##Earl and Countess of Enniskillen
##Earl and Countess of Erne
##Earl and Countess of Errol
##Earl and Countess of Essex and Dowager Countess of Erroll
##Earl of Euston
##Earl and Countess of Feversham
##Earl and Countess of Fingall
##Earl of Fortescue
##Earl and Countess of Gainsborough
##Earl and Countess of Galloway
##Earl and Countess of Glasgow
##Countess of Gosford
##Earl and Countess of Granard
##Countess of Granville
##Earl and Countess of Grey
##Countess of Grosvenor
##Countess of Guilford
##Earl and Countess of Harewood and Dowager Countess of Harewood
##Earl and Countess of Harrington
##Earl and Countess of Hopetoun
##Earl and Countess of Huntingdon
##Earl and Countess of Harrowby
##Countess of Hohenau
##Countess of Howe
##Earl and Countess of Iddesleigh
##Earl and Countess of Jersey
##Earl and Countess of Kenmare
##Earl of Kerry
##Earl and Countess of Kilmorey
##Earl of Kimberley
##Earl and Countess of Kingston
##Earl of Kinnoull
##Josephine, Countess Kinsky
##Earl and Countess of Kintore
##Countess of Leitrim
##Earl and Countess of Lanesborough
##Countess of Lathom
##Earl and Countess of Lauderdale
##Countess of Leicester
##Earl and Countess of Leven and Melville
##Earl and Countess of Lichfield
##Earl and Countess of Limerick
##Earl and Countess of Lindsay
##Earl and Countess of Lisburne
##Earl and Countess of Listowel
##Earl and Countess of Londesborough
##Earl and Countess of Longford
##Earl and Countess of Lonsdale and Dowager Countess of Lonsdale
##Earl and Countess of Loudoun
##Earl and Countess of Lovelace
##Earl and Countess of Lucan
##Countess of Lytton
##Countess of Macclesfield
##Earl and Countess of Malmesbury and Dowager Countess of Malmesbury
##Earl and Countess of Mar
##Earl and Countess of Mar and Kellie and Dowager Countess of Mar and Kellie
##Earl and Countess of Mayo and Dowager Countess of Mayo
##Countess of Meath
##Countess of Metaxas
##Earl and Countess of Mexborough
##Earl and Countess of Minto
##Earl of De Montalt
##Earl and Countess of Morley
##Earl and Countess of Morton and Dowager Countess of Morton
##Earl of Nelson
##Earl and Countess of Norbury
##Earl of Northbrook
##Earl and Countess of Northesk and Dowager Countess of Northesk
##Earl and Countess of Onslow
##Earl of Orford
##Countess of Oxford
##Earl and Countess of Pembroke
##Countess of Percy
##Earl and Countess of Portarlington
##Earl and Countess of Portsmouth
##Earl and Countess of Powis
##Earl and Countess of Radnor
##Earl and Countess of Ravensworth
##Earl and Countess of Roden
##Earl and Countess of Romney
##Lawrence, [[Social Victorians/People/Zetland|Earl of Ronaldshay]]
##Earl of Rosebery
##Earl and Countess of Rosse
##Earl and Countess of Rosslyn and Dowager Countess of Rosslyn
##Earl of Sandwich
##Earl of Scarbrough
##Earl and Countess of Selborne
##Countess of Selkirk
##Countess of Shaftesbury
##Dowager Countess of Shrewsbury and Talbot
##Earl and Countess of Spencer
##Earl and Countess of Stamford
##Earl and Countess of Stanhope
##Earl and Countess of St. Germans
##Earl of Stradbroke
##Earl of Strafford
##Earl and Countess of Suffolk and Berkshire
##Earl and Countess of Temple (of Stowe)
##Earl and Countess of Verulam
##Earl and Countess of Waldegrave
##Earl and Countess of Warwick
##Earl and Countess of Westmeath
##Earl and Countess of Wharncliffe
##Elizabeth, Dowager Countess of Wilton and Isabella, Dowager Countess of Wilton
##Earl and Countess of Winchilsea and Nottingham
##Earl and Countess of Winterton
##Earl and Countess of Yarborough and Dowager Countess of Yarborough
#Viscounts&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 3c / Col. 4a) and Viscountesses
##Viscount and Viscountess of Boyne
##Viscountess of Cantelupe
##Viscount and Viscountess of Castlerosse
##Viscount and Viscountess of Chelsea
##Viscount and Viscountess of Chetwynd
##Viscountess of Chewton
##Viscount and Viscountess of Clifden
##Viscount and Viscountess of Cobham
##Viscount and Viscountess of Coke
##Viscount of Corry
##Viscount and Viscountess of Cranborne
##Viscount of Crichton
##Viscount and Viscountess of Cross
##Viscount of Curzon
##Viscount and Viscountess of Dalrymple
##Viscount and Viscountess of Deerhurst
##Viscount and Viscountess of De Vesci
##Viscount and Viscountess of Dillon
##Viscount of Doneraile
##Viscount and Viscountess of Duncannon
##Viscount of Dungarvan
##Viscount and Viscountess of Ebrington
##Viscount and Viscountess of Emlyn
##Viscount of Encombe
##Viscount and Viscountess of Exmouth
##Viscount and Viscountess of Falkland
##Viscount and Viscountess of Falmouth
##Viscount of Fitz Harris
##Viscount and Viscountess of Folkestone
##Viscount and Viscountess of Frankfort de Montmorency
##Viscount and Viscountess of Gage
##Viscount and Viscountess of Galway
##Viscount and Viscountess of Garnock
##Viscount and Viscountess of Gough
##Viscount of Gort
##Viscount and Viscountess of Halifax
##Viscount and Viscountess of Hardinge
##Viscount of Harrington
##Viscount and Viscountess of Hood
##Viscount and Viscountess of Kilcoursie
##Viscount and Viscountess of Knutsford
##Viscount and Viscountess of Lifford
##Viscount of Llandaff
##Viscount and Viscountess of Maitland
##Viscount and Viscountess of Marsham
##Viscount and Viscountess of Massereene and Ferrard
##Viscount and Viscountess of Melville
##Viscount and Viscountess of Midleton
##Viscount and Viscountess of Milton
##Viscount and Viscountess of Monck
##Viscount and Viscountess of Morpeth
##Dowager Viscountess of Mountmorres
##Viscount and Viscountess of Newark
##Viscount and Viscountess of Newport
##Viscount and Viscountess of Oxenbridge
##Viscount of Parker
##Viscount of Peel
##Viscount and Viscountess of Portman
##Viscount and Viscountess of Powerscourt
##Viscount and Viscountess of Raincliffe
##Viscountess of Sherbrooke
##Viscount of Sidmouth
##Viscount of St. Cyres
##Viscount of Southwell
##Viscount of Suirdale
##Viscount and Viscountess of Templetown
##Viscountess of Torrington
##Viscount and Viscountess of Trafalgar
##Viscount and Viscountess of Valentia
##Viscount of Valletort
##Viscount of Villiers
##Viscountess of Wolseley
#Bishops — Auckland, Barry, Bath and Wells, British Colombia, Chichester, Durham, Ely, Exeter, Gloucester and Bristol, Gibraltar, Hereford, London, Lichfield, Lincoln, Manchester, Newcastle, Norwich, Oxford, Peterborough, Rochester, Ripon, Stepney, Southwark, St. Albans, Salisbury, Sodor and Man, Southwell, Sydney, Sierra Leone, Worcester, Winchester, Wellington
#Baronesses — Burdett-Coutts, Macdonald
#Lords and Ladies&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 4b / Col. 5a) — 
##Lord and Lady Abercromby
##Lord and Lady Aberdare
##Lord Aberdour
##Lady Abinger
##Lady Alexandra Acheson
##Lady Adam
##Lady Adderley
##Lord and Lady Addington
##Lady Adye
##Lady Agnew
##Lady Alderson
##Lord and Lady Alington
##Lady Alison
##Lady Mildred Allsopp
##Lord and Lady Amherst of Hackney
##Lady Heathcoat Amory
##Lord and Lady Ampthill
##Lady Agnes Anderson
##Lady Bertha Anson
##Lady Arbuthnot
##Lady Alice Archer Houblon
##Lord Ardee
##Lord and Lady Ardilaun
##Lady Armstrong
##Lady Arnold
##Lady Arnott
##Lord and Lady Ashbourne
##Lord and Lady Ashburton and Dowager Ashburton
##Lord and Lady Ashcombe
##Lady Alice Ashley
##Lady Edith Ashley
##Lady Ashmead-Bartlett
##Lord and Lady Ashton
##Lord and Lady Ashtown
##Lady Florence Astley
##Lady Gertrude Astley-Corbett
##Lady Austin
##Lord Bagot
##Lady Bailey
##Lady Blanche Baillie
##Lady Baird
##Lady Baker
##Lord Balcarres
##Lord and Lady Balfour of Burleigh, Lady Nina Balfour and Lady Betty Balfour
##Lord Balvaird
##Lord Bangor
##Dowager Lady Barclay
##Lord and Lady Barnard
##Lady Florence Barnardiston
##Lady Constance Barne
##Lady Barran
##Lady Barrington
##Lord and Lady Basing
##Lord and Lady Bateman
##Lady Evelyn Bathurst
##Lord and Lady Battersea
##Lady Steuart Bayley
##Lady Violet Beauchamp
##Lord Osborne Beauclerk and Lady Beauclerk (2)
##Lady A. Beaumont
##Lady Bedford
##Lord and Lady Belhaven and Stenton and Dowager Belhaven and Stenton
##Lord and Lady Bellew and Dowager Bellew
##Lord and Lady Belper
##Lady Charles Beresford
##Lady William Beresford (Lilian Duchess of Marlborough)
##Lady Bergne
##Lord and Lady Bertie and Lady Elizabeth Bertie
##Lady Biddulph, Lady Elizabeth Biddulph and Lady Wilfreda Biddulph
##Lady Bigge
##Lord and Lady Bingham
##Lord and Lady Binning
##Lord Blackwood, Lord Basil Blackwood. Lady Hermione Blackwood and Lord Terence Blackwood
##Lady Bloomfield
##Lady Blythswood
##Lord and Lady Bolton
##Lady Maud Bootle-Wilbraham, Lady Bertha Bootle-Wilbraham and Lady Edith Bootle-Wilbraham
##Lord Borthwick
##Lady Margaret Boscawen
##Lord and Lady Boston
##Lady Boughey
##Lady Albreda Bourke and Lady Florence Bourke
##Lady Bowen
##Lady Bower
##Lady Muriel Boyle and Lady Boyle (2)
##Lady Mary Brabazon
##Lady Brackenbury
##Lady Braddon
##Lady Bramwell
##Lady Bramston
##Lord Brassey, Lady Idina Brassey and Lady Violet Brassey
##Lord and Lady Braye
##Lady Mary Bridgeman
##Lady Eleanor Brodie
##Lady Hilda Brodrick
##Lady De Capel Brooke and Dowager Brooke
##Lady Cunliffe Brooks
##Lord and Lady Brougham and Vaux
##Lord and Lady Ulick Browne, Lady Browne and Lady Crichton Browne
##Lady Brownlow
##Lord and Lady F. Brudenell-Bruce
##Lady Brunner
##Dowager Buchanan-Riddeil
##Lady Audrey Buller
##Lady Burdett
##Lord and Lady Burghclere
##Lord Burghley
##Lady Agnes Burne
##Lady Burrell
##Lord and Lady Burton
##Lady Butler and Lady Butler (2)
##Lord and Lady Arthur Butter
##Lady Buxton and Lady Victoria Buxton
##Lady Susan Byng
##Lord and Calthorpe
##Lady C. Cameron and Lady Margaret Cameron
##Lord and Lady Archibald Campbell and Lady A. Campbell
##Lord and Lady George Campbell
##Lady Campbell-Bannerman
##Lord and Lady Camoys
##Lord and Lady Carbery and Dowager Carbery
##Lady Carbutt
##Lady Cardon
##Lord and Lady Cardross
##Lord and Lady Carew
##Lady Carmichael
##Lord and Lady Carnegie
##Lord and Lady Castlemaine
##Lord and Lady Castletown
##Lady Eva Cathcart and Lady R. Cathcart
##Lady Frederick Cavendish, Lady Myra Cavendish, Lady Evelyn Cavendish and Lady Harriet Cavendish
##Lord Charles Cavendish-Bentinck, Lord and Lady Henry Cavendish-Bentinck, Lord William Cavendish-Bentinck, Lady Ottoline Cavendish-Bentinck
##Lord and Eustace Cecil, Lord Hugh Cecil, Lord and John Cecil, Lord and Edward Cecil, Lord and Lady Robert Cecil, Lord W. Cecil, Lady Gwendolen Cecil, Lady Florence Cecil, Lady William Cecil, Lady Louisa Cecil
##Lady Francis Cecil-Dallas
##Lady Chamberlain
##Lady Chelmsford
##Lord and Lady Chesham
##Lady Chetwode
##Lord Cheylesmore
##Lord and Lady Fitzwarine Chichester
##Lady Chitty
##Lady Cholmeley
##Lady Henry Cholmondeley
##Lady Clements (2)
##Lady Churchill, Lady Randolph Churchill, Dowager Churchill, Lady Spencer Churchill (2)
##Lord Edward Spencer-Churchill, Lady Alfred Spencer-Churchill
##Lord and Lady Churston
##Lord and Lady Clifford of Chudleigh
##Lady Marshal Clarke, Lady E. Clarke
##Lady Isabel Clayton
##Lord and Lady Clinton
##Lord and Lady Clonbrock
##Lord Cloncurry
##Lady Muriel Close
##Lady Evelyn Cobbold
##Lady Cochrane, Lady Gertrude Cochrane, Lady Adela Cochrane
##Lady Coddington
##Lady Mabel Coke
##Lord and Lady Colchester
##Lady Cole (2)
##Lady Colebrooke
##Lord and Lady Coleridge
##Lady Collins
##Lady Colomb
##Lady Colvile, Lady Colville
##Lord and Lady Colville of Culross
##Lady Jane Seymour Combe, Lady Constance Combe
##Lady Commerell
##Lord and Lady Alwyne Compton
##Lady Dowager Congleton
##Lord and Lady Connemara
##Lady Conyers
##Lady Blanche Conyngham
##Lady Cooper
##Lady Evelyn Cotterell
##Lord and Lady Cottesloe
##Lady Couch
##Lord and Lady Courtenay
##Lady Coventry (2)
##Lady Cowell
##Lady Helen Craven
##Lord and Lady Crawshaw
##Lady Evelyn Crichton, Lady Emma Crichton
##Lord Crofton
##Lady Cromer
##Lady Mary Crosse
##Lady Crossley
##Lady Mary Cuffe
##Lady Culme-Seymour
##Lady Cunliffe
##Lady Georgiana Curzon
##Lady Elizabeth Cust
##Lady Ida Dalzell
##Lady Mary Dashwood
##Lord and Lady Davey
##Lady Victoria Dawnay, Lady Evelyn Dawnay, Lady Adelaide Dawnay
##Lady Decies
##Lord and Lady De Freyne
##Lord and Lady De L’Isle and Dudley
##Lord De Manley
##Lady Mildred Denison, Lady Elinor Denison
##Lord Deramore
##Lord and Lady De Ramsey
##Lady Dering
##Lady De Ross
##Lord and Lady De Saumarez
##Lady Des Voeux
##Lady De Trafford, Lady Agnes De Trafford
##Lady De Winton
##Lord and Lady Digby
##Lady Dorchester
##Lady Dorington
##Lady Margaret Douglas, Lady Edith Douglas
##Lady H. Douglas-Hamilton
##Lady Dowell
##Lady Drummond, Lady Edith Drummond
##Lady Du Cane
##Lady Duckworth
##Lady Eva Dugdale
##Lord Dunally
##Lady Florence Duncombe, Lady Ulrica Duncombe, Lady Caroline Duncombe
##Lady Alice Dundas
##Lord and Lady Dunleath
##Lord Dunglass
##Lady Dunn
##Lord Dunsandle and Clanconal
##Lady Durand
##Lord Dynevor
##Lord Ebury
##Lady Edmonstone
##Lady Edwards, Lady J. B. Edwards, Lady Blanche Edwards
##Lady Ernestine Edgcumbe
##Lady Egerton (2)
##Lord Egerton of Tatton
##Lady Grey-Egerton
##Lord and Lady Elcho
##Lord and Lady Elibank
##Lady Ellenborough
##Lady Ellis
##Lord and Lady Elphinstone
##Lady Winifred Cary-Elwes
##Lady Engleheart
##Lord Erskine, Lady Erskine (2), Lady Horatia Erskine, Lady Erskine
##Lord and Lady Esher
##Lady Evans
##Lady Evelyn Ewart, Lady Mary Ewart
##Lady Evelyn Eyre
##Lady Fairbairn
##Lady Fairfax
##Lady Anne Fane, Lady Augusta Fane
##Lady Farquhar
##Lord and Lady Farrer
##Lady Fayrer
##Lady Louisa Feilding
##Lady Helen Munro Ferguson
##Lady Fergusson
##Lady Ffolkes
##Lady Finlay
##Lady Fisher
##Lady Dorothea Fitz-Clarence, Lady Maria Fitz-Clarence, Lady Dorothy Fitzclarence
##Lord and Lady Henry Fitz-Gerald, Lady B. Fitz Gerald, Lady M. FitzGerald, Lord Seymour Fitz-Gerald
##Lady Beatrix Fitzmaurice
##Lord and Lady F. FitzRoy, Lady C. Fitz-Roy
##Lady Mary Fitzwilliam
##Lady FitzWygram
##Lady Fletcher
##Lady Flower, Lady Flower
##Lord Foley, Lady Mary Foley
##Lady Gertrude Foljambe
##Lady Angela Forbes, Lady Forbes (2), Dowager Helen Forbes
##Lord and Lady Forester
##Lady Forrest
##Lady Susan Fortescue
##Lady Forwood
##Lady Foster
##Lady Fowler
##Lady Edith Franklin
##Lady Fremantle, Lady Fremantle
##Lady Frere
##Lady Fulton
##Lady Gardiner, Lady Lynedoch Gardiner
##Lord Garioch
##Lady Galton
##Lady Katharine Gathorne-Hardy
##Lady Garvagh
##Lord and Lady Gerard
##Lady Gilbey
##Lady Gillford
##Lady Susan Gilmour
##Lady Gipps
##Lord and Lady Glamis
##Lord and Lady Glenesk
##Lady Glyn, Lady Mary Carr Glyn
##Lady D'Arcy Godolphin-Osborne
##Lady Gordon
##Lady Margaret Ormsby Gore, Lady Constance Gore
##Lady Gore Langton (2)
##Lord Walter Gordon-Lennox, Lord Algernon Gordon-Lennox
##Lady Evelyn Goschen
##Lord R. S. Gower
##Lady Graham, Lady Margaret Graham, Lady Helen Graham
##Lady Charlotte Graham-Toler
##Lady Grant, Lady Florence Grant
##Lady Grant-Duff
##Lady Green
##Lord Greenock
##Lady Grenfell
##Lady Frances Gresley
##Lady Victoria Grey, Lady Grey
##Lady Jane Grey-Trefusis
##Lady Griffin
##Lady Helen Grimston
##Lord and Lady Arthur Grosvenor, Lady Grosvenor (2)
##Lady Gull
##Lady Haldon
##Lady Haliburton
##Lady Basil Hall
##Lady Halle
##Lord and Lady Halsbury
##Lord and Lady E. Hamilton, Lord F. Hamilton, Lady F. Douglas Hamilton, Lady Alexandra Hamilton, Lady Baillie Hamilton (2), Lady C. Hamilton, Lady Victoria Hamilton, Lady George Hamilton
##Lady Hanson
##Lady Harcourt
##Lady Cicely Hardy, Lady Hardy
##Lady Beatrice Hare
##Lord Harlech
##Lady Constance Harris, Lady Harris
##Lady Harrison, Lady Harriet Harrison
##Lady Hart
##Lady Emily Hart-Dyke
##Lady Dixon-Hartland
##Lady Hartopp
##Lord and Lady Hastings
##Lord and Lady Hatherton
##Lady Alice Havelock-Allan
##Lady Hawke
##Lord and Lady Hawkesbury
##Lady John Hay, Lady Hay
##Lady Blanche Haygarth
##Lady Hayter
##Lady Hely-Hutchinson (2)
##Lady Hemming
##Lord and Lady Heneage
##Lord and Lady Henley
##Lord Henniker
##Lady Beatrix Herbert, Lady Herbert (2)
##Lord and Lady Herries
##Lord and Lady Herschell
##Lord Francis Hervey, Lady Augustus Hervey
##Lady Hervey-Bathurst
##Lady Fermor Hesketh
##Lady Hibbert
##Lady Lucy Hicks-Beach
##Lord and Lady Arthur Hill, Lady Clement Hill, Lady Stock Hill
##Lord and Lady Hillingdon
##Lord and Lady Hindlip
##Lord and Lady Hobhouse
##Lady Norah Hodgson
##Lady Holdich
##Lady Mary Holland
##Lady Beatrix Douglas Home
##Lady Maria Hood
##Lady Hood of Avalon
##Lady Hooker
##Lady Mary Hope
##Lady Hoskins
##Lord and Lady Hotham
##Lord and Lady Hothfield
##Lady Houldsworth
##Lady Eleanor Howard, Lady Agnes Howard, Lady Howard (2), Lady Mabel Howard, Lady Rachel Howard
##Lord and Lady Howard of Glossop
##Lady Howarth
##Lady Mary Hozier
##Lady Florentia Hughes
##Lady Seager Hunt
##Lady Hunter
##Lord Hyde
##Lady Hylton
##Lord and Lady Inchiquin
##Lord Inverurie
##Lord and Lady Iveagh
##Lady Jackson
##Lord James of Hereford
##Lady Margaret Jenkins, Lady Jenkins
##Lady Jenner
##Lady Jephson
##Dowager Jessel, Lady Jessell
##Lady Jeune
##Lady Hill Johnes
##Lady Joicey
##Lady Alice Jolliffe
##Lady Burn Jones
##Lady Caroline Lister Kaye, Lady Beatrice Lister Kaye, Lady Lister Kaye
##Lady Isabella Keane
##Lady Keith-Falconer (2)
##Lord and Lady Kelvin
##Lady Kemball
##Lady Beatrice Kemp
##Lady Kennard
##Lady Kennaway
##Lady Aline Kennedy
##Lady Kennett-Barrington
##Lord Kenyon
##Lady Mabel Kenyon-Slaney
##Lord Kensington
##Lady Mary Stuart Keppel
##Lady Innes-Ker (2)
##Lady Kerr (2)
##Lord Kilmarnock
##Lady King
##Lady Florence King King
##Lady Emily Kingscote
##Lady Edith King-Tenison
##Lord and Lady Kinnaird
##Lady Kitson
##Lady Laking
##Lady Frances Lambart, Lady Ellen Lambart
##Lady Victoria Lambton
##Lady Adela Larking
##Lady Isabel Larnach
##Lady Mary Lascelles
##Lord and Lady Lawrence
##Lady Lawson
##Lord and Lady Leconfield
##Lady Elliott Lees, Lady Lees
##Lady Leese
##Lady Legard
##Lord and Lady Leigh
##Lady Henry Gordon-Lennox, Lady Walter Gordon-Lennox, Lady Algernon Gordon-Lennox, Lady Caroline Gordon-Lennox
##Lady Katharine Le Poer Trench
##Lady Constance Leslie
##Lady Susan Leslie-Melville
##Lady Lewis
##Lady Lilian Liddell
##Lady Lindley
##Lady Harriet Lindsay, Lady Jane Lindsay, Lady Jane Lindsay
##Lord and Lady Lingen
##Lord and Lady Lister
##Lady Gwendolen Little
##Lady Margaret Littleton
##Lord and Lady Llangattock
##Lady Llewelyn
##Lord and Lady Loch
##Lady Lockwood
##Lady Louise Loder
##Lady Catherine Loftus
##Lady Doreen Long
##Lady Longley
##Lady Albertha Lopes
##Lady Loraine
##Lord and Lady Lovat
##Lady Drury Lowe, Lady Lucy Drury Lowe
##Lady Lowry-Corry (2)
##Lady Mary Loyd
##Lady Lubbock
##Lord and Lady Lurgan and Dowager Lurgan
##Lady Lyall
##Lady Lyell
##Lady Mary Lygon
##Lady Lyons
##Lady Lysons
##Lady Lyttelton
##Lady Emily Lytton
##Lady MacCormac
##Lord and Lady Macdonald
##Lady Macgregor, Lady MacGregor, Lady Helen MacGregor
##Lady Mackenzie, Lady Mackenzie
##Lady Mackworth
##Lady Maclean
##Lord and Lady Macnaghten
##Lady Macpherson-Grant
##Lady Caroline Madden, Lady Madden
##Lady Louisa Magenis
##Lady Magheramorne, Dowager Magheramorne
##Lady Nora Maitland
##Lady Margaret Crichton-Maitland
##Lady Margaret Majendie
##Lord Cecil Manners, Lord Edward Manners, Lord Manners, Lady Victoria Manners, Lady Manners
##Lady Blundell Maple
##Lady Mappin
##Lady Marjoribanks
##Lady Markham
##Lady Marriott
##Lady Martin, Lady Martin
##Lady Evelyn Mason
##Lady Maude (2)
##Lady H. Maxwell, Lady Maxwell, Lady Maxwell, Lady Maxwell
##Lady Heron-Maxwell
##Lady M'Clintock
##Lady Evelyn M'Donnell
##Lady Meade (2)
##Lord and Lady Medway
##Lady Methuen
##Lady Meysey-Thompson
##Lord and Lady Middleton, Lady Middleton
##Lady Mary Milbanke
##Lady Miller
##Lady Milner
##Lady Clementina Mitford
##Lady Lady M'lver
##Lady Hilda M'Neile
##Lady Monckton
##Lord Moncreiff, Lady Scott Moncrieff
##Lady Moncreiffe
##Lord and Lady Monkswell
##Lady Monson
##Lord Charles Montagu, Lady Cecil Scott Montagu, Lady S. Montagu, Lady Agneta Montagu
##Lord Montagu of Beaulieu
##Lord and Lady Monteagle
##Lady Edith Montgomerie, Lady Sophia Montgomerie
##Lady Charlotte Montgomery
##Lady More-Molyneux
##Lord and Lady Moreton
##Lady Morgan
##Lord and Lady Morris
##Lady Blanche Morris
##Lady Mary Morrison
##Lady Moseley
##Lord and Lady Mostyn
##Lord and Lady Mowbray and Stourton, Dowager Mowbray and Stourton, Lady Mowbray
##Lord and Lady Muncaster
##Lady Anne Murray
##Lady Murray (2)
##Lady Georgiana Mure, Lady Georgiana Mure [sic]
##Lord and Lady Napier and Ettrick
##Lord and Lady Napier of Magdala and Dowager Napier of Magdala
##Lady Naylor-Leyland
##Lady Nelson
##Lord and Lady Henry Nevill
##Lord and Lady Newton
##Lord and Lady Newtown-Butler
##Lady Nicolson
##Lady Augusta Noel, Lady Agnes Noel
##Lady Norman
##Lord and Lady Norreys
##Lord and Lady North, Lady Muriel North
##Lady Northcote, Lady Northcote (2)
##Lord Norton
##Lady Elizabeth Nugent
##Lady O'Brien, Lady O'Brien [sic]
##Lady O'Hagan
##Lady Olpherts
##Lord and Lady O'Neill
##Lady Gwendoline O'Shee
##Princep [sic] Alice Packe
##Lord and Lady Berkeley Paget
##Lady Alfred Paget
##Lady Paget of Cranmore
##Lady Katherine Pakenham
##Lady Palgrave
##Lady Sophia Palmer, Lady Palmer
##Lady Evelyn Parker
##Lady Parratt
##Lady Maude Parry
##Lady Muriel Parsons
##Lord and Lady Pearson, Lady Pearson
##Lady Peel, Lady Georgiana Peel
##Lady Constance Childe-Pemberton
##Lord and Lady Penrhyn
##Lady Mary Pepys
##Lady Perceval
##Lady Percy (2)
##Lady Petre
##Dowager Lady Peyton
##Lady Phillimore
##Lady William Phipps
##Lord and Lady Pirbright
##Lord and Lady Playfair
##Lady Chichele Plowden
##Lady Anna Chandos-Pole
##Lady Pollock
##Lord and Lady Poltimore
##Lady Pontifex
##Lady Alice Portal
##Lady Powell, Lady Powell [sic]
##Lady Baden-Powell
##Lady Dickson-Poynder
##Lady Poynter
##Lord and Lady George Pratt
##Lady Priestley
##Lady Probyn
##Lady Eva Wyndham-Quin, Lady Wyndham-Quin (2)
##Lord and Lady Raglan, Dowager Raglan
##Lady Ramsay
##Lord and Lady Rathdonnell
##Lady Rathmore
##Lord and Lady Rayleigh, Dowager Rayleigh
##Lord and Lady Reay
##Lady Reid
##Lord and Lady Rendel
##Lord Rendlesham
##Lady Jane Repton
##Lord Revelstoke
##Lord and Lady Ribblesdale
##Lady Laura Ridding
##Lord and Lady Robartes
##Lady O. Roberts
##Lady Roberts of Kandahar
##Lady Robinson
##Lord and Lady Rodney
##Lord Romilly
##Lord and Lady Rookwood
##Lord and Lady Rossmore
##Lord Rowton
##Lady Roxburgh
##Lord and Lady Rothschild
##Lady Victoria Russell, Lady Arthur Russell, Lady G. Russell, Lady W. H. Russell, Lady Alexander Russell
##Lord and Lady Russell of Killowen
##Lord and Lady Ruthven
##Lady Jane Ryan
##Lady Mary Sackville
##Lady Salmon
##Lord and Lady Saltoun
##Lady Samuelson, Lady S. Samuel
##Lady Mary Saurin
##Lord and Lady Savile, Lady Marie Savile
##Lady Savory
##Lord George Scott, Lord Henry Scott, Lord Herbert Scott, Lady Sophie Scott, Lady Charles Scott, Lady Louisa Scott, Lady Scott (2)
##Lord and Lady Seaton
##Lord and Lady Settrington
##Lady Seymour, Lady Albert Seymour, Lady William Seymour, Lady Seymour (2)
##Lord and Lady Shand
##Lady Shaw
##Lady Constance Shaw-Lefevre
##Lady Octavia Shaw-Stewart, Lady Alice Shaw-Stewart
##Lady Mary Shelley
##Lord and Lady Sherborne
##Lady Shippard
##Lady Shute
##Lady Kay-Shuttleworth
##Lady Simeon
##Lady Simmons
##Lady Simpson of Windsor
##Lord and Lady Sinclair
##Lord and Lady Skelmersdale
##Lady Esther Smith, Lady Barbara Smith, Lady Smith, Lady Blanche Smith, Lady Sybil Smith, Lady Euan Smith, Lady D. Smith
##Lady Smyth
##Lady Catherine Somerset, Lady Geraldine Somerset, Lady Henry Somerset
##Lord and Lady Southampton, Dowager Southampton
##Lady Edward Spencer-Churchill
##Lady Margaret Spicer
##Lady Sprigg
##Lady Stafford
##Lord Stalbridge
##Lady Stanhope (2)
##Lord Stanmore
##Lord Stanley, Lady Alice Stanley, Lady Isobel Stanley
##Lady Stansfield
##Lord Stavordale
##Lady Stephenson
##Lady Stevenson
##Lady Helen Stewart, Lady Mary Stewart, Lady Mark Stewart, Lady Stewart, Lady Houston Stewart, Lady Stewart [sic], Lady Isabel Stewart
##Lady Stewart of Grantully
##Lady Edith St. Aubyn
##Lord and Lady St. Levan
##Lady St. Leonards
##Lord and Lady St. Oswald
##Lady Stone
##Lady Charlotte Stopford
##Lord and Lady Stratheden and Campbell
##Lady Mary Stuart-Richardson
##Lord Suffield
##Lady Sutherland
##Lady Evelyn Sutton, Lady Susan Sutton
##Lord and Lady Swansea
##Lady Swinnerton Dyer
##Lady Kathleen Swinnerton-Pilkington
##Lord and Lady E. Talbot, Lady Emma Talbot
##Lady Jane Taylor
##Lady Taylour (2)
##Lady Tatton Sykes
##Lord Herbert Vane-Tempest, Lord Henry Vane-Tempest
##Lord and Lady Templemore
##Lady Tennant
##Lord and Lady Tennyson
##Lady Tenterden
##Lord Tewkesbury
##Lord and Lady Teynham
##Lord and Lady Thring
##Lady E. Thornton
##Lady Thursby
##Lady Ulrica Thynne
##Lord and Lady Tollemache
##Lady Agnes Townshend
##Lady Mary Trefusis
##Lady Tredegar
##Lady Trevelyan, Lady Trevelyan [sic]
##Lord and Lady Trevor
##Lady Troubridge
##Lady Turner
##Lady Henrietta Turnor
##Lady Tuson
##Lord and Lady Tweedmouth
##Lady Tyler
##Lady Emily Van De Weyer
##Lady Jane Van Koughnet
##Lord and Lady Ventry
##Lady Villiers (2), Lady Edith Villiers
##Lady Howard Vincent, Lady Helen Vincent, Lady Vincent
##Lady Vivian, Lady Jane Vivian
##Lady Mary Waldegrave
##Lady F. F. Walker, Lady James Walker
##Lady Walrond
##Lady Clementine Walsh
##Lord Wandsworth
##Lady Wantage
##Lord Warksworth
##Lady Leucha Warner
##Lady Warrender
##Lord and Lady Watson
##Lady Cecilia Webb
##Lady Rose Weigall
##Lord Welby
##Lady Willes
##Lady Willis
##Lady Arthur Wellesley
##Lord and Lady Wenlock
##Lord and Lady Westbury and Dowager Westbury
##Lady Isabella Whitbread
##Lady White
##Lady Whitehead
##Lady Whiteway
##Lady Elizabeth Williamson
##Lady Williams-Wynn
##Lady Willoughby (2)
##Lord Willoughby de Broke
##Lord Willoughby de Eresby
##Lady Willshire
##Lady Wilson, Lady Sarah Gordon Wilson
##Lord and Lady Wimborne
##Lady Windeyer
##Lord and Lady Windsor
##Lady Winnington
##Lady Constance Wodehouse
##Lord and Lady Wolverton
##Lady Julia Wombwell
##Lady Wood, Lady Mary Wood
##Lady Woods
##Lord Wrottesley
##Lady Hugh Wyndham
##Lady Barbara Yeatman
##Lady Lilian Yorke
##Lord Zouche
#Right Honourables
##H. H. Asquith
##E. Ashley
##A. H. Dyke Acland
##J. Atkinson
##J. B. Balfour
##Sir G. Bowen
##G. W. Balfour
##Sir Hicks-Beach
##A. J. Balfour
##James Bryce
##Sir H. Campbell-Bannerman
##A. H. Smith-Barry
##E. Carson
##H. Chaplin
##Sir J. Chitty
##Jesse Collings
##Sir R. Couch
##G. N. Curzon
##J. Chamberlain
##L. Courtney
##Sir M. Grant-Duff
##A. Akers-Douglas
##Sir W. Hart Dyke
##Sir H. Elliot
##F. Foljambe
##Sir H. Fowler
##Sir A. B. Forwood
##Sir J. Fergusson
##Herbert Gladstone
##Sir J. Gorst
##G. J. Goschen
##W. E. Gladstone
##Sir G. Grey
##C. H. Hemphill
##Charles Seale-Hayne
##R. W. Hanbury
##Lord George Hamilton
##Staveley Hill
##Sir J. T. Hibbert
##Sir W. Harcourt
##lon Hamilton
##Sir Arthur Hayter
##Sir F. Jeune
##W. L. Jackson
##Sir John Kennaway
##G. Shaw-Lefevre
##W. Lidderdale
##Sir Massey Lopes
##James Lowther
##Sir J. Lubbock
##Sir H. Lopes
##Walter Long
##Sir N. Lindley
##J. W. Mellor
##Sir G. O. Morgan
##John Morley
##Arnold Morley
##Sir J. Mowbray
##A. J. Mundella
##J. H. Macdonald
##F. Max Müller
##Sir W. Marriott
##Graham Murray (the Lord Advocate)
##Sir E. Monson
##Sir P. O'Brien
##Sir A. Otway
##Sir F. Peel
##Sir R. Paget of Cranmore
##W. J. Pirrie
##J. P. Robertson
##Sir. J. Rigby
##C. T. Ritchie
##Sir S. H. Strong
##Sir B. Saunderson
##Sir J. Stansfeld
##Sir A. Smith
##C. R. Spencer
##Sir C. Kay-Shuttleworth
##Sir R. Temple
##Sir R. Thompson
##Sir E. Thornton
##Lord Henry Thynne
##Sir G. O. Trevelyan
##C. P. Villiers
##Sir Algernon West
##Sir C. L. Wyke
##C. B. Stuart-Wortley
##S. J. Way
#Honourables&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5a / Col. 5b) and Honourable Ladies&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5b / Col. 5c)
##Mrs. Acland
##Mrs. Alexander
##H. Allsopp, Mrs. Allsopp, George Allsopp
##Mrs. Anstruther
##Mrs. Armytage
##[Hon. Lady] Vere Annesley
##Mrs. Bagot, Mrs. Bagot [sic 2x]
##Mrs. Baillie of Dochfour
##Mrs. Balfour
##[Hon.] Coplestone and [Hon.] Mrs. Bampfylde
##John Baring, Susan Baring, Lilian Baring
##Mrs. Barker
##Mrs. Barlow
##Eric Barrington, Mrs. Barrington
##Mrs. Hamar Bass
##Misses Bateman-Hanbury (2)
##Allen B. Bathurst
##Mrs. Benyon
##[Hon. Lady] Beresford
##[Hon.] R. Chetwynd
##Arthur Chichester
##Lady Biddulph
##C. E. Bingham, Mrs. Bingham, Albert Bingham, Mrs. Bingham [sic x2]
##Lady Birkbeck
##Ivo Bligh, Mrs. Bligh
##Diana Sclater-Booth
##O. Borthwick
##J. Boscawen
##Henry Bourke, Mrs. H. Bourke, Charles Bourke, Terence Bourke, Mrs. T. Bourke, Algernon Bourke, Mrs. A. Bourke, Mrs. E. R. Bourke
##Charles Brand, Arthur Brand, Mrs. Brand, Mrs. T. Brand
##T. Brassey, Mrs. A. Brassey
##Mrs. Stapleton Bretherton
##Reginald Brett, Mrs. Brett
##Mrs. F. Bridgeman, Misses Bridgeman (2)
##Mrs. Britten
##W. St. John Brodrick, Albinia Brodrick
##Emmeline Brownlow
##Mrs. T. C. Bruce, Misses Bruce (2)
##Misses M'Clintock Bunbury (2)
##Mary Byng
##T. J. Byrnes
##Arthur Cadogan, Mrs. A. Cadogan, Mrs. C. Cadogan, Ethel Cadogan
##Mrs. Gough-Calthorpe, Rachel (Gough) Calthorpe, Misses Gough Calthorpe (2)
##Mrs. Candy
##G. H. Campbell, K. Campbell, Hugh Campbell, Mrs. H. Campbell, Mrs. Ronald Campbell, Misses Campbell (2), Mrs. J. B. Campbell, Mildred Campbell
##Mrs. Carington
##Mrs. Carpenter
##Emily Cathcart
##W. Cavendish, Mrs. W. Cavendish, Mrs. Cavendish
##Eleonora Chetwynd, Mrs. R. Chetwynd
##Mrs. A. Chichester, Hilda Chichester
##Mrs. Clowes
##T. H. Cochrane
##Audrey Coleridge
##George Colville
##Mrs. Corbett
##Mrs. H. Corry
##Caroline Courtenay
##Henry Coventry
##Osbert Craven
##Misses Cross
##Mrs. P. Crutchley
##Henry Cubitt, Mrs. Cubitt
##Hamilton Cuffe, Mrs. Otway Cuffe
##Lady Cunningham
##Montagu Curzon, Darea Curzon, Mrs. Curzon
##Hew Dalrymple
##John Dawnay, Eustace Dawnay, W. Dawnay, Mrs. Dawnay (2)
##Misses de Montmorency (2)
##Mrs. H. Dennison
##R. C. Devereux, Mrs. R. C. Devereux
##Mrs. Digby
##Conrad Dillon, Mrs. C. Dillon, Edith Dillon
##Misses Douglas-Pennant (2)
##A. Hay Drummond, Mrs. Hay Drummond, Frances Drummond, Mrs. M. Drummond
##Hubert V. Duncombe, Cecil Duncombe, Mrs. C. Duncombe
##C. T. Dundas, Mrs. C. T. Dundas, W. Dundas, Mrs. W. Dundas, Mrs. John Dundas
##Lady Du Cane
##Herbert Eaton, Mrs. H. Eaton
##F. Egerton, Mrs. A. F. Egerton, Lady Grey Egerton, Tatton Egerton, Mrs. T. Egerton
##Arthur Elliot, Mrs. Arthur Elliot, Lady Elliot, Mrs. Eliot
##Lilian Elphinstone
##Mrs. Ellis
##Muriel Erskine
##H. Escombe, Mrs. Escombe
##Mrs. Evans
##Mrs. C. Keith-Falconer
##Sir S. Ponsonby Fane
##Mrs. W. Farquhar
##Ailwyn Fellowes, Mrs. A. Fellowes
##Mrs. Ferguson of Pitfour
##Everard Fielding
##N. Fitzgerald, Mrs. N. Fitzgerald, Mrs. Fitzgerald, , Mrs. F. G. FitzGerald, Lady FitzGerald
##R. Fitzwilliam, W. H. Fitzwilliam
##Mary Forester
##Sir John Forrest
##Mrs. W. H. Forster
##Mrs. Lionel Fortescue
##Sir C. Fremantle, Mary Fremantle
##Sir Malcolm Fraser, Misses Fraser (2)
##Mrs. Charles Keith-Fraser
##Violet Gibson
##Evelyn Giffard
##Mrs. Henry Gladstone
##Lady Godley
##George Ormsby Gore
##F. Leveson-Gower
##Mrs. Gough
##Mrs. Alaric Grant
##Ronald Greville, Mrs. R. Greville, Louis Greville, Mrs. L. Greville, Sidney Greville, Mrs. A. Greville, Mrs. A. H. F. Greville
##Robert Grosvenor, Algernon Grosvenor, Mrs. A. Grosvenor, Maud Grosvenor, Elizabeth Grosvenor
##Lady Hamilton Gordon, [Hon. Lady] Nevil Gordon
##Misses Guest (2)
##Geoffrey Browne Guthrie
##Mrs. Gye
##Mrs. A. Haig
##Mrs. Halford
##, Misses Hamilton (2)
##Mrs. North Dalrymple-Hamilton
##Mrs. Hobart Hampden
##Mrs. Assheton Harbord, Mrs. C. Harbord, Judith Harbord, Bridget Harbord, Mrs. Harbord
##C. Hardinge, Mrs. C. Hardinge, A. Hardinge
##A. E. Gathorne-Hardy, Nina Gathorne-Hardy
##Misses Hawke (2)
##C. G. Hay
##Misses Heneage (2)
##Helen Henniker, Mrs. Henniker
##Robert Herbert, Sir Robert Herbert, Mrs. R. Herbert, Mrs. Herbert
##A. Holland Hibbert, Mrs. A. Holland Hibbert
##Lady Higginson
##Mrs. Hill
##Lionel Holland, Sydney Holland
##Grosvenor Hood, Dorothy Hood
##Lady Acland-Hood
##Fanny Hood of Avalon
##Mrs. Curzon Howe
##[Hon.] Evelyn Hubbard, Mrs. E. Hubbard, Alice Hubbard
##Mary Hughes
##Mrs. Meynell Ingram
##G. Jolliffe, Sydney H. Jolliffe, Mrs. Jolliffe
##Lady Johnston
##G. Keppel, Mrs. Keppel, Derek Keppel, Mrs. William Keppel
##Mrs. Alfred Ker
##Constance Kerr
##Mrs. Kingscote
##C. C. Kingston
##Lady Knollys
##Bertha Lambart
##F. W. Lambton, Mrs. Lambton
##Mary Lascelles
##Charles Laurence, Herbert Laurence
##Wilfrid Laurier
##Mrs. Lawley
##Mrs. C. Lawrence, Misses Lawrence (2), Mrs. H. Lawrence
##Mrs. Legge
##T. W. Legh, Mrs. Legh, Sybil Legh
##F. D. Leigh, Mrs. F. D. Leigh, E. Chandos Leigh, Mrs. E. C. Leigh, Cordelia Leigh
##C. Hanbury Lennox, Mrs. Hanbury Lennox
##G. W. Leslie
##R. l’Estrange
##Atholl Liddell, Mrs. A. Liddell
##Mrs. H. Gore-Lindsay
##Reginald Lister
##Henry Littleton, Misses Littleton (2)
##Misses Loch (2)
##William Lowther, Mrs. W. Lowther, L. Lowther, Mrs. L. Lowther
##Mrs. E. H. Loyd
##Mrs. Lumley
##Alfred Lyttelton, Mrs. A. Lyttelton, Misses Lyttelton (2), Mrs. Lyttelton
##Flora Macdonald, Lady Macdonald
##Mrs. Mackinnon
##Mrs. Maclagan
##Mrs. Magniac
##Mrs. Maguire
##W. Massey-Mainwaring, Mrs. Massey-Mainwaring
##Mrs. Fuller-Maitland
##Aline Majendie
##Misses Henniker Major (2)
##Mrs. Mallet
##Archibald Marjoribanks
##Misses Constable Maxwell (2)
##Mrs. M'Calmont
##Schomberg M'Donnell
##Charles Mills, Violet Mills, Mrs. Mills
##Mrs. Percy Mitford
##Maud de Moleyns
##Mrs. C. Molyneux
##Annette Monck, Mrs. Monck
##Violet Monckton
##Mrs. Monson
##John Scott Montagu
##[Hon.] Evelyn Moore
##R. Moreton, Mrs. R. Moreton
##Mrs. Mostyn, Misses Mostyn (2)
##Mrs. G. H. Murray, Alice Murray
##Lady Musgrave
##[Hon. Lady] Napier, Emilia Napier, Mrs. Scott Napier
##Mrs. Neeld
##Sir Hugh Nelson
##[Hon.] R. Nevill
##Mrs. Newdigate
##Sir H. S. Northcote
##Misses O'Brien (2)
##Mary O'Hagan
##Mrs. Okeover
##Mrs. Oliphant
##R. Terence O'Neill, Henrietta O'Neill
##Misses Palk (2)
##Cecil Parker, R. Parker, F. Parker, Mrs. F. Parker, Mrs. Parker
##Mabel Parnell
##[Hon.] C. B. Parsons, Mrs. Parsons
##Mrs. W. Paton
##[Hon.] Sydney Peel, Misses Peel (2)
##Mrs. Anderson Pelham
##E. S. Douglas-Pennant, Mrs. E. S. Douglas-Pennant
##Mrs. Heber Percy
##Albert Petre, Mrs. A. Petre
##Harriet Phipps
##Mrs. Pirie
##Thomas Playford
##Horace C. Plunkett
##[Hon.] Ashley Ponsonby, Mrs. Ponsonby, Misses Ponsonby (2)
##H. Orde Powlett, Mrs. Orde-Powlett, Myra Orde-Powlett
##E. W. B. Portman, Mrs. Portman, Mary Portman
##Mrs. Pretyman
##C. Ramsay, Mrs. C. Ramsay
##G. H. Reid
##Misses Rendel (2)
##Misses Rice (2)
##Lady White Ridley
##Mrs. Ritchie
##F. Roberts, Mrs. Phillips Roberts
##Misses Roberts (of Kandahar) (2)
##J. M. Rolls, Eleanor Rolls
##W. Rothschild, Evelina Rothschild
##W. Rowley, Mrs W. Rowley, Lady Thelluson Rowley
##A. Russell, Misses Russell (2)
##Gustavus Hamilton-Russell, Misses Hamilton Russell (2)
##the Master of Ruthven, Mrs. Ruthven
##Mrs. J. D. Ryder
##Sir Saul Samuel
##A. Saumarez, Mrs. A. Saumarez
##Mrs. E. J. Saunderson
##J. Maxwell Scott, Mrs. Maxwell Scott
##R. J. Seddon
##Mary Sidney
##Lady Simeon
##Misses Skeffington (2)
##Sir Donald Smith, Mrs. A. H. Smith, [Hon.] W. F. D. Smith
##Granville Somerset, Mrs. G. Somerset, Arthur Somerset, Mrs. A. Somerset, R. Somerset, Violet Somerset
##Mrs. C. R. Spencer
##Sir J. Gordon Sprigg
##Lyulph Stanley, F. C. Stanley, George Stanley, Mrs. E. J. Stanley, Mrs. Stanley, Mrs. V. A. Stanley, Maude Stanley
##Lady Cowell-Stepney
##Randolph Stewart, Mrs. R. Stewart, FitzRoy Stewart, Mrs. Stewart
##Mabel St. Aubyn
##Misses St. Clair (2)
##Mrs. Stirling
##Horatia Stopford
##[Hon. Lady] Alison Stourton
##Mrs. Strutt, Misses Strutt (2)
##Hilda Sugden
##Alfred Talbot, Mrs. Talbot, Mrs. R. A. J. Talbot
##Sir D. Tennant
##S. R. Thayer
##Misses Thellusson (2)
##Edward Thesiger, Mrs. E. Thesiger, Frederick Thesiger, Mrs. F. Thesiger, Mary Thesiger
##Lady Thorold
##Katharine Thring
##Misses Tollemache (2)
##R. Marsham-Townshend, Mrs. Marsham-Townshend
##Alice Hanbury-Tracy
##Charles Grey Trefusis, Misses Trefusis (2)
##Mrs. Trelawny
##Mrs Tremayne
##Mrs. W. le Poer Trench
##Charles Trevor
##George Hill-Trevor, Marcus Hill-Trevor, Mrs. Hill-Trevor, Misses Hill-Trevor (2)
##Mrs. C. W. Trotter
##Lady Tryon
##Rosamond Tufton
##Sir G. Turner
##Rev. L. Tyrwhitt
##Misses Tyssen Amherst (2)
##Misses Vereker (2)
##R. Greville-Verney, Mrs. R. G. Verney, Misses Verney (2)
##F. Villiers, Mrs. F. Villiers
##Misses Vivian (2)
##Arthur Walsh
##Mrs. P. E. Warburton
##Robert Ward, Mrs. Dudley-Ward
##Mrs. West
##Mrs. Whateley
##Sir W. Whiteway
##F. Bootle-Wilbraham
##Ella Williamson
##Tatton Willoughby
##Lady Wilson
##[Hon.] Armine Wodehouse, Mrs. Wodehouse
##Frances Wolseley
##F. Wood, Misses Wood (2)
##Mrs. G. Wrottesley, Evelyn Wrottesley
##Percy Wyndham, Mrs. P. Wyndham, Misses Wyndham (2)
##Maud Wynn
##Lois Yarde-Buller
##Alex. G. Yorke, Mrs. J. Yorke, Mrs. E. C. Yorke
#Sirs&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5c–6a)
##Augustus Adderley
##Edwin Arnold
##John Austin
##George Arthur
##John Heathcoat-Amory
##A. Armstrong
##Andrew Agnew
##Frederick Abel
##Henry Acland
##A. Arnold
##Alexander Arbuthnot
##John Barran
##G. Bower
##J. W. Bonser
##J. Crichton-Browne
##Joseph Bailey
##E. Ashmead-Bartlett
##Henry Barkly
##R. Beauchamp
##Raymond Burrell
##Charles Barrington
##David Baird
##Arthur Birch
##Edward Birkbeck
##W. Cunliffe Brooks
##A. de Capel Brooke
##Courtenay Boyle
##F. Burton
##F. Buxton
##Steuart Bayley
##John Bramston
##John Baker
##H. Bullard
##J. T. Brunner
##H. Bellingham
##Henry Bergne
##Thomas Boughey
##F. J. Bramwell
##E. Burne-Jones
##James Blyth
##Seymour Blane
##Henry Chamberlain
##Roderick Cameron
##Hugh Cholmeley
##John Conroy
##Edward Clarke
##C. Cameron
##E. Carbutt
##W. Coddington
##Marshal Clarke
##Reginald Cathcart
##Savile Crossley
##Edward Colebrooke
##Reginald Cust
##Charles Crosthwaite
##John Colomb
##Daniel Cooper
##F. Astley-Corbett
##Donald Currie
##Henry Cunningham
##Robert Cunliffe
##Henry Cotterell
##T. D. Gibson Carmichael
##F. Curden,
##George Dallas
##James Drummond
##Mortimer Durand
##G. Des Vieux
##Henry Dering
##J. N. Dick
##Dyce Duckworth
##T. Swinnerton Dyer
##E. Hastings Doyle
##John Dorington
##William Dunn
##Humphrey de Trafford
##Charles Dalrymple
##G. Dashwood
##Gardner
##Engleheart
##Francis Evans
##A. Edmonstone
##Whittaker Ellis
##W. H. Flower
##Horace Farquhar
##Joseph Fayrer
##H. Fletcher
##William Ffolkes
##William Fraser
##Bartle Frere
##Gerald Seymour Fitz-Gerald
##Robert Finlay
##B. Walter Foster
##Gerald FitzGerald
##R. FitzGerald
##Maurice FitzGerald
##Forrest Fulton
##William Flower
##Andrew Fairbairn
##John Gilbert
##E. T. Gourley
##Edward Grey
##W. Gull
##Walter Gilbey
##Lepel Griffin
##G. Macpherson-Grant
##Reginald Graham
##Philip Grey Egerton
##Douglas Galton
##R. Glyn
##Arthur Godley
##Charles Grant
##R. Gresley
##Alexander Acland-Hood
##T. G. Fermor Hesketh
##Arthur Haliburton
##Brydges Henniker
##F. Dixon-Hartland
##R. Hanson
##Alfred Hickman
##W. Houldsworth
##Henry Howorth
##F. Seager Hunt
##Charles Hall
##E. W. Hamilton
##Reginald Hardy
##Clement Hill
##Basil Hall
##Joseph Hooker
##Charles Hunter
##Charles Hartopp
##Victor Houlton
##Augustus Hemming
##Henry Irving
##Frederic Johnstone
##W. Jenner
##J. Jenkins
##James Joicey
##Charles Jessell
##Harry Johnston
##Edward Jenkinson
##James Hill Johnes
##John Jackson
##H. Seymour King
##James Kitson
##J. Lister-Kaye
##V. Kennett-Barrington
##George Kekewich
##John Leslie
##Thomas Dick Lander
##T. Villiers Lister
##James Linton
##Charles Lees
##Charles Legard
##Thomas Lea
##Wilfrid Lawson
##Elliott Lees
##A. C. Lyall
##J. T. D. Llewelyn
##Joseph Leese
##Leonard Lyell
##F. Laking
##Godfrey Lushington
##F. Lockwood
##Henry Longley
##George Lewis
##F. Milner
##Herbert Maxwell
##Francis Montefiore
##Graham Montgomery
##Robert Moncreiffe
##Musgrave
##Colin Scott Moncrieff
##Francis Mowatt
##Evan MacGregor
##J. G. Miller
##F. D. Maclean
##J. Blundell Maple
##Allan Mackenzie
##Lewis M'lver
##F. Mappin
##Theodore Martin
##Samuel Montagu
##William MacCormac
##Hubert Miller
##Lewis Morris
##Clements Markham
##A. C. Mackenzie
##John Monckton
##J. Stirling-Maxwell
##J. Heron Maxwell
##Kenneth Matheson
##J. S. Montefiore
##Acquin Martin
##W. Maxwell
##Oswald Moseley
##Arthur Nicolson
##Terence O'Brien
##Reginald Ogilvy
##Herbert Oakeley
##Hush Owen
##G. G. Petre
##Walter Parratt
##Frederick Pollock
##Herbert Perrott
##Douglas Powell
##Weetman Pearson
##Joseph Pease
##Francis S. Powell
##Reginald Palgrave
##W. Priestley
##E. G. Poynter
##G. S. Baden-Powell
##Charles Pontifex
##J. Dickson-Poynder
##James Paget
##C. M. Palmer
##C. Lennox Peel
##James B. Peile
##Westby Perceval
##Charles Pigott
##John Puleston
##W. Plowden
##Richard Quain
##George Russell
##C. Lister Ryan
##W. H. Russell
##J. Ramsay
##Owen Roberts
##R. T. Reid
##Charles Robinson
##J. Thellusson Rowley
##James Reid
##C. Euan-Smith
##J. Barrington Simeon
##J. B. Stone
##M. Shaw-Stewart
##Edward Sieveking
##T. H. Sanderson
##Augustus K. Stephenson
##Thomas Sutherland
##Mark Stewart
##Andrew Scoble
##Joseph Savory
##Douglas Straight
##Charles Shelley
##S. Shippard
##E. Sassoon
##A. Condie Stephen
##E. Sullivan
##Arthur Sullivan
##S. Scott
##H. Simpson
##E. Stafford
##Ernest Satow
##Tatton Sykes
##John Tyler
##Charles Tennant
##John Tenniel
##J. Thorold
##John Thursby
##Thomas Troubridge
##Charles Turner
##H. Meysey-Thompson
##W. Vincent
##Edgar Vincent
##Arthur Vicars
##W. Williams-Wynn
##James Walker
##R. Webster
##George Wombwell
##C. Rivers Wilson
##W. H. Wills
##Donald Mackenzie Wallace
##George Warrender
##F. Winnington
##James Whitehead
##Arthur Willshire
##Henry Wood
##Hugh Wyndham
##W. White
##Sidney Waterlow
##Hedworth Williamson
##Jacob Wilson
##W. Windeyer
##Albert Woods (Garter)
##Allen Young
#Chairman of County Council (Dr. Collins)
#Counts and Countesses
##Count Cassini
##Count and Countess De Ganay
##Count Gurowski
##Count Hohenau
##Count Theodor Bolesta Koziebrodski
##Count Leon Mniszeek
##Count and Countess Potocki
##Count and Countess Raben
#Barons and Baronesses
##Baroness Emile Beaumont d'Erlanger
##Baroness De Brienen
##Baron De Onethau and Baroness D’Onethan [sic]
##Baron and Baroness Alphonse de Rothschild
##Baron Ferdinand Rothschild
##Baron and Baroness Schröder
##Baron and Baroness von Deichmann
##Baron von Heeckeren van Wassenaer
##Baroness von Hügel, Baroness Gertrud von Hügel [sic]
##Baron and Baroness Campbell von Laurentz
##Baroness Wilhelm von Rothschild
#Rev. the Moderator of the General Assembly of the Church of Scotland
#Deans — Christ Church, St. Paul's, Westminster, Windsor
#The Provost of Eton
#Master of Trinity (Mr. Butler)
#The Sub-Dean of the Chapels Royal
#Canons — Blundell, Dalton, Duckworth, Fleming, Hervey, Teignmouth Shore, Wilberforce
#Dr. Adler (Chief Rabbi)
#Dr. M'Cormick
#Chaplain of the Fleet
#Chaplain General
#Reverend Doctors — Edmund Warre, C. J. Welldon
#Reverends — Prebendary Hawkshaw, Albert Baillie, W. H. Bliss, M. Ebrington Bisset, Lord W. Cecil, Lord Charles Fitzroy, J. H. Ellison, H. Haweis, W. R. Jolly, G. J. Martin, Newton Mant, Marquis of Normanby, A. Robins. W. Gunion Rutherford, Clement Smith, Montagu Villiers
#Doctors — Lennox Browne, J. V. Bridge, Barlow, Robert Farquharson, J. F. Fox, Surgeon-Major Kilkelly, John Lowe, C. H. H. Parry, G. V. Poore, Dorrien Smith, S. Wilks
#Messieurs&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 6b–7a),  Mesdames (4, Col. 7a–b) and Misses&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7c – 5, Col. 1a)
##Mme Abdy
##Mr C. T. Dyke-Acland, Mme A. H. Dyke Acland, Mme Dyke Acland
##Mme Adair
##Misses Adam (2)
##Mr and Mme Adeane
##Misses Adye [?] (2)
##Mme Agar
##Mr Hamilton Aidé
##Mr John Aird, Misses Aird (2)
##Miss Akers-Douglas
##Mr Edward Alderson
##Mr George Alexander, Mme Alexander, Miss Alexander
##Miss Alison
##Mr and Mme Allhusen
##Mme Alma-Tadema
##Mr W. Ambrose
##Miss Heathcoat-Amory
##Mr R. Anderson, Miss Florence Anderson
##Mr E. H. Anson
##Mr H. T. Anstruther, Miss Rosomond Anstruther
##Mme Antrobus
##Mr Arbuthnot, Miss Arbuthnott [sic]
##Miss Archer-Houblon
##Mme Argles
##Mme Arkwright, Miss Arkwright
##Misses Armytage (2)
##Miss Arnott
##Mr and Mme Ascroft, Miss Ascroft
##Mr Arthur Ash
##Mr A. Asher
##Mme Ashton
##Mme Asquith
##Mr Astor, Mr W. Astor
##Mr B. F. Astley
##Mme Evelyn Atherley
##Mr and Mme Alfred Austin, Misses Austin (2)
##Mr and Mrs C. H. Babington
##Mr and Mrs Bagge
##Mrs Charles Bagot, Mrs J. F. Bagot, Miss Alice Bagot
##Mr James Bailey, Mrs J. Bailey, Mrs Bailey, Misses Bailey (2)
##Mrs Duncan Baillie, Misses Duncan Baillie (2)
##Mr Baillie of Dochfour
##Mr and Mrs W. A. Baillie-Hamilton
##Mr E. Bainbridge
##Mr and Mrs H. R. Baird, Mr and Mrs J. G. A. Baird, Misses Baird (2)
##Mr and Mrs Baldwin
##Mr and Mrs E. Balfour, Mr and Mrs Charles Balfour, Miss Balfour
##Mr and Mrs Banbury, Miss Banbury
##Mr and Mrs S. B. Bancroft [actor &quot;Bancroft and his wife accepted with becoming grace the congratulations with which they were well-nigh overwhelmed&quot;&lt;ref name=&quot;:3&quot; /&gt; (5, Col. 6b)]
##Bandanaratke [?]
##Mrs Bankes
##Mr Banks
##Mr and Mrs Walter Baring, Miss Baring
##Miss Barker
##Mr J. Emmott Barlow, Mrs Barlow, Mrs Barlow [sic 2x]
##Misses Barnardiston (2)
##Miss Barne
##Mr and Mrs F. G. Barnes, Mr and Mrs Barnes, Misses Barnes (2)
##Miss Barran (2)
##Mr and Mrs J. Wolfe Barry, Mr and Mrs F. Tress Barry, Mrs A. Barry
##Misses Bartlett (2)
##Mr and Mrs D. P. Barton, Mr and Mrs Barton
##Mr Hamar Bass
##Mrs Bates, Miss Bates
##Mr and Mrs H. Bathurst, Misses Bathurst (2)
##Mr and Mrs Baxendale, Miss Baxendale
##Miss Mariot [?] Bayley
##Mr and Mrs W. W. Beach, Miss Beach
##Misses Hicks-Beach (2)
##Mr R. M. Beachcroft
##Mr and Mrs Wentworth Beaumont, Mr Wentworth B. Beaumont, Mrs Beaumont, Miss Hilda Beaumont
##Mr and Mrs Rupert Beckett, Mr E. W. Beckett
##Mr and Mrs Beer
##Mr and Mrs F. F. Begg
##Mr  Charles Bell, Mr and Mrs Bell, Misses Bell (2)
##Miss Bellingham
##Mr and Mrs R. Benson, Mr and Mrs Benson
##Miss Berens
##Mr and Mrs Beresford, Miss Beresford
##Miss Berkeley, Misses Berkeley (2)
##Mr and Mrs Bertier, Miss Bertier
##Mr and Mrs Cosmo Bevan, Mr and Mrs F. Bevan, Miss Bevan
##Mr M. M. Bhownaggree
##Mr and Mrs F. Bibby
##Mr Leonard Biddulph, Mr Biddulph, Mr Victor Biddulph, Mr M. Biddulph, Mrs H. M. Biddulph, Misses Biddulph (2), Miss Biddulph, Miss Freda Biddulph
##Mr and Mrs Bigham
##Mr Bigwood
##Mrs C. Bill, Miss Bill
##Miss Birch
##Mrs Birch-Reynardson, Misses Birch-Reynardson (2)
##Mr A. Birrell, Mrs Birrell
##Mr and Mrs Bischoffsheim
##Mrs Ebrington Bissett
##Misses Blackwood (2)
##Mr and Mrs R. G. Blennerhassett
##Mrs W. H. Bliss
##Mrs Blundell, Miss Blundell
##Misses Blyth (2)
##Mr and Mrs Bolitho, Miss Bolitho
##Mr H. C. O. Bonsor, Mrs Bonsor, Miss Bonsor
##Mrs W. Borsel
##Mrs Griffith-Boscawen
##Mr and Mrs Boulnois
##Miss Bourke
##Mr W. R. Bousfield
##Mrs Bowden-Smith, Misses Bowden-Smith (2)
##Miss Bowen (2)
##Mr T. G. Bowles, Mrs Bowles
##Mr Edmund R. Boyle
##Miss Mabel Brackenbury
##Mrs Bradley, Miss Bradley
##Miss Beryl Bradford
##Miss Braddon
##Miss Bramwell
##Mr H. L. C. Brassey, Mrs H. A. Brassey, Misses Brassey (2), Misses Brassey (2) [sic 2x]
##Mr Stapleton Bretherton, Misses Stapleton Bretherton (2), Mr F. Stapleton Bretherton
##Mrs Bridge
##Mr G. Bridgman, Mr and Mrs C. G. O. Bridgeman
##Mr Brigg
##Mrs Brocklehurst
##Misses Brodie (2)
##Mr and Mrs Brookfield, Miss Brookfield
##Miss Bromley-Davenport
##Miss Brooke
##Miss Rhoda Broughton
##Mr and Mrs A. H. Brown, Miss Brown
##Mrs Browne, Misses Browne (2), Misses Browne (2) [sic 2x]
##Mrs Brownrigg, Miss Brownrigg
##Mr A. O. Bruce, Mrs A. C. Bruce [sic], Misses Bruce (2)
##Miss Brunner
##Mrs Bryce
##Mr Brymer
##Mr and Mrs Buchanan
##Mrs C. E. Buckle
##Mr Bucknill
##Miss Budgett
##Miss Mary Bulteel
##Miss Burdett
##Mr and Mrs Burges, Misses Burges (2)
##Mrs C. K. Burn
##Mr and Mrs F. C Burnand
##Miss Evelyne Burne
##Mr and Mrs W. Burns, Miss Burns
##Misses Burrell (2)
##Mr J. G. Butcher
##Mrs Butler, Mrs Butler, Miss Butler
##Mr Sydney Buxton, Mrs S. Buxton, Misses Buxton (2)
##Mr P. H. Calderon
##Mrs Calley
##Mrs Archibald Calvert, Miss Calvert
##Mr Cameron, Miss Cameron, Misses Cameron (2)
##Mr and Mrs J. D. Campbell, Mr J. A. Campbell, Miss J. A. Campbell, Mrs F. Campbell, Mrs W. Campbell, Mrs Hastings Campbell, Mrs W. Campbell [sic 2x], Mrs F. L. Campbell, Mrs D. B. O. Campbell, Miss Lilah Campbell, Miss Campbell, Miss Ronald Campbell, Misses Campbell (2)
##Miss Grace de Capell-Brooke
##Miss Carden
##Miss Carleton
##Mr and Mrs W. W. Carlile, Miss Carlisle
##Mrs Rivett Carnac
##Mrs Carnegy
##Mrs Boyd Carpenter, Misses Boyd Carpenter (2)
##Mrs Carson
##Mr and Mrs D'Oyly Carte
##Mrs Carter
##Mrs Castance
##Mr R. K. Causton, Mrs Causton, Miss Causton
##Mrs Cavaye
##Mr and Mrs C. Tyrall Cavendish, Mr Victor Cavendish, Mr Henry Cavendish, Mr Cavendish, Mrs Cavendish
##Mr and Mrs F. Cavendish-Bentinck, Mr Cavendish-Bentinck, Mrs W. G. Cavendish-Bentinck
##Mr F. Cawley
##Mr and Mrs Cayzer, Miss Cayzer
##Mr and Mrs W. M. Cazalet
##Mr F. Cazenove
##Mr Evelyn Cecil, Miss Cecil
##Mrs Chaine
##Mrs Chaloner
##Mr Austen Chamberlain, Mrs Chamberlain, Misses Chamberlain (2)
##Misses Chaning (2)
##Mr and Mrs Channing
##Mr and Mrs Cecil Chaplin, Misses Chaplin (2), Miss Edith Chaplin, Miss Chaplin
##Mrs Chapman
##Misses Chetwode (2)
##Mrs W. Chetwynd, Miss Chetwynd (2)
##Mr Childe-Pemberton
##Miss Chitty
##Miss Leila Crichton
##Miss Cholmeley (2)
##Miss Cholmondeley
##Miss Chrichton-Maitland
##Mrs H. Churchill
##Miss Spencer Churchill
##Mr J. D. Clark, Mr and Mrs Atkinson Clark, Mr Clark, Mrs B. F. Clark, Mrs G. D. Clark, Stanley Clark, Miss Clark
##Mr Purdon Clarke, Mr Ernest Clarke, Miss Clarke, Miss Stanley Clarke
##Mrs Clerk
##Mr and Mrs Henry Pelham Clinton
##Mrs Clive, Misses Clive (2)
##Mrs Close
##Mr Clough
##Mr Clowes, Misses Clowes (2)
##Mr Cobbold
##Mr T. B. Cochrane, Miss Cochrane
##Mr and Mrs W. A. Cockerell, Miss Cockerell, Miss Cockerell [sic 2x]
##Mr and Mrs D. Coghill
##Mr B. Cohen
##Mr Wentworth Cole
##Miss Colomb
##Mr and Mrs Colston
##Miss Colville
##Mr Richard Combe
##Miss Commerell, Miss Commerell [sic 2x]
##Mr and Mrs Compton
##Mr and Mrs Consett, Miss Vera Consett
##Mr and Mrs F. L. Cook, Mr Ward Cook, Miss Cook
##Mr and Mrs Kinloch Cooke, Mr Cooke, Mr and Mrs C. Kinloch Cooke
##Mr and Mrs Daniel Cooper, Mrs E. H. Cooper, Misses Cooper (2), Miss Cooper
##Mr and Mrs Cameron Corbett, Miss Corbett
##Mr and Mrs V. Seymour Corkran, Miss Corkran
##Mr and Mrs F. S. W. Cornwallis
##Mr and Mrs Cory
##Mrs Armar Corry, Mrs Clifford Corry, Miss Corry
##Mr J. R. G. Cotterell, Miss Cotterell (2)
##Mrs Stapleton Coton
##Mr and Mrs George Courroux
##Mrs Courtney
##Mr Burdett-Coutts
##Mrs Coventry
##Miss Cowell
##Miss Cowell-Stepney
##Mr and Mrs R. Cox, Mrs Cox, Miss Cox
##Mrs Crabbe, Misses Crabbe (2)
##Mrs Craik
##Mr and Mrs Crawshay
##Mrs Creignton, Miss Lucia Creighton
##Mr C. A. Cripps, Mr and Mrs Wilfrid Cripps
##Mr and Mrs Critchett
##Mr and Mrs Croombie
##Mrs A. B. Crosbie
##Mr and Mrs Shepherd Cross, Mr A. Cross, Miss Crosse
##Mr and Mrs Cruddas, Misses Cruddas (2)
##Mr and Mrs Percy Crutchley, Misses Crutchley (2)
##Miss Cuffe
##Miss Culme-Seymour
##Mrs Cuninghame
##Miss Cunliffe
##Mrs Dick-Cunynghame
##Mrs Curzon
##Misses Cust (2)
##Miss Custance
##Mrs Dalbiac
##Miss Gladys Dalgety [?]
##Mr C. B. Dalison
##Miss Dalrymple
##Mrs Dalton
##Mrs Denis Daly
##Mr and Mrs Darling
##Miss Dashwood
##Mr W. Bromley-Davenport
##Miss Davey
##Mr and Mrs Louis Davidson, Mrs Randall Davidson
##Mr W. Rees Davies, Mr Ben Davies, Mr and Mrs Vaughan Davies
##Mrs Davis
##Miss Dawnay (2)
##Mrs de Arcos
##Misses De Brienen (2)
##[Miss] La Baronne de Friesen
##Mrs R. C. de Grey Vyner
##[Miss] La Baronne Sirtema de Grovestins [?]
##Mr and Mrs J. de la Cour
##Mr and Mrs Edwin de Lisle
##Mr W. E. Denison
##Mrs Denny
##Miss De Perpigna
##Mrs de Salis
##Mr de Soria
##Mr De Trafford, Miss De Trafford
##Mr Deverell, Miss Deverell
##Mr and Mrs W. de Winton, Miss De Winton
##Mr and Mrs Gerard Dicconson
##Mr and Mrs Dicken
##Mr and Mrs C. S. Dickson, Mrs Dickson
##Mr J. K. Digby, Kenelm E. Digby, Mrs Digby, Misses Digby (2), Miss Digby
##Mr and Mrs J. Diggle
##Mr Lee Dillon, Misses Dillon (2)
##Mr and Mrs Coningsby Disraeli, Mr and Mrs R. Disraeli, Miss Disraeli
##Mrs Domvile, Miss Domvile
##Mr Greville Douglas, Mrs A. L. Douglas, Misses Douglas (2)
##Mrs Akers-Douglas
##Miss Dowell
##Mr and Mrs Doxford, Miss Doxford
##Mrs Geoffrey Drage
##Mr A. Drummond, Mr and Mrs G. Drummond, Mrs A. Hay Drummond, Mrs Lawrence Drummond, Mrs Drummond, Miss Edith Drummond,  Misses Drummond (2), Miss Mary Drummond, Miss Adelizs [?] Drummond, Misses Drummond (2) [sic 2x]
##Misses Du Cane (2)
##Miss Du Chair
##Mr W. H. Dudley-Ward, Miss Sybil Dudley-Ward
##Mr F. Dugdale
##Misses Duncombe (2)
##Mrs Dundas, Miss May Dundas
##Miss Dunn
##Mrs Dunne, Miss Marion Dunne
##Mr Du Plat Taylor, Mrs G. Du Plat Taylor
##Mrs Durnford
##Mr and Mrs Thiselton Dyer
##Mrs East, Misses East (2)
##Mr F. Eaton
##Mr R. Edgcumb
##Mrs Edis, Misses Edis (2)
##Mr Bevan Edwards, Miss Bevan Edwards (2), Mr C. C. Edwards, Mrs Edwards
##Mrs Egerton, Miss Egerton (2), Miss Egerton
##Miss Grey Egerton
##Mr and Mrs M. Eliot, Misses Eliot (2)
##Miss Ellaby
##Mrs Ellicott, Miss Ellicott
##Mr and Mrs F. Elliot, Mr T. H. Elliott, Miss Gertrude Elliot
##Mr T. E. Ellis, Miss Ellis (2), Miss Evelyn Ellis
##Mrs Ellison, Miss Ellison
##Misses Elphinstone (2)
##Mr Cary-Elwes
##Mr Erskine, Miss Rachel Erskine
##Mr Maurice Euphrussi
##Mr W. H. Evans, Misses Evans (2)
##Mr H. P. Ewart, Mrs C. B. Ewart
##Mr Eyre
##Mr Cecil Fane, Mr G. H. Fane, Mr Fane
##Mr Dyafer Fakhry
##Misses Keith Falconer (2)
##Mrs Fane
##Mrs Fanshawe, Miss Fanshawe
##Mr and Mrs Fardell, Misses Fardell (2)
##Mr and Mrs Farmer, Mrs Lancelot Farmer, Miss Farmer
##Mrs Farnham
##Mr Alfred Farquhar, Mr W. Farquhar, Mr and Mrs E. Farquhar, Mrs G. M. Farquhar
##Mr J. N. Farquharson, Miss Amelia Farquharson, Miss Henrietta Farquharson
##Mr and Mrs Farquharson of Invercauld, Misses Farquharson of Invercauld (2)
##Misses Feilding (2)
##Mrs Fellowes
##Mrs Fenn
##Mrs Fenwick, Misses Fenwick (2)
##Mr and Mrs Johnson-Ferguson
##Mr Munro-Ferguson
##Misses Ferguson of Pitfour (2)
##Miss Fergusson
##Miss Dorothy Ffolkes
##Mrs Field
##Mr and Mrs Fielden, Misses Fielden (2)
##Mrs G. H. Finch, Mrs Wynne Finch, Misses Finch (2)
##Mr and Mrs Firbank
##Mr Herbert Fisher, Mr and Mrs Hayes Fisher, Misses Fisher (2)
##Mr and Mrs Fison, Miss Fison
##Miss FitzClarence (2)
##Mrs FitzGeorge, Miss Olga FitzGeorge
##Mr Fitzgerald, Mr F. G. Fitzgerald, Miss Fitz Gerald
##Mr and Mrs Almeric Fitzroy, Miss Ethel Fitz-Roy
##Mrs R. Fitzwilliam, Misses Fitzwilliam (2)
##Mr Flannery
##Mr E. Flower, Miss Flower, Miss Flower [sic 2x]
##Mrs Floyd
##Mrs H. Fludyer
##Mr H. St. George Foley
##Mrs Barrington Foote
##Mr J. S. Forbes, Mr Forbes
##Mr John Ford
##Mr H. W. Forster
##Mr and Mrs Arnold-Forster
##Mr and Mrs Bevill Fortescue
##Misses Forwood (2)
##Mr W. S. Foster, Mrs W. H. Foster, Mrs H. S. Foster, Miss Foster
##Misses Fowler (2)
##Mr Franklin
##Mrs Houston French
##Misses Frere (2)
##Mr L. Fry
##Mrs Fullerton, Misses Fullerton (2)
##Mr Gadson
##Mr Wilhelm Ganz
##Miss Gardiner, Miss Gardiner [sic 2x]
##Mrs Gardner
##Mr and Mrs Garfit [?]
##Mrs Gathorne-Hardy, Miss Gathorne-Hardy
##Mr Hamilton Gatliff
##Mr and Mrs Scott Gatty
##Mr and Mrs Sydney Gedge
##Mr Geoffrey Drage [sic; does this belong here?]
##Mr F. W. Gibbs, Misses Gibbs (2)
##Mr and Mrs Walter Gibson
##Miss Gilbey
##Mr and Mrs Tyrell Giles
##Mr W. Gillett
##Mr and Mrs Gilliat, Misses Gilliat (2)
##Mr Henry Gladstone, Mrs Gladstone, Miss Helen Gladstone
##Miss Glyn
##Misses Godley (2)
##Mrs Godson
##Mr and Mrs Goelet, Miss Goelet
##Mr Charles Gold, Miss Gold
##Mr G. P. Goldney
##Mr and Mrs S. Hoffnung Goldnung Goldsmid
##Mrs A. Goldsmid, Miss Goldsmid
##Mr Otto Goldsmidt
##Mrs Goldsworthy
##Mrs Goodden, Miss Gurrney Goodden
##Mrs Goodenough
##Mr and Mrs John Gordon, Mr and Mrs J. E. Gordon, Mrs Gordon, Mrs G. G. Gordon, Mrs S. Gordon, Mrs Gordon [sic 2x], Miss Hamilton Gordon, Misses Gordon (2)
##Mr and Mrs Frank Gore, V. Gore, Mr and Mrs S. W. Gore, Mrs Gore, Miss Gore
##Mr and Mrs Goschen, Misses Goschen (2)
##Mr and Mrs A. Gosling, Miss Gosling
##Mr and Mrs F. R. Gosset
##Misses Gough-Calthorpe (2)
##Mr E. A. Goulding
##Mr G. Leveson-Gower
##Mr F. Graham, Mr Graham, Mr H. R. Graham, Mr and Mrs C. C. Graham
##Mrs Grant, Miss Grant
##Miss Victona Grant-Duff
##Mr and Mrs Henry Graves, Miss Graves
##Mr Ernest Gray
##Mrs Green
##Mr H. D. Greene, Mr W. R. Greene
##Mrs Gregory, Miss Gregory
##Mr and Mrs W. H. Grenfell, Mrs H. Grenfell, Miss Maud Grenfell
##Mr J. A. Gretton
##Mr Howard of Greystoke
##Mr Grifflth-Boscawen
##Mr and Mrs W. H. Kendal Grimston
##Mr George Grossmith [&quot;George Grossmith was not a little lionised by titled ladies&quot;&lt;ref name=&quot;:3&quot;&gt;“The Queen’s Garden Party. Buckingham Palace Grounds. A Brilliant Scene. The Queen’s Cup of Tea.” ''Daily News'' (London) 29 June 1897, Tuesday: 5 [of 10], Col. 6a [of 7] – 6, Col. 2a. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970629/021/0005. Print pp. 5–6.&lt;/ref&gt; (5, Col. 6b)]
##Mr Montagu Guest
##Mrs Gunter, Misses Gunter (2)
##Mr Gurdon
##Mrs Gurney
##Mrs Guy-Pym
##Mr and Mrs Gye
##Mr and Mrs Carl Haag, Miss Carl Haag
##The Munshi Abdul Hafiz Karim
##Mr and Mrs Haggard
##Miss Haig
##Mr R. B. Haldane
##Mr Halford, Misses Halford (2)
##Mr and Mrs Lewis Hall, Mrs Hall, Miss Hall, Miss (Lewis) Hall
##Mr and Mrs Thomas Halsey, Misses Halsey (2)
##Mr Francis Hamilton, Mrs R. W. Hamilton, Mrs Ian Hamilton, Misses Hamilton (2), Miss Hamilton
##Mrs Hammet
##Mr and Mrs Hanbury, Miss Dora Hanbury, Mrs Hanbury
##Miss V. Hanson
##Mr L. V. Harcourt
##Mr and Mrs Hardcastle, Misses Hardcastle (2)
##Mr and Mrs Hardy, Misses Hardy (2)
##Mr Cozens-Hardy
##Mr T. Hare, Mr Augustus Hare, Mr and Mrs John Hare, Mrs Marcus Hare, Mrs Marcus Hare [sic 2x], Miss Hare, Misses Hare (2), Misses Hare (2) [sic 2x]
##Mrs Harford
##Mrs Hargreaves-Rogers
##Mr C. Harrison, Miss Harrison
##Miss Hart
##Misses Hart-Dyke (2)
##Mr and Mrs Hartmann
##Mr George Harwood, Misses Harwood (2)
##Mr Hatch
##Mrs Hatton
##Mrs Haweis
##Mr and Mrs Claude Hay, Misses Hay (2), Misses Hay (2) [sic 2x]
##Mrs Arthur Heath, Mrs Heath
##Miss Louisa Heathcote
##Mr J. Henniker Heaton
##Misses Hemming (2)
##Mr Philip Henriques
##Mrs Heneage, Miss Heneage
##Mrs Henderson
##Miss (Brydges) Henniker
##Mrs Philip Henriques
##Mrs Herbert, Miss Herbert
##Mr and Mrs Hermon-Hodge
##Miss Heron Maxwell (2)
##Mr G. T. Hertslet
##Mrs Hervey, Miss Hervey
##Misses Hervey-Bathurst (2)
##Mr and Mrs Heseltine, Miss Heseltine
##Miss Hickman
##Mr H. Higgins, Mr Cecil Higgins, Mrs Higgins
##Mrs Platt-Higgins
##Miss Gladys Higginson
##Mrs Hildyard
##Mrs Staveley Hill, Miss Hill
##Misses (Stock) Hill (2)
##Mrs Hills
##Mrs Hippisley
##Mr and Mrs E. Brodie Hoare, Misses (Brodie) Hoare (2), Mr G. Hoare, Mrs S. Hoare, Misses Hoare (2)
##Mr and Mrs H. Hobhouse
##Mr R. K. Hodgson
##Mr and Mrs C. D. Hohler
##Mr R. R. Holmes, Mrs Holmes
##Mr R. Hallett Holt
##Mr Maurice Holzman
##Misses Hood (2)
##Miss Margaret Acland Hood
##Miss Hooker
##Mr and Mrs E. Hope, Mr and Mrs Adrian Hope, Misses Adrian Hope (2), Mr and Mrs James Hope, Mr Hope, Miss Mary Hope, Miss Hope
##Mr and Mrs Beresford-Hope, Miss Agnes Beresford Hope
##Mr and Mrs W. H. Hornby, Misses Hornby (2)
##Mr and Mrs Horner
##Mr and Mrs Hornyold [sic]
##Mr J. C. Horsley
##Miss Jean Hotham
##Misses Houldsworth (2)
##Mr R. P. Houston
##Mr E. S. Howard, Mr and Mrs A. C. Howard, Mr Joseph and Mrs J. Howard, Mr and Mrs H. Howard, Mrs Howard, Misses Howard (2), Miss Howard, Misses Howard (2) [sic 2x]
##Mr J. Hozier
##Mr and Mrs G. B. Hudson
##Hughes, Misses Hughes (2)
##Mr and Mrs A. C. Humphreys-Owen
##Mr and Mrs Hungerford
##Miss M. Carew Hunt
##Mrs W. G. G. Hutchinson, Miss Hutchinson
##Mr and Mrs G. M. Hutton, Mr John Hutton, Mr A. E. Hutton, Mrs G. Hutton
##Mrs Inglefield
##Mr and Mrs Wootton Isaacson
##Mrs Jackson, Miss Grace Jackson
##Mr Jacobs
##Mrs Jacoby
##Mr and Mrs W. James, Mr Arthur and Mrs A. James, Miss Helena James
##Mrs J. E. Jameson, Misses Jameson (2)
##Mr and Mrs Jebb
##Mr and Mrs A. F. Jeffreys
##Mr E. R. Jenkins, Missess Jenkins (2)
##Mrs Jenkinson
##Miss Jenner
##Mr and Mrs H. C. Jervoise, Miss Jervoise
##Mrs Jessel, Miss Jessel
##Mrs Cotton-Jodrell, Miss Cotton-Jodrell
##Mr and Mrs J. H. Johnstone, Mrs G. Johnstone, Miss Johnstone, Miss S. L. Johnstone
##Mrs Joicey, Miss Joicey
##Misses Jolliffe (2)
##Mr and Mrs Atherley-Jones
##Mr and Mrs Brynmor-Jones
##Mrs Inigo Jones
##Mr Philip Burne Jones
##Mrs Pryce Jones
##Mr Henry Joslin
##Mr and Mrs Kearley
##Mrs Keeley
##Misses Keith-Falconer (2)
##Miss Kemball
##Mr George Kemp
##Mr C. Kempe
##Mr and Mrs A. Kennard, Mrs Hegan Kennard, Miss Kennard, Misses Kennard (2)
##Misses Kennaway (2)
##Mrs Kennedy, Miss Kennedy
##Mrs Kennion [?]
##Mrs Kennison
##Mr and Mrs W. Kenny, Miss Ethel Kenny
##Mr J. Kenyon
##Mrs Colin Keppel, Miss Keppel
##Misses Ker (2)
##Misses Kerr (2), Miss Nona Kerr
##Mrs Kilkelly
##Mr and Mrs Kimber, Miss Kimber
##Mr King King, Miss King King
##Mr Nigel Kingscote, Mr T. Kingscote
##Mrs Kingston
##Mrs Kitching
##Misses Kitson (2)
##Mr Lees Knowles and Mrs Knowles
##Mr Knowles
##Mr and Mrs Kuhe
##Mrs A. P. Lake
##Misses Lambart (2)
##Miss Aline Lambton
##Mr Landon
##Mrs Lane
##Mrs Langenbach
##Miss Larking
##Mrs Lascelles
##Mr W. F. Laurence
##Mr Edwin Laurence
##Misses Laurie (2)
##Mrs Laurier
##Mr and Mrs E. Law
##Mrs E. Lawrence, Misses Lawrence (2)
##Mrs Lawrie
##Mr J. Grant Lawson, Miss J. Lawson
##Mr and Mrs Lecky
##Mrs Hanning Lee, Misses Hanning Lee (2)
##Miss C. Lees
##Miss Leese
##Miss Violet Leigh
##Mr and Mrs S. Leighton, Misses Leighton (2)
##Mrs Leslie
##Mr L’Estrange, Miss l’Estrange
##Mr Letchworth
##Mrs Lewis, Misses Lewis (2)
##Mrs Naylor Leyland
##Mrs Liddell, Miss Liddell, Misses Liddell (2)
##Mrs Lidderdale, Misses Lidderdale (2)
##Misses Lindley (2)
##Mr Henry Gore Lindsay, Miss Gore Lindsay, Mr H. B. Lindsay, Mr W. A. Lindsay, Miss Lindsay, Miss Lindsay [sic 2x]
##Mr Leonard Lindsey
##Misses Linton (2)
##Miss Lister
##Mr Cecil Lister-Kaye
##Miss Llewelyn
##Mr E. Lloyd, Mrs Lloyd, Misses Lloyd (2)
##Miss Alice Loch, Miss Emily Loch
##Mrs Lockhart
##Mrs Lockwood, Miss Lockwood
##Mr Loder
##Miss Loftus
##Mr Heathcote Long and Mrs Long
##Mr H. T. Lopes, Misses Lopes (2)
##Mr Drury Lowe and Mrs Lowe, Miss Drury Lowe
##Mr and Mrs J. W. Lowther, Miss Aimee Lowther
##Mr E. H. Loyd, Mr and Mrs A. K. Loyd
##Mr and Mrs H Lubbock, Miss Lubbock
##Mr Reginald Lucas, Mrs Lucas, Mrs F. A. Lucas
##Mrs Lucas-Shadwell, Miss Lucas-Shadwell
##Mrs Luck
##Mr and Mrs Fairfax Lucy
##Mr H. Luttrell, Mr W. C. F. Luttrell, Mrs Luttrell, Miss Luttrell
##Miss Lyall
##Misses Lyell (2)
##Mr and Mrs Lyon
##Miss Lyson
##Misses Lyte (2)
##Mr W. G. E. Macartney
##Mr and Mrs  J. C. Macdona
##Miss Macdonald
##Mr Alpin Macgregor, Misses MacGregor(2), Miss Macgregor
##Mr and Mrs Muir Mackenzie, Miss Margaret Muir MacKenzie, Miss Mackenzie
##Mr Mackinnon
##Misses Mackworth (2)
##Mr and Mrs J. Maclean
##Mr and Mrs J. W. Maclure, Miss Maclure
##Mr and Mrs Frederick Macmillan
##Miss Macnaghten
##Miss Macpherson-Grant
##Mr Madden, Miss Madden, Misses Madden (2)
##Misses Magniac (2)
##Mr R. Maguire
##Mr Fuller Maitland
##Mr lan Z. Malcolm
##Mrs Malet, Miss Malet
##Mr B. Mallet
##Mr and Mrs G. Manners
##Mrs Newton Mant
##Mr and Mrs Marjoribanks, Mrs Majoribanks
##Mr T. C. March
##Mrs Markham, Misses Markham (2)
##Mr and Mrs H. H. Marks
##Mrs Marshall
##Mr and Mrs W. A. M’Arthur
##Mr Martin, Mrs R. B. Martin
##Miss Martyn
##Mr Mason
##Miss Massey-Mainwaring
##Mr C. Maud
##Mr and Mrs F. W. Maude, Mrs C. Maude, Miss Constance Maude
##Mrs Maurice
##Misses Maxwell (2)
##Mr and Mrs Maxwell-Lyte
##Mrs May
##Mr H. L. B. MCalmont, Mrs J. M'Calmont
##Miss M'Clintock
##Mrs J. M'Donald
##Mrs Meeking, Miss Meeking
##Mrs Mellor, Misses Mellor (2)
##Mr and Mrs Beresford Melville
##Mr and Mrs T. G. Menzies, Miss Menzies
##Mr and Mrs M'Ewan
##Mr and Mrs P. C. Milbank, Miss May Milbank
##Mr F. Bingham Mildmay, Mr and Mrs Bingham Mildmay, Miss Beatrice Mildmay
##Mr and Mrs Arundel St. John Mildmay, Miss St. John Mildmay
##Mrs Napier Miles
##Mrs Millett
##Mr and Mrs A. Milman, Miss Lena Milman, Misses Milman (2)
##Mr and Mrs Milvain
##Mrs Victor Milward, Miss Milward
##Mr A. B. F. Mitford, Miss Mitford
##Mr and Mrs M’Laren
##Mrs M'Neile
##Mr and Mrs C. M’Neill, Mrs M’Neill
##Mrs W. C. F. Molyneux
##Mrs G. Moncrieff
##Mr Monk, Misses Monk (2)
##Mr E. P. Monckton
##Mr Ronald Moncrieffe
##Miss Cicely Monson
##Mr V. Montagu, Misses Montagu (2)
##Mrs Montefiore
##Mr Montgomerie
##Mrs Montgomery, Miss Montgomery
##Mr Moon
##Miss Mary Moore
##Mrs Moorhouse
##Mrs Manvers Moorson
##Mr R. J. More
##Miss More-Molyneux
##Miss Evelyn Moreton
##Mr Charles Morley
##Mr and Mrs Morrell
##Mrs Ashurst Morris, Misses Morris (2), Miss Ethel Morris
##Mr Hugh Morrison
##Misses Moseley (2)
##Mrs Mostyn
##Mr and Mrs Mount, Misses Mount (2)
##Misses Mowatt (2)
##Miss Mowbray
##Mrs Max Muller
##Miss Mundella
##Mr and Mrs Campbell Munro, Miss Campbell Munro
##Mr and Mrs Muntz, Miss Muntz
##Mr and Mrs Murdoch, Miss Murdoch
##Mr and Mrs W. J. Mure, Mr Mure, Miss Mure
##Mr C. J. Murray, Mr G. H. Murray, Mrs Graham Murray, Miss Graham Murray, Mrs J. Murray, Mrs Wyndham Murray, Miss Murray
##Mr W. H. Myers
##Mr M. Myther
##Misses Nelson (2)
##Misses Nevill (2), Miss Nevill
##Mrs F. Neville, Miss Neville
##Mrs Nevul [?]
##Mr F. A. Newdigate
##Mrs Newhouse
##Mrs H. F. Nicholson
##Mr and Mrs Nicol, Miss Nicol
##Mr G. and Mrs Noel, Misses Noel (2)
##Mrs Nugent
##Mr T. W. Nussey
##Mrs Oakley, Miss Oakley
##Misses O’Brien (2)
##Mr J. C. O'Dowd
##Miss Ogilvy
##Miss V. A. Okeover
##Mrs H. H. Oldham
##Mr and Mrs M. Oldroyd
##Mr and Mrs Arthur Oliphant
##Misses Olpherts [?] (2)
##Mr and Mrs Oppenheim, Miss Linda Oppenheim
##Mr and Mrs Charles Orde
##Mr C. L. Orr-Ewing
##Miss Alina O'Shee
##Mr and Mrs R. A. Oswald, Mr and Mrs Oswald
##Miss Phoebe Otway
##Miss Humphreys Owen
##Mr Hussey Packe, Misses Packe (2)
##Mrs A. Paget, Mrs Paget, Miss Alice Paget, Miss Paget
##Misses Paget of Cranmore (2)
##Mrs Pakenham, Mrs Pakenham
##Miss Palgrave
##Mrs Dampier Palmer, Miss Palmer
##Mr Paoli
##Miss Parker
##Mr E.  and Mrs Parkes
##Mrs H. Parr
##Miss Parry
##Mr Paton
##Mr and Mrs J. L. Pattison, Miss Pattison
##Mr J. Balfour Paul
##Mr J. M. Paulton
##Mr Walter Peace, Miss Peace
##Mrs Peacocke [?]
##Mr Godfrey and Mrs G. Pearse
##Mr Joseph and Mrs J. Pease, Mr Arthur and Mrs A. Pease, Mr A. E. Pease, Miss Pease, Miss Pease
##Mr A. Peckover, Misses Peekover [?]
##Mr Archibald Peel, Mr Algernon Peel, Mrs A. Peel, Misses Peel (2), Miss Cecilia Peel
##Mrs Aldrich Pelham, Miss Anderson Pelham
##Misses Pelly (2)
##Mr J. and Mrs Pender
##Mr J. Penn, Miss Penn
##Mr Pennefather
##Mr Heber Percy
##Miss Pereira
##Mr and Mrs Perks
##Mrs Perowne, Miss Perowne
##Mr Henry Petre
##Mrs Peyton, Miss Peyton
##Mrs N. G. Philips
##Mr and Mrs Phellps, Miss Pollock Phellps
##Mr and Mrs Lort Phillips, Miss [?] Phillips
##Mr B. Faudel-Phillips, Mr L. Faudel-Phillips
##Mrs Phillpotts
##Mr and Mrs Constantine Phipps, Mr Charles Phipps, Miss Phipps, Mr and Mrs Wilton Phipps, Miss Wilton Phipps
##Mrs Pipon
##Mrs Pirie [?]
##Mrs Pitman
##Mrs Fox Pitt
##Mr Platt-Higgins
##Mrs Poe
##Miss Pole, Miss Chandos Pole
##Mr and Mrs Pollock
##Mr and Mrs John Ponsonby, Miss Julia Ponsonby, Miss Ponsonby
##Mr and Mrs Wyndham Portal
##Mrs F. Post, Miss Post
##Mr and Mrs Powell, Miss Baden Powell, Miss Powell (2)
##Mrs Powlett, Miss Powlett, Miss Orde Powlett
##Mr Herbert Praed
##Mr and Mrs Price, Mr and Mrs Montagu Price
##Miss Priestley
##Mr H. W. and Mrs Primrose
##Mrs Upton Prior
##Mr and Mrs Leslie Probyn, Miss Charlotte Probyn
##Mr Roland and Mrs R. Protheroe
##Mr Provand
##Mr A. V. Pryor
##Mr Guy Pym
##Miss Quain
##Mr and Mrs Quilter, Misses Quilter (2)
##Mr Henry and Mrs H. Raikes, Mrs Raikes, Miss Lucy Raikes
##Mr Pandeli Ralli
##Mr and Mrs Alexander Ramsay, Miss Ramsay
##Mr and Mrs J. Rankin, Miss Rankin
##Mr Read
##Mr and Mrs Rebow, Miss Rebow
##Mr G. A. Redford and Mrs Redford
##Miss K. Reiss
##Mr and Mrs J. Rennell Rodd, Mrs Rennell Rodd
##Mr and Mrs Renshaw
##Mr J. A. Rentoul
##Mr Repton
##Mrs I[?]. Ricardo, Misses Ricardo (2), Miss Ricardo
##Mrs Rice, Miss Beatrix Rice
##Mr H. C. Richards
##Mr T. Richardson, Mr and Mrs Richardson
##Mrs Riddel
##Miss Rigby
##Alderman and Sheriff Ritchie, Misses Ritchie (2)
##Mr A. T. Phillips Roberts
##Mr Forbes Robertson, Mr Edmund Robertson, Mrs Robertson, Miss Sibyl Robertson
##Mrs Robins, Miss Robins
##Mr Brooke Robinson, Mrs Robinson
##Miss Frances Rod
##Sheriff Hargreaves Rogers
##Mrs Rolfe
##[[Social Victorians/People/Fanny Ronalds|Mrs Ronalds]]
##Mrs James Ronand
##Mrs Carl Rosa
##Mrs Harcourt Rose
##Mr and Mrs Leopold Rothschild, Mr Alfred Rothschild
##Mr James Round, Misses Round (2)
##Mr Bowen Rowlands
##Mr and Mrs Royds, Misses Royds (2)
##Mrs Arnold Royle [? Royce?], Mrs G. Royle
##Mr Hugo von Ruffer
##Mr G. W. E. Russell, Mr H. J. H. Russell, Mr T. W. RusseII, Mrs J. C. Russell, Mrs F. Russell, Miss Russell, Misses Russell (2), Misses Russell (2)
##Misses Russell of Killowen (2)
##Mrs W. W. Russon[?]
##Mr John Rutherford and Mrs Rutherford
##Miss Jane Ryan
##Mr G. L. Ryder, Mr J. D. Ryder
##Mrs Salmon [?]
##Mrs Salmond, Misses Salmond (2)
##Mr and Mrs Salting
##Mr and Mrs H. S. Samuel
##Mr Albert and Mrs A. Sandeman, Mrs Sandeman, Misses Sandeman (2)
##Miss [Sanderson?]
##Mrs Sandford
##Mr and Mrs Sant, Miss Sant
##Misses Sar[?] (2)
##Misses Sartorius (2)
##Mr and Mrs R. Sassoon, Mr and Mrs Arthur Sassoon, Misses Sassoon (2)
##Miss Saumarez Smith
##Miss Truda Saunderson
##Miss Saurin
##Mrs Graves Sawle
##Mr and Mrs Scaramanga
##Mr Leo Schuster
##Mr P. L. Sclater, Miss P. L. Sclater
##Mrs Scobell
##Mr J. Murray Scott, Mrs Scott, Miss Maxwell Scott
##Mrs Seddon, Misses Seddon (2)
##Mr and Mrs C. H. Seely
##Mr and Mrs Senhouse
##Mrs Sergison [?]
##Mr and Mrs H. Seton-Karr
##Mrs Settle, Mrs Settle, 
##Miss Lily Severn
##Mr Horace and Mrs H. Seymour, Mrs L. Seymour, Miss M. Seymour, Misses Seymour (2), Miss Mabel Seymour
##Mr Lucas Shadwell
##Mr W. E. T. Sharpe
##Mr H. H. Shaw, Mr C. E. Shaw, Mr and Mrs Shaw, Miss Shaw
##Mr Michael Shaw-Stewart, Miss Shaw-Stewart
##Miss Shelley
##Mr and Mrs Shelley-Bontein
##Mrs Edgar Shephard
##Miss [Sheppart?]
##Mrs Brinsley Sheridan
##Miss Maud S[?]hey
##Mrs Teignmouth [?] Shore
##Miss Shute
##Misses Kay Shuttleworth [?] (2)
##Mr W. Sidebottom, Mr and Mrs T. H. Sidebottom
##Mr and Mrs Louis Sinclair
##Mr and Mrs T. Skewes-Cox
##Mrs Bridgman Simpson
##Mr and Mrs Skefflngton Smyth
##Mrs Slade, Mrs Frederick Slade
##Mrs P. L. Slater
##Mrs Hawley Smart
##Mr and Mrs Abel Smith, Miss Abel Smith, Mr and Mrs J. P. Smith, Mr A. H. Smith, Mr and Mrs H. C. Smith, Mr and Mrs T. Smith, Mr Smith, Mr G. D. Smith, Mr and Mrs Dudley Smith, Miss Dudley Smith, Mrs Graham Smith, Mrs C. Smith, Miss Smith, Miss Dorrien Smith, Smith (2), Miss [?]-Smith, Miss Smith (Clement), Miss Rachel Smith
##Mrs Smith-Barry
##Mr Philip Somers-Cocks
##Mr H. Somerset
##Mr Augustus Spalding
##Mr and Mrs E. B. Sparke
##Mr and Mrs A. Spicer
##, , , , , , , , , , , , , , , , , , , , E. Strachey, J. Stern, Steward, Stibbert, H. M. Stanley, J. A. Swettenham, Stevenson, J. H. Stock, J. Sturgis, C. J. Stewart, Leslie Stephen, Eames Storey, Christopher Sykes, E. J. Stanley, F. Sutton, C. E. Tritton. W. E. M. Tomlinson, H. F. Tollemache, A. M. Torrance, Tarleton, Edward Tighe, Alma-Tadema, W. H. Wilson-Todd, P. Thornton, F. Taylor, Beerbohm Tree, Dan Tupper, Montagu Tharp, Abel Thomas, Algernon Turnor, Tudway, C. W. Trotter, H. J. Tennant, J. C. Thynne, H. D. Trelawny, C. E. Thynne, F. J. Thynne, Montagu Thorold, Tremayne, H. Graham Toler, John Taylor, A. J. R. Trendell, Tosti, Christopher Tower, T. Usher, A. Ure, T. Usborne, Chas van Raalte, Graham Vivian, R. C. de Grey Vyner, Hope Vere, F. E. Villiers, Von André, Venning, L. Van Loon, Van De Weyer, Val Prinsep, Walter, Thomas Wayman, Hwfa Williams, Cornwallis West, R. G. Webster, Sackville West, Wanklyn, A. S. Wiison, G. Fleetwood Wilson, A. F. Warr, F. W. Wilson, Piers Egerton Warburton, S. Wombwell, Weigall, Powell Williams, John Welby, Wingfleld, Whitbread, J. W. Wilson, Walton, D’Arcy Wyvill, Wodehouse, Wylie, A. Wilson, John Wilson, C. H. Wilson, Herbert Whiteley, Wynne, Lee Warner, W. West, G. Whiteley, Spencer Walpole, H. C. Woods, M.D., Deputy Inspector-General, Charles Wyndham, J. Humphrey Ward, F. Walker, Whateley, W. Woodall, Wyndham, Godfrey Webb, J. Welby, Charles Waldstein, H. Yorke and Yerburgh
#Mesdames&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7a–b) — , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Stopford, St. Clair, H. M. Stanley, Stevenson, Swaine, Sullivan, J. H. Stock. J. Sturgis, J. Stern, A. C. Stewart, E. Strachey, Napier Sturt, Steward, Eames Storey, Starkie, Leslie Stephen, R. F. Synge, Stevenson, Sterling, Stewart, J. A. Swettenham, Surtees, Synge, Thomson, M. Thorold, H. Graham Toler [?], J. W. Taylor, Christopher Tower, Tosti, Temple, Beerbohm Tree, Dan Tupper, R. T. Thynne, Montagu Tharp, Trotter, Anstruther Thomson, Tupper, Taylor, C. E. Tritton, C. F. Anstruther Thomson, Edward Tighe, F. Taylor, Tillard, Tillbrook, Brook Taylor, Tudway, C. E. Thynne, J. C. Thynne, H. Thomas, Thwaites, Tarleton, A. Ure, Usher, R. Vivian, Val Prinsep, Edmund Vaughan, E. Villiers, C. van Raalte, Von André, Verschoyle, F. E. Villiers, Vance, Hope Vere, Villiers, Venning, Sackviile West, Whatman, Williams Wynn, Watson, Wharton, John Wilson, Williams, Stuart Wortley, Wood, C. H. Wilson, S. J. Way, Walton, H. Whiteley, G. Whiteley, Ellis Williams, Wilson, Weywan, E. F. Wodehouse, John Welby, Wray, Wickham, Whatley, Spencer Walpole, Hwfa Williams. J. Woodford, Charles Wyndham, Wingfield, Charles Wood, Lee Warner, Warre, Humphrey Ward, Wallis, Wilberforce, Wynne, J. Welby, Eardley Wilmot, A. S. Wilson, C. [?] E. Ward, Walter, Warner, R. G. Webster, Wells, Cornwallis-West, F. Charteris Wemyss, Yerburgh
#Misses&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7c – 5, Col. 1a) —  , , , , , , , , , , , , , , , , , , , , , , , , , ,  , , , , Satyendra Bala '''Tagore''', , , , , Stafford, Stevenson, Stopford (2), Evelyn Starling[?], Stewart, Magaret Stanley, Swaine, Stephenson (2), Stewart (2), Dora Stone, Sparkes, [Stanley?], Nita Houston Stewart, Evelyn Stanley, Swinburne, [Sullivan?], R. Sterling, Stern (2), Sparke, Hilda Stewart, Sprigg (2), [?] Stephen, Ruby Spencer Churchill, [?], Tremayne (2), Ellen Terry, Ethel Thomas, Muriel [?], Taylor, Mary Talbot, Tomlinson, G. le M. Tupper, [?], Ella Taylor, Thorold, Taylor (2), E. Tuson, Trelawny [?], Adela[?] Trefusis, Rachel Thynne, Tritton (2), Thomson (2), [?], Thesiger, Thynne, I. C. (2), Thynne (2), Thornton (2), [Temple?], Turner, Talbot, Thynne, Usher, Van de Weyer (2), [Vivian?] (2), Dorothy Villiers, Freda Villiers, Verschoyle, Van [der Byl?], Villiers, Venning, Hilda von Deichmann, Wood[ford?], Fleetwood Wilson, Eardley-Wilmot, Maud Walpole, [?hend?] Wilson, Wilson, Wilberforce, Warren (2), [W?vil?] (2), Wills (2), Warrender (2), Walrond (2), Wynd[ham?] (2), Webster (2), Watson, Wombwell, Whitehead (2), [W?Ieyer?] (2), Evelyn Wellesley, Cornwallis West, Whatman {2), [?] (2), Rachel Weigall, F. Walker, Smart Walker, Wood (2), de la Wood[?], Ward, Wilbraham, Wilberforce (2), Walker, Williams, [Workham?] (2), Yeatman
#Admirals of the Fleet [initial large caps, rest sm caps] — Earl of Clanwilliam, Lord John [Hay?], the Hon. Sir H. Keppel
#Admirals — H. G. Andoe, C. E. Buckle, Sir F. Bedford, Britten, the Hon. W. Carpenter, H. F. Cleveland, Sir H. Chads, Close, [?], Carr, E. J. Church, Sir W. Dowell, R. G. Douglas, A. L. [?], C. E. Domvile, A. T. Dale, D’Eyncourt, Field, Sir A. [Farquhar?], Fitzgerald, Fellowes, Fanshawe, Sir H. Fairfax, Sir [?] Fisher, C. J. Fane, Fullerton, the Hon. Sir E. Fremantle, [?] FitzGeorge, Woods Pasha, Sir W. Hunt-Grubbe, Sir Anthony [?] Hoskin, Lord Hood of Avalon, Sir Leopold Heath, Sir [?] [F.?] Hotham, Sir Algernon Heneage, R. H. Hamond, the Right Hon. Sir [J.?] Hay, St. G. C. D’Arcy Irvine, Jones, Kennedy, Sir A. [?s], A. P. Lake, R. M. Lloyd, Sir L. Loraine, A. H. Markham, [Sir?] R. More-Molyneux, Sir F. L. M'Clintock, Sir R. Macdonald, [the?] Hon. V. Montagu, Nicholson, Noel, Marquis of Northampton, Sir E. Ommaney [?], Sir Augustus Phillimore, A. T. Powlett, [?], [?. ?.] Rowley, Sir F. Richards, Lord Charles Scott, [? St.? John?], W. H. C. St. Clair, Bowden Smith, Sulivan, E. H. Sey[mour?], H. Stephenson, Sir Nowell Salmon, Sir W. Houston [Stewart?], Sir M. [Cuhne?]-Seymour, E. W. Turnour, E. W. Van[?] Wharton, Sir G. Willes, the Hon. W. J. Ward
#Captain, R.N. — W. A. D. Acland, C. J. Barlow, F. R. Board[?], H. Bainbridge, Hon. T. Brand, Bickford, Lord Charles [B?ford?], B. F. Clark, Colville, Carter, Hon. S. Cecil Colville, [?ford?], A. G. Douglas, Sir C. Domville, Hon. A. Hay Dru[?], [?] [W.?] [?] Gordon, Hammet, Hon. Curzon Howe, Hender[?], [?] Ingles, Jellicoe, Jephson, Johnstone, Jeffreys, H. C. [?], Hon. A. Littleton, Hon. Hedworth Lambton, Moore, May, [? Net?], Poe, Pipon [?], Aldrich Pelham, Alfred Paget, [Bi.idcl?], Rolleston, John Sinclair, Bridgeman Simpson, [?], Van Koughnet [?], Burges Watson, Eardley-Wilmot, [?ham, Winsloe, Hon. J. Yorke
#[Lieutenants???] — Anson, G. R. Bethell, Blair, Bayley, Cave[?], [?] Cave,Hon. Cecil Cadogan, de Salis, Fraser, Floyd, Hon. [?] [F?], Alaric Grant, Morgan, Moore, Marescaux, [?] Stuart, Tupper, Wells, Williams, G. J. S. Warrender
#[Lieutenants?] R.N. — Alton, Murray Aynsley, Boyle, Bather, [?], [R. F.?] Boyle, Chaytor, Sir Charles Cust, G. W. Davy, [?] Wyndham-Fiennes, Fair, Godfrey Faussett, Garforth, [L?]ord Clifford, Hopkinson, Henderson, Keyes, Keppel, [?] Lloyd, Majendie, Mitchell, Morant, Kerr-Pearse, [?] Richmond, Rae, Stewart, Hon. Victor Stanley, [?] [Calta?]-Seymoar, Trye, Thring, Hon. Cyril Ward, W[?], R. E. Wemyss, Woolcombe
#[Captain?] Trinity House, Sir J. Sydney Webbe
#[Field?] Marshall — Sir F. P. Haines, Sir Lintorn Simmons, Sir [?] Stewart, Lord Roberts of Kandahar, Viscount Wolseley
#[Generals?] —Sir J. Ardagh, Sir A. Alison, Sir H. J. Alderson, [?n] Annesley, J. Alleyne, Sir J. M. Adye, Sir C. G. [Arbuth?]not, Sir H. Havelock-Allan, R. Bateson, Sir W. F. [B?er, Sir H. Brackenbury, H. M. Bengough, the Right Hon. [?] Buller, Sir Owen Tador-Burne, H. J. Buchanan, Sir C. H. [Brown?low], Sir S. Browne, Sir M. Biddulph, Viscount Bridport, [?. O.?] Barnard, E. F. Chapman, Lord Clarina, C. F. Clery, the Hon. S. Gough-Calthorpe, E. H. Clive, Godfrey Clerk, Lord [Ch?]sford, the Hon. Sir Andrew Clarke, Sir E. Du Cane, Crutchley [?], Lord de Ros, Sir John Donelly, J. H. Dunne, Sir Martin Dillon, Sir Collingwood Dickson, Sir H. de Bathe, Davis, Sir F. de Winton, Sir T. Dennehy, Sir H. Ewart, Sir J. B. Edwards, C. B. Ewart, Cecil East, Arthur French, Sir T. Fitz-Wygram, the Hon. Sir P. Feilding, Sir T. E. Gallwey, Sir T. Goldsmid, Sir R. Gipps, Sir R. Grant, Sir F. W. Grenfell, Coleridge Grove, Goldsworthy, J. J. H. Gordon, Sir E. A. Holdich, Sir E. W. Higginson, Sir R. J. Hay, Sir R. Harrison, Julian Hall, Earl Howe, the Hon. W. Home, J. Jameson, Sir Arnold Kemball, Kelly-Kenay, Lord Mark Kerr, F. T. Lloyd, Sir D. Lysons, Sir Drury Lowe, G. Luck, J. W. Laurie, F. Marshall, the Hon. R. Monck, Crichton Maitland, Sir J. M'Neill, Montgomery, the Hon. S. Mostyn, G. Moncrieff, E. Markham, Sir W. A. Mackinnon, Bryan Milman [?], H. M’Calmont [?], M'Donnell, W. C. F. Molyneux, Lord [Methuen?], J. F. Maurice, Sir F. Middleton, O. H. Nicolls, Sir E. [?] Newdegate, Sir H. N[orman?], Sir W. Olpherts, F. Peyton [?], G. [?] Upton Prior, T. H. Pakenham, G. W. T. Rich, Lord [?der] Russell, Robinson, Rowlands, J. C. Russell, F. [Russell?], A. C. Stewart, Sir Henry Smyth, Sterling, Sir C. [?] Shute, N. Stevenson, Swaine, Lord William Seymour, [?] [Sahmond?], Sir Frederick Stephenson, Sir John Stokes, Sir R. [?], Sir H. B. Tuson, the Hon. R. A. J. Talbot, G. le M. [Tupper?], Taylor, Hon. C. Thesiger, R. T. Thynne, Upperton, [?]H. Utterson, Sir J. Watson, Sir C. W. Wilson, Sir F. F. Walker, Sir Evelyn Wood, Sir C. Warren, Albert Williams, the Hon. G. Wrottesley, Sir G. H. Willis, Sir H. Wilmot
#Colonels — Armytage, Arkwright, Pat Boyle, Burges, the Hon. [?] Byng, H. B. H. Blundell, M. S. Brownrigg, Sir E. Bradford, Sir A. [Blyge? Bigge?], the Hon. F. Bridgeman, Brassey, Lord William Beresford, St. John Barne, N. Barnardiston, Lord Blythswood, [?] Cunynghame, F. H. Custance, Clayton, Sir Henry Colville, [?] Carnac [?], Cavaye, Seymour Corkran, the Hon. Charles [?], W. Campbell, Chaloner, Archibald Calvert, the Hon. [?] Campbell, the Hon. Wenman C. Coke, the Hon. W. [?ton], the Hon. Sir W. Colville, Chaine, A. B. Crosbie, [T.?] [R?] Crosse, Lord Edward Pelham Clinton, the Hon. Henry [C?hton], E. H. Cooper, the Hon. H. Corry, John Clerk, Lord Dorchestcr, C. R. Dease, the Hon, Lewis Dawnay, [the?] Hon. H. Denison, Denny, Dalbiac, A. Davidson, the Hon. Cathbert Edwards, the Right Hon. Sir F. Edwards, [?son], R. Edis, the Hon. Charles Edgecumbe, Aubone Fife, [?], Wynne Finch, Ferguson of Pitfour, Forster, Lancelot [?r] H. Frudyer, Barrington Foote, Goldsmid, Gore, Grenfell, [?n], C. G. Gordon, R. Gunter, Alan Gardner, Hon. G. Gough, [?] [?iton], the Hon. A. Hood, the Earl of Home, Lord Claud [Hamilton?], Harford, Herbert, the Earl of Haddington, Haygarth, G. Hatton [?], Hillyard, Arthur Haig, Sir E. Stock Hill, R. Hennell, Archer Houblon [?], the Hon. Cospatrick Home, the Hon. C. Gathorne-Hardy, Johnstone, Cotton-Jodrell, Hegan, [H?nard], Sir N. Kingscote, H. A. Lascelles, the Hon. Heneage [L?], Hanning Lee, F. A. Lucas, the Hon. H. Lyttelton, Lockwood, L. V. Loyd, C. W. Long, Ronald Lane, Lucas, J. Leslie, the Hon. Caryl [?]Molyneux, John Murray, Sir A. W. Mackworth, J. M'Calmont [?], Milward, the Hon. F. C. Morgan, J. J. Mellor, Meeking, Manvers [?], Moorsom, H. Malet, the Earl of Mount Edgecumbe, the [Earl?] of March, Wyndham Murray, Sir V. Majendie, the Hon. G. [Napper?], H. H. Oldham, L. J. Oliphant, A. Paget, Dampier Palmer, [Earl?] Percy, George Paget, C. D. Patterson, Arthur Peel, [Birch?] [Richardson?], the Hon. F. W. Stopford, Sir W. G. Stirling, E J. [Sanderson?], T. M. Sandys, H. Smith, J. F. Sandeman, Renyon-[Surrey?], C. E Stewart, E. H. Sartorius, the Hon. Walter [Stewart?], L. Seymour, Settle, Stevenson, Starkie, C. H. Seafe, the Hon. Sir W. P. Talbot, J. Du Plat[?] Taylor, H. Thomas, A. W. [T?], the Hon. W. Ie Poer Trench, H. P. Vance, Sir C. E. Howard Vincent, M.P.; R. Vivian, A. P. Vivian, E. Villiers, the Duke of Westminster, the Earl of Wemyss, Lord Wantage, Ward, [Waring?], [Earle?] Welby, Lord Arthur Wellesley, Robert Williams, the Hon. H. L. Wood, Sir W. H. Walroud, F. Smart Walker, A. [Williams?] Wynn, Wardrop
#Majors — Anne, Atherley, Ashton, F. H. Bowles, the Hon. [?] R. Bourke, Carnegy, H. Candy, Close, the Hon. F. Colborue, the Hon. Wenman Coke, Lawrence Drummond, Alfred [Edgecombe?], G. Egerton, E. H. Elliot, the Hon. A. Henniker, J. [H?a?h], the Hon. Assheton Harbord, the Hon. North Dalrymple [Hamilton?], Jameson, Pryce Jones, Larnach, the Hon. Osbert [Lumley?], C. Little, Marindin, the Hon. J. Scott Napier, Wyndham Quin, F. C Rasch, the Hon. A. Sidney, the Hon. J. T. St. Aubyn, Sir Edgar Sebright, Stirling, T. E. M. Swinnington-Parkington, [?.] M. Temple, Tillbrook, Anstruther Thomson, [E.?] [L.?] Woodhouse, and the Marquis of Winchester
#Captains — O. Ames, J. Acland, Alan Boisragon, Bates, H. M. [Biddulph?], the Hon. Baring, Butler, the Hon. J. Byng, the Hon. [N.?] Yarde-Butler, E. W. Blunt, J. F. Bagot, the Hon. W. Bagot, Seymour Combe, W. Chetwynd, Dundas, Denis Daly, Cecil Drummond, M. Drummond, Ellison, Houston French, Gye, R. G. [Gilmour?], P. Green, W. G. Grice-Hutchinson, Ahmed Hussain, G. [L.?] Holford, Jessel, the Hon. W. Lambton, the Hon. G. H. [L?], Sir H. Naylor-Leyland, G. Lister, Matthews, A. D. Miller, [?],M. M'Neill, C. Norton, Phillpotts, N. G. Philips, Prety[man?], Duncan Pirie, Pitman, Fox Pitt, Petre, Harcourt Rose, [W.?] [J.?] Stopford, Sir Eyre Shaw, H. G. D. Shute, Spicer, the Hon. [?.] St. Aubyn, Sutton, Tillard, Webbe, Wray, and Gordon [Watson?]
#Lieutenants — Baun, A. Cowell, the Hon. E. C. Lennox, F. Ponsonby, J. Ponsonby, Vandeleur, the Hon. C. Willoughby, and the Hon. C. S. H. D. Willoughby
===Entertainment===
&quot;The Bands of the 1st Life Guards, Grenadier Guards, and Royal Artillery played a selection of music during the afternoon.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)

==Anthology==

====Quote Intro====
&lt;quote&gt;&lt;/quote&gt; ()

== Notes and Questions ==
# 

==References==
*
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      <text bytes="127311" sha1="ofs8vxegxbn14chyakwuerlro5iti8y" xml:space="preserve">=Event=
On Monday 28 June 1897, Queen Victoria hosted a garden party at Buckingham Palace, inviting between 5,000 and 6,000 people. This party was the final official event of the London Diamond Jubilee celebrations. The Queen released to the press the names of people invited, which means the newspapers could print some or all of this list. The very long article in the London ''Morning Post'', for example, prints what may be the comprehensive list of those invited, although two columns are illegible in some places.

The original newspaper account seems to have been published by the ''Court Circular'', and then the popular newspapers reprinted pieces of that story, many adding contextualizing paragraphs of their own. Some of these later reports are quite long, perhaps 5 or more full columns. Sometimes the newspapers included short descriptions of the women's dresses, suggesting that for the list of people invited, the source was the ''Court Circular'', but the parts of the stories devoted to context, history or fashion might have been written by a reporter present at the event.

==Logistics==
* 28 June 1897, Monday, in the gardens at Buckingham Palace, hosted by Queen Victoria.
* Between 5,000 and 6,000 guests were invited.
* Many visitors from the empire who were in town for the Jubilee celebrations were invited to this garden party.
* The weather was fine, having improved since the day before.
* The garden party was held in the grounds around Buckingham Palace, and the Palace itself was open and available for guests to visit:&lt;blockquote&gt;Great preparations had been made in the splendid grounds adjoining the Royal Palace for the party, the whole scene presenting a fascinating appearance. The beautifully-kept grounds were partially covered with tents and marquees for the convenience of the many guests, and the lovely lake was really in the hands of the Queen’s bargemen, who had charge of the many boats which had been placed on the extensive ornamental waters for the use of guests. There was also plenty of music, several regimental bands being in attendance, while for those who wished to become acquainted with the valuable pictures and works art which are to be found at the Royal residence, all the State and reception rooms of the Palace were thrown open.&lt;ref name=&quot;:2&quot;&gt;“The Queen’s Garden Party. Brilliant Scene at Buckingham Palace.” ''Globe'' 29 June 1897, Tuesday: 6 [of 8], Col. 3a–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18970629/050/0006. Print p. 6.&lt;/ref&gt; (6, Col. 3a)&lt;/blockquote&gt;
*The streets around the entrances to Buckingham Palace were lined with spectators beginning hours before the Queen was to arrive:&lt;blockquote&gt;Although the Garden Party was not timed to commence until after five o’clock, the Mall from Marlborough House to Buckingham Palace was well lined by two o’clock, and an hour afterwards large crowds, for the most part composed of ladies, had taken up their positions. This was also the case along Constitution-hill, where the assembly which had gathered to witness the Queen’s arrival at the Palace from Windsor nad [sic] to a large extent remained. The heat was somewhat oppressive, but the trees along the Mall and the Green Park afforded welcome shelter. Many ladies had evidently come prepared for a long wait, as they had provided themselves with the now familiar camp stool, which is always prominent on these occasions. On the other hand, the police were waging war against the men who frequent such places with stools and forms, and as soon as any of them put in an appearance they were quickly pounced upon by the officers, who at once proceeded to destroy the intended stands before the eyes of the helpless owners. Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.&lt;ref name=&quot;:2&quot; /&gt; (6, Col. 3a)&lt;/blockquote&gt;
==Related Events==
This garden party was the culminating event of the official celebrations for Queen Victoria's Diamond Jubilee, and more specific events led up to it:

# Trip from Windsor to Paddington Station Queen Victoria and a large retinue traveled by train from Windsor to Paddington Station the day before, preceded on an earlier train by &quot;the royal equipages sent from Buckingham Palace for the use of the Queen and her suite,&quot; which were&lt;blockquote&gt;First came the splendied semi-state landau in which the Queen made her now famous journey on June 22d. It was preceded by scarlet-coated outriders, and horsed by four magnificent bays driven by postilions in navy blue and white uniforms. Two similar carriages followed, and these were in turn succeeded by a number of pair-horse clarences for the conveyance of the household and suite, and several breaks and ‘buses for luggage. A captain's escort, furnished by the 2d Life Guards, and commanded by Captain Ellison, clattered along in rear of the carriages, and took up a position opposite the spot where, by prior arrangement, Her Majesty’s saloon was to be brought to a standstill. These magnificent troops, riding their great black horses, and with the sunlight dancing upon their nodding plumes, and reflected by their burnished helmets, cuirasses, and trappings, made a very fine show indeed. The escort did not carry the colour, as it did on the 21st, nor was it accompanied by the regimental trumpets.&lt;ref name=&quot;:0&quot;&gt;&quot;Jubilee Festivities. The Queen Again in London. Interesting Functions. A Visit to Kensington. The Garden Party.&quot; ''North British Daily Mail'' 29 June 1897, Tuesday: 5 [of 8], Col. 3a–7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002683/18970629/083/0005. Print p. 5.&lt;/ref&gt;{{rp|5, Col. 3b}}&lt;/blockquote&gt;
# Reception at Paddington
# Visit to Kensington
# Kensington to Buckingham Palace
# The Garden Party
# Return to Windsor by Way of Paddington

=== Foreign Admirals ===
On 29 June 1897, the day after the garden party, the ''North British Daily Mail'' reports that, after the Queen's garden party, the foreign admirals would return to Spithead for a tour around the dockyard and luncheon:&lt;blockquote&gt;THE FLEET AT SPITHEAD&lt;p&gt;
The fleet at Spithead was again illuminated last night, the railway companies having duplicated the ordinary train service to bring visitors down. The Koenig Wilhelm was to have sailed on Sunday evening, but her departure has been deferred, and last night her officers gave a private dinner party aboard for the anniversary of the Queen’s Coronation. All the commissioned ships in the harbour were dressed at noon. A royal salute was fired. The [Col. 6c–7a] foreign admirals will return from their visit to London on the occasion of the Queen’s garden party to be conducted round the dockyard to-day, and they will be entertained to luncheon.&lt;ref name=&quot;:0&quot; /&gt;{{rp|5, Col. 6c–7a}}&lt;/blockquote&gt;

=== Colonial Premiers ===
The day of the garden party the colonial premiers attended a meeting with Secretary of State for the Colonies, [[Social Victorians/People/Chamberlain|Joseph Chamberlain]]:&lt;blockquote&gt;THE COLONIAL PREMIERS

The whole of the Colonial Premiers went to the Colonial Office yesterday for further conference with Mr Chamberlain, who received them in his private room, attended by Mr F. H. Wilson, legal assistant, Mr Reid and the Hon. T. Cochrane, M.P., assistant private secretaries. The conference lasted hours, and was of a strictly private and confidential character, the matters discussed involving several points of high State policy.

Premiers will be entertained at Warwick Castle by the Earl and Countess of Warwick on July 15th. On the same occasion the Attorney General of Queensland will present a loving cup from Warwick, in Queensland, to the old county town of Warwick, from which it takes its name. He will be accompanied by the Colonial troopers.&lt;ref name=&quot;:0&quot; /&gt;{{rp|5, Col. 7a}}&lt;/blockquote&gt;

For these visitors to London during the Diamond Jubilee, the next major social event was on 15 July, at Warwick Castle, hosted by [[Social Victorians/People/Warwick|Daisy, Countess of Warwick and Francis, 5th Earl of Warwick]], although perhaps some attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]].

== Who Was Present ==
In the absence of a copy of the report about the garden party in the ''Court Circular'', the newspaper account with the fullest list of names is from the ''Morning Post'', although people further down the list can be impossible to identify, and two full columns are damaged (Col. 7 on p. 4 and Col. 1 on p. 5).&lt;ref name=&quot;:1&quot;&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005.&lt;/ref&gt; Whenever possible, then, what is here has been amended with other newspaper reports that have names to help decipher the illegible ones in the ''Morning Post'' account. The names in the Morning Post are grouped, mostly by rank and name.

=== People of Color at This Event ===

One purpose of a closer look at this event is to get a more precise list of names of people of color from the various countries in the empire, who were not recognized and thus not named in newspaper descriptions of other events. For example, the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]] was said to include a number of South Asian dignitaries, but because the Duchess did not release to the newspapers the names of those who were invited, those dignitaries went mostly unnamed in the newspaper reports, if their presence was noted at all. Besides the South Asian guests invited to this garden party, some South Asian visitors to London were spectators as well:&lt;blockquote&gt;Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.&lt;ref name=&quot;:2&quot; /&gt; (6, Col. 3a)&lt;/blockquote&gt;In a section on what people — mostly women — wore, the reporter for the ''Daily News'' said,&lt;blockquote&gt;Suffice to say, the modistes had done their best, and that their achievements excited general admiration. Here and there, however, was an Eastern beauty whose golden lace drapery, loosely enveloping a figure that owed nothing to the corset, challenged comparison, we will not say with what success, with the European model. In the almost entire absence of uniforms or Court dress, the costumes of the East Indian notables lent colour to the assemblage, while their pearls and diamonds, the wealth of Ormuz and of Ind, were not allowed to pass unobserved.&lt;ref name=&quot;:3&quot; /&gt; (5, Col. 6b)&lt;/blockquote&gt;

=== People Invited ===
# Queen Victoria, with escort and attendants
## Captain's Escort of the 2nd Life Guards
## The Duchess of Buccleuch, Mistress of the Robes
## The Dowager Lady Churchill, Lady in Waiting
## The Hon. Harriet Phipps, Woman of the Bedchamber
## Maids of Honour in Waiting
### The Hon. Mary Hughes
### The Hon. Aline Majendie
## the Earl of Kintore, Lord in Waiting
## Captain Drummond, Groom in Waiting
## Equerries in Waiting
### Major-General Sir John M'Neill, V.C.
### Lieutenant Colonel Davidson, M.V O. [sic]
#Grand Duke and Grand Duchess Serge of Russia
#Princess Henry of Battenberg, with attendants
##Miss Minnie Cochrane
##Colonel John Clerk, C.S.I., C.V.O.
#Her Imperial Majesty the Empress Frederic, attended by 
##the Dowager Lady Ampthill
##Lord Harris
##Colonel S. Waller
##Princess Hatzfeldt Trachenberg
##Count Seckendorff
##Baron and Baroness Reischach
#Their Royal Highnesses the Prince and Princess of Wales, with attendants
##Lady Suffield, Lady in Waiting
##Miss Knollys, Woman of the Bedchamber
##Lord Colville of Culross, K.T., G.C.V.O., Chamberlain to the Princess of Wales
##The Earl of Gosford, K.P., Lord in Waiting
##General Sir D. Probyn, G.C.V.O., K.C.B., K.C.S.I., V.C, Comptroller
##Sir Francis Knollys, K.C.M.G., C.B., Groom in Waiting
##Major-General Stanley Clarke, C.M.G., Equerry in Waiting
#Princess Victoria of Wales
#Their Royal Highnesses Prince and Princess Charles of Denmark
#Their Royal Highnesses the Grand Duke and Grand Duchess of Mecklenburg-Strelitz, attended by
##Lady Caroline Cust
##Mr. Hugo Erskine Wemyss
##Count Reventlow Criminil
##Baron von der Wense
#Their Royal Highnesses Prince and Princess Christian, attended by
##Baroness von und zu Egloffstein
##Colonel the Hon. Charles Eliot
#Her Highness Princess Victoria
#His Highness Prince Christian Victor
#His Highness Prince Albert of Schleswig-Holstein
#Her Royal Highness Princess Louise Marchioness of Lorne and the Marquis of Lorne, attended by
##Lady Sophia Macnamara
##[[Social Victorians/People/Arthur Collins|Colonel Arthur Collins]], M.V.O.
#Their Royal Highnesses Prince and Princess Henry of Prussia, attended by
##Admiral of the Fleet Sir Edmund Commerell
##Baron and Baroness Seckendorff
##Count Hahn
##Captain Muller
#Their Royal Highnesses the Duke and Duchess of Saxe-Coburg and Gotha, attended by
##The Hon. Mrs. Monson
##His Excellency Herr von Schön
##Captain the Hon. D. J. Mouson [sic, s/b Monson?], M.V.O.
##Mr. A. D. J. Monson
##Captain von Ruxleben
#Princess Beatrice of Saxe-Coburg and Gotha
#The Hereditary Prince of Saxe-Coburg and Gotha
#Their Royal Highnesses the Duke and Duchess of Connaught and Strathearn, attended by
##Colonel and the Hon. Mrs. A. Egerton
#Her Royal Highness the Duchess of Albany, attended by
##Sir Robert and Lady Collins
##Miss Potts
#Her Royal Highness Princess Frederica of Hanover and Baron von Pawel Raminingen, attended by
##Mr. and Mrs. Charles Wood
#His Royal Highness the Duke of Cambridge, attended by
##Colonel A. C. FitzGeorge, C.B.
#Her Royal Highness the Duchess of Teck and his Highness the Duke of Teck, attended by
##Lady Katherine Coke
##The Hon. A. Nelson Hood
#Her Royal Highness Princess Louise Duchess of Fife and the Duke of Fife
#His Highness the Prince and her Royal Highness Princess Frederic Charles of Hesse, attended by
##The Hon. A. Hay
##Fraulein von Tasmund
##Baron von Kotwitz
#Their Highnesses Prince and Princess Aribert of Anhalt, attended by
##Miss Deverell
##Major Evan Martin
#Her Royal Highness the Hereditary Princess of Saxe-Meiningen and her Serene Highness Princess Feodore of Saxe-Meiningen, attended by
##The Hon. Aubrey FitzClarence
##Miss von Dreskan
##Baron von Roeder
#His Serene Highness the Prince of Schaumburg-Lippe
#Their Highnesses Prince and Princess Edward of Saxe-Weimar
#Her Serene Highness Princess Victor of Hohenlohe
#Countess Gleichen (x2)
#Their Serene Highnesses Prince and Princess Adolphus of Teck
#The Prince Francis and Prince Alexander of Teck
#His Highness Prince Augustus Leopold of Saxe-Coburg
#Their Serene Highnesses Prince and Princess Blucher von Wahlstatt
#Their Serene Highnesses Prince and Princess Joachim Murat
#Their Serene Highnesses [[Social Victorians/People/Pless|Prince and Princess Hans Henry Pless]]
#Prince and Princess Loewenstein
#Their Serene Highnesses the Duke and Duchess of Arenberg
#Prince Victor Duleep Singh
#Prince Frederick Duleep Singh
#Princess Duleep Singh (x2)
#ARGENTINE REPUBLIC — M. Florencio Dominguez and M. Carlos Dominguez
#BADEN — Herr yon Brauer, Mr. Brook Taylor, and Baron Bohlen Halbach
#BAVARIA — His Royal Highness the Prince Rupert, General Sir L. Gardiner, K.C.V.O., C.B., Major Fairholme, Lieutenant-Colonel Emile von le Bret Nucourt, and Captain Othon von Stettin
#BELGIUM — His Serene Highness the Prince Charles de Ligne, Princess de Ligne, Madlle. de Ligne, Mr. C. lnnes Ker, Count de Jonghe d'Ardoye, and the Marquis d’Asshe
#BOLIVIA — M. Caso, Mr. Conway Seymour, M. Pedro Suarez, Madame Suarez, and M. Adolfo Bolivian
#BRAZIL — M. [[Social Victorians/People/Souza Correa|de Souza Correa]] [Corréa?]
#BULGARIA — Their Royal Highnesses the Prince and Princess of Bulgaria, Colonel J. R. Slade, C.B., Madame Petrow Tchomakoff, Count Robert de Bourboulon, Lieutenant-Colonel Marcoff, Major Petrew, Captain Stoïanow, and Mr. Martin Furth
#CENTRAL AMERICA (Greater Republic) — M. Medina and Miss Medina
#CHILI — M. Ramon Subercasseaux and Mr. Raglan Somerset
#CHINA — His Excellency Chang Yen Hoon, Colonel Mark Bell, V.C,. Mr. Liang, Mr. Jui, and Mr. Koo
#COREA — His Excellency Min Young Hwan, Major A. Cavendish, Mr. Min Young Chan, Mr. Min Shangho, and Mr. von Rautenfeld
#COSTA RICA — Senor Don Demetrio Iglesias, Mr. C. Alban Young, Dona Eudoxia Castro, Señorita Maria Iglesias, Don Ricardo Fernandez Guardia, and Dona Christina Castro Keith
#DENMARK — His Royal Highness the Prince Waldemar, Major-General Arthur Ellis, C.S.I., M. Charles Rothe, and Captain Evers
#EGYPT — Prince Mohammed Ali Pasha, Colonel Larking, Tigrane Pasha, Colonel Aziz Bey, Mr. George Smart, Said Zoulfikar Bey
#ECUADOR — M. Navares, Colonel Concha
#FRANCE — General Davoust, Duc d'Auerstadt, Duchesse d'Auerstadt, and Madlle. Davoust, Colonel Brabazon, Colonel Dawson, General Hagron, M. Crozier, Colonel Humbert, and Captain Riviers de Mauny
#GERMANY — His Royal Highness the Prince Albert of Prussia, Prince Regent of Brunswick, Major-General Sir C. du Piat, K.C.B., Colonel Grierson, Lieutenant-General von Plessen, Colonel von Arnim, Captain Fischel, Count von der Schulenberg (Hofmarschall), Major Freiherr von Stein, Dr. Schreibe, Captain von Unzer
#GREECE — M. Rangabi, Mr. R. D. Norton
#GUATEMALA — Dr. Cruz, Madlles. Cruz (2), Señor Estrada
#HAWAIIAN ISLANDS — Mr. S. M. Damon, Captain the Hon. H. Napier, Major Curtis P. Jaukea
#HESSE — Their Royal Highnesses the Grand Duke and Grand Duchess of Hesse, Colonel the Hon. H. Byng, C.B., Baroness de Grancy, Baron Riedesel zu Eisenbach, Baron de Genadius Grancy
#ITALY — Their Royal Highnesses the Crown Prince and Princess, the Earl of Clarendon, Colonel Needham, Countess Giulia Trigona, Lieutenant-General Terzaghi, Major Cavaliere Viganoni, Captain Cavaliere Merli Miglietti, Count Romnaldo Trigona, Cavaliere F. Comotto
#JAPAN — His Imperial Highness the Prince Arisugawa, Mr. R. F. Synge, Captain Beaumont, R.N., Marquis Ito, Mr. S. Saito, Marquis Kido, Captain Funaki, Lieutenant-Colonel Murata, Lieutenant Kato, Mr. Nabeshima
#LIBERIA — Mr. H. Hayman
#LUXEMBURG — His Royal Highness the Hereditary Grand Duke of Luxemburg, Colonel H. D. Browne, Baron Ritter yon Grünstein
#MECKLENBURG-SCHWERIN — His Excellency Herr D. yon Vietinghoff, Mr. Eyre A. Crowe
#MEXICO — Don Antonio Mier y Celis, Mr. Arnold Royle, C.B., Don Francisco R. Gallardo, Don Eustagino dc Escaudon, and Captain Don Ponfirio Diaz
#MONTENEGRO — His Highness the Prince Danilo, Major the Hon. C Harbord, Colonel Djurcovitch, and Captain Pejanovitch
#NETHERLANDS — Count van Lynden, Countess van Lynden, Mr. Horace West, and Count W. de Bylandt
#PARAGUAY — M. E. Machain and Madame Machain
#PERSIA — His Imperial Highness the Prince Amir Khan, General Sir Thomas Gordon, K.C.I.E., C.B., C.S.I.[,] Mr. Harry Churchill, General Karim Khan, Mirza Ahmad Khan, Mirza Ohaness Khan, Mirza Mohamad Ali Khan
#PERU — Senor Canevaro, Duchesse de Zoagli Canevaro, Dr. Don A. N. Puente, Don Alfredo Elster, and Don Carlos von der Heyde
#PORTUGAL — His Royal Highness the Duke of Oporto, Major the Hon. H. C Legge, M.V.O., Colonel Duval Telles, Captain Moreira de Sà, Major d'Albuquerque, and Lieutenant Jose de Melie[?]
#ROME — Right Rev. Monsignore Sambucetti, [[Social Victorians/People/Stonor|Hon. Harry Stonor]], Right Rev. Monsignore Belmont, the Right Rev. Monsignore de Vaz, Marchesi and Marchesa Muccioli, of the Noble Guard
#ROUMANIA — General Pancovici, Colonel G. P. Georgescu
#RUSSIA — Their Imperial Highnesses the Grand Duke Serge and Grand Duchess Feodrowna, the Grand Duke Cyril, Lord Churchill, Lieutenant-Colonel Waters, Countess Olsouffiew, Princess Youssoupoff, Princess Lobanoff de Rostow, General Stépanoff, Colonel Gadon, and Prince Youssoupoff, Colonel Clements, Mr. Alexander Gordon Ross, and Sub-Lieutenant N. Coubé (A.D.C. to Grand Duke Cyril)
#SAXE-COBURG — His Royal Highness the Prince Philip of Saxe-Coburg, Captain Walter Campbell, and Herr von Schön
#SAXE-WEIMAR — His Highness the Prince Hermann of Saxe-Weimar, Mr. Frederick Campbell, and Count Zeppelin
#SAXONY — His Royal Highness the Prince Frederick Augustus, Duke of Saxony, Colonel Howard, Freiherr yon Reitzenstern, First Lieutenant von Metzsch, and Baron von Oppell
#SERVIA — M. Mijatovich and Madame Mijatovich
#SIAM — His Royal Highness the Crown Prince and the Prince Mahit of Siam, Colonel E. H. Sartorius, V.C., Lieutenant-Colonel Rajavallabha, Lieutenant-Colonel C. Vernon Hume, Colonel Indaraty, Surgeon-Major Yarr
#SPAIN — Duke of Sotomayor, Captain the Hon. A. Greville, Señor José Caro, Señor Alfonso Merry del Val, and Señor Benitez al Villar
#SWEDEN AND NORWAY — His Royal Highness the Prince Eugène of Sweden and Norway, Captain G. L. Holford, Count G. Gyldenstolpe, Captain Roeder, Captain Baron Cederstrom
#TURKEY — Munir Pasha, Major Surtees, Brigadier-General Nassir Pasha, Captain Enver Bey, Colonel Gordon Ponsonby
#UNITED STATES — His Excellency the Hon. Whitelaw Reid, Mrs. Whitelaw Reid, Colonel Hallam Parr, Major-General Nelson A. Miles, Mrs. Nelson Miles, Rear-Admiral Joseph N. Miller, Captain M. P. Maus, Mr. Ogden Mills, Mrs. Ogden Mills, Mr. G. Creighton Webb, Mr. Erskine Hewett, Commander W. H. Emory, Lieutenant Philip Andrews, Lieutenant T. S. Rogers
#URUGUAY — Dr. Alberto Nin, Madlle. Nin, Don Alfonso Saenz de Zumaran, Don Luis Posadas, Colonel C. Robido
#WURTEMBURG— His Royal Highness the Duke Albert of Wurtemburg, Colonel C. Swaine, Lieutenant-General von Bilfinger, First Lieutenant Count von Degenfeld- Schonburg; five officers of the Queen's German Regiment: Major C. R. Burn (in attendance), Lieutenant-Colonel von Falkenhayn, Major von Arnim, First Lieutenant Baron von Moeller-Lilienstern, First Lieutenant von Gerlach, Second Lieutenant von Studnitz
#&quot;Native Princes, and gentlemen and ladies accompanying them&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##His Highness the Raja of Kaparthala
##His Highness the Thakur Sahib of Morvi, K.C.I.E.
##His Highness the Thakur Sahib of Gondal, C.I., and her Highness the Maharani of Gondal, C.I.
##Colonel Maharaj Dhiraz
##Sir Pratab Singh, K.C.S.I.
##Thakur Hari Singh[,?]
##Kunwar Dhokal Singh
##Rajah Ajit Singh of Khetri, attended by
##Rajkumar Unmaid Singh of Shahpura, attended by
###Colonel Trevor (in attendance upon the Rajah Ajit Singh of Khetri and the Rajkumar Unmaid Singh of Shahpura)
##Bijey Singh
##Sir Jamaetjee Jejeebhoy, Bart., C.S.I., Miss Jejeebhoy, Mr. Jejeebhoy
##Mr. and Mrs. Powrala
##Major J. G. Turner and Mrs. Turner
##Mr. A. R. Wood and Mrs. Wood
#The &quot;officers of the Imperial Service Troops, with British officers and ladies&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Captain Mir Hashim Ali Khan Hyderabad-Resaldar
##Major Sunayat Singh, Kashmir
##Commandant Abdul Ganny, Gwalior
##Commandant Gooind, Rao Matkar, Indore
##Commandant Mirza Kurim Beg, Bhopal
##Rai Bahadur Dhunpat Rai, Jeypore
##Commandant Nand Singh, Patiala
##Commandant Rai Bahadur Thakur Dip Sing, Bikanir
##Commandant Chatru Singh, Bhartpur
##Resaldar Abdul Majid Khan, Babawalpur
##Commandant Daud Khan, Ulwar
##Commandant Nazir Khan, Rampur
##Risalda-Major Didar Singh, Sindi
##Risaldar-Major Kishan Singh, Nabha
##Risaldar Hara Singh, Karpurthala
##Risaldar Dhan Singhi, Bhavnagar
##Colonel H. Melliss, C.S.I., and Mrs. Melliss
##Major F. H. R. Drummond and Mrs. Drummond
##Captain F. Angelo
##Lieutenant H. Coape-Smith
##Captain G. F. Chenevix-Trench
#The &quot;officers of Native Cavalry Corps with British officers and ladies&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Risaldar-Major Baha-ud-din-Khan
##Sardar Bahadur, A.D.C. to Viceroy
##Risaldar-Major Sayyid Abdul Aziz
##Risaldar-Major Khan Bahadur
##Risaldar-Major Izzat Khan
##Risaldar-Major Hukam Singh
##Risaldar-Major Sher Singh
##Risaldar-Major Husain Khan
##Risaldar-Major Mangal Singh
##Risaldar-Major Kesar Singh
##Risaldar- Major Faiz Khan
##Risaldar-Major Muhammad Umar Khan
##Risaldar-Major Ali Mahomed Khan
##Risaldar-Major Mihrab Ali Khan
##Risaldar Kaddam Khan
##Risaldar Jahanzir Khan
##Risaldar Nadir Khan
##Risaldar Mir Haidar Shah Khan
##Risaldar Makbul Khan
##Risaldar Net Ram
##Ressaidar Gurdatt Singh
##Subadar Muhammed Beg Junadar
##Abdul Karin Khan
##Lieutenant-Colonel J. C. H. Gordon and Mrs. Gordon
##Major A. Phayre and Mrs. Phayre
##Captain C. F. Campbell
##Captain P. Melville, in attendance on his Highness Thakur Sahib of Morvi
##Captain M'Cartney Filgate, in attendance on their Highnesses the Thakur Sahib and Maharani of Gondal
##Mr. Nowroz
##M. Parveez
##Sir M. Mansherjee Bhownaggree, M.P.
##Mr. Percy Armytage and Mrs. Armytage
##Mr. Frank Cook, C.I.E., and Mrs. Frank Cook
#The &quot;commanding officers of Colonial contingents, with the ladies accompanying them&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Colonel the Hon. M. and Mrs. Aylmer, Canada
##Colonel and Mrs. Lassetter, New South Wales
##Major Reay, Victoria
##Colonel Pitt, New Zealand
##Major and Miss King, Queensland
##Lieutenant and Mrs. Phillips, Cape of Good Hope
##Lieutenant-Colonel Rowell, South Australia
##Major Strickland, Western Australia
##Captain Shepstone, Natal
##Major and Miss Reeves, Ceylon
##Mr. Badeley, Hong Kong
##Colonel Walker, C.M.G., and Mrs. Walker, Straits Settlements
##Captain Lucie Smith, Jamaica
##Lieutenant-Colonel E. B. M'lnnis, C.M.G., and Mrs. M'lnnis, British Guiana
##Major Rooks, Trinidad
##Captain Bernard, Malta
##Captain Kershaw, Cyprus
##Captain and Mrs. Middlemist, Gold Coast
##Inspector Hook, Lagos
##Captain Blakeney, Sierra Leone
##Lieutenant Festing, Royal Niger Company
##Captain Flint, British North Borneo Company
##The Hon. M. Gifford, Rhodesian Horse
##The following British officers attached: Lieutenant-Colonel Boulton, Lieutenant-Colonel Prior, Lieutenant-Colonel Tucker, Lieutenant-Colonel Domville, Lieutenant-Colonel Gibson, and Lieutenant-Colonel Tyrwhitt
#The &quot;gentlemen representing the various races in the Island of Ceylon&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)
##Maha Mudaliyar don Solomon Dias Bandaranaihe
##The Hon. Alexander Dealius Sonewiratne
##M. E. Rowland Goonoratne
##M. Charles de Soysa Dessanayaka
##Panabokko Jikiri Banda
##Nugawela Kuia Banda
##Kobbokeduwe Loku Banda
##M. E. S. W. Senathi rajah [sic] and Mrs. Senathi
##M. J. H. de Saram and Miss de Saram
##M. P. Ramanathan
##M. Saunders and Miss Saunders
#The &quot;members of the Corps Diplomatique and other foreigners of distinction&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)
##The Russian Ambassador, Madame de Staal, Madlle. de Staal, Madame de Stoeckl, Princess de San Donato, Madame Yermoloff, Madlle. Yermoloff, the Councillor, three Secretaries, and four Attachés of Embassy
##The German Ambassador, Countess Paul Hatzfeldt-Wildenburg, her Serene Highness Princess Hans Hohenlohe-Oehringen, Baroness yon Eckardtstein, the Councillor, two Secretaries, three Attachés of Embassy, and the Director of the Chancery
##The Austro-Hungarian Ambassador, Countess Deym, Countess Isabella Deym, Countess Clary Aldringen, Baroness Ferstel, the Councillor, two Secretaries, and four Attachés of Embassy
##The French Ambassador, Baroness de Courcel[,] Madlle. de Courcel, Madame Geoffray, the Minister Plenipotentiary, five Secretaries, and three Attachés of Embassy
##The Italian Ambassador, Princess Ruspoli, three Secretaries, and three Attachés of Embassy
##The Spanish Ambassador, Countess de Casa Valencia; Mesdlles. de Alcala Galiano (2), Marquise de Guiria, Donna de Zea Bermudez, Countess de Morella, Donna de Ia Camara y Livermore, three Secretaries, and four Attachés of Embassy
##The Turkish Ambassador, Madame Antbopoulos, the Councillor, and two Secretaries of Embassy
##The United States Ambassador, Mrs. Hay, Miss Hay, Mrs. Henry White, Mrs. Carter, Mrs. Colwell, two Secretaries, one Attaché of Embassy, and the Private Secretary to the Ambassador
##The Argentine Minister, Madame Dominguez, Mesdlles. Dominguez (3), and the Secretary of Legation
##The Persian Minister, and one Secretary of Legation
##The Danish Minister, Madame de Bille, Madame Gosch, and the Secretary of Legation
##The Siamese Minister, Mrs. Verney, Miss Verney, Mrs. Loftus, the Councillor, the Secretary, the Attaché, and the Interpreter to the Legation
##The Liberian Minister
##The Roumanian Minister and the Councillor of the Legation
##The Netherlands Minister, Baroness de Goltstein d'Oldenaller, Baroness Schimmelpenninck van der Oye, and the Councillor of Legation
##The Belgian Minister, the Councillor, and two Secretaries of Legation
##The Mexican Minister, Madame Yturbe, Madame Romero, Madame Farias, Madame Garcia, two Secretaries and three Attachés of Legation
##The Japanese Minister, Madame Kato, two Secretaries, and three Atachés [sic] of Legation
##The Minister for Sweden and Norway, Countess Lewenhaupt, and the Attaché of Legation
##The Chinese Minister, Lady Macartney, the English Secretary, three Secretaries, and four Attachés of Legation
##The Portuguese Minister, Madlle. de Quilinan, three Secretaries, and one Attaché of Legation
##The Swiss Minister, Madame Bourcart, Madame de Salis, the Secretary, and the Attaché of Legation
##The Haytian Chargé d’Affaires
##The Chargé d’Affaires of Greece, Madame Metaxas, and the Attaché
##The Chargé d’Affaires of Chile and Madame Bascunan
##Two Secretaries and one Attaché of the Brazilian Legation
##Count E. van Rosen
##Mr. Hippolyte de Aranjo
##Vice-Admiral Montt
##Mr. Pinto, Mrs. Pinto
##Mr. and Mrs. Scaramanga
##Vicomte de Galard
##Dr. Arnold, and Madlle. von Rappoport
##Mrs. John Meiggs, Miss Meiggs
##Miss Margaret Butler
##Mrs. Henry Morgan
##Hon. Chauncey Depew
##Mr. and Mrs. James Taylor
##Mr. and Mrs. Charles Marshall
##Mr. and Mrs. Edmund Bayliss
##Mrs. Colgate
##Miss Furniss
##Miss Wells
##Miss Harris
##Hon. Levi P. Morton, Mrs. Morton, and the Misses Morton
##The Bishop of Illinois and Mrs. Leonard, Miss Leonard
##The Bishop of Albany and Mrs. Doane
##The Bishop of New York and Mrs. Potter
##the Bishop of Minnesota and Mrs. Whipple
##Mr. and Mrs. Walter Burns
##Mrs. Douglas Grant
##Miss Scott
##Mrs. Grace, Miss Margarita Grace
##Mrs. Wentworth
##Miss van Wart
##M. Valentin de Courcel
##Madame la Marquise de Talleyrand Perigord
##Comte Boson de Perigord
##Vicomte d'Espenilles
##Madame and Madlle. Thierry Delanoue
##Madlle. de la Cherè
##M. Cellerier
##M. and Madame Delawarre
##Madame Evelina Fenzi
##Count A. Zannini
##M. and Madame Jules Cottran
##Chevalier E. Mazzuechi
##Signor A. Tedeschi
##Signor A. Mariotti
##Captain Lucian von Ziegler
##Chevalier Lieutenant von Barry
##Baron Georg Rothschild
##Privy Councillor Count Berchtold
##Baron G. E. Levi, Baroness Levi
##Commander E. Philipson, Mrs. E. Philipson
##The Duke and Duchess of San Germano Calabritto
##The Marquis of San Vito
##Donna Lidia Serramezzana
##Donna Margherita Chigi
##Marchioness Vitelleschi
##Chevalier Elia
##Count de Franqueville
##Count Urbain Chevrau
##M. Marcel Fonquier
##M. Baudon de Mony, Madame Baudon de Mony
##Duchess de Rohan
##Marquis de Lastorgrie, Marchioness de Lastorgrie
##Count de Boisgelin, Countess L. de Boisgelin
##M. Stern, Madame Stern, Madlle. Stern
##Count Charles du Luart
##General de Saucy
##M. E. Seydoux
##Count Jean de Madre
##M. de Monbrison
##Baron de la Chevrelière
##Count de la Villestreux, Countess de la Villestreux
##Count Urbain de Maille, Countess Urbain de Maille
##General Faveret de Kerbrich
##Monsieur de la Haye Jousselin
##Baronne Faveret de Kerbrich
##Colonel Matton
##M. Ferinier Didet
##Madame Ferinier Didet
##Donna Isabella Colonna, Donna Victoria Colonna
##Pom-k-Soh
##Madame Reyntiens
##Marquis de Fuente Hermosa
##Herr Rudolf Swobody
##M. Lauritz Tuxen
##Duchesse de Baiten
##M. de Marcoarti
##Comte de Heeren, Madlle. de Heeren
##Monsieur M. de Mauny Talvande
##Senor Don Nicolas Campero
##Lieutenant Charny
##Lieutenant Sanders
##Madame and Madlle. de Mouni
##Comtesse de Montsoulmin
#&quot;Foreign Admirals and Commanding Officers and Staffs&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 3a / Col. 3b)
##Austrian Admiral Baron von Spaun, Commander von Ziegler, Lieutenant Retter yon Barry, Lieutenant Mitchell, R.N. (attached)
##Danish Admiral H. H. Koch, Captain Waudel, Lieutenant Middelboc, Lieutenant Majendie, R.N. (attached)
##French Admiral C. F. E. De Courthille, Captain Germinet, Commander Poidlone, Lieutenant Perdriel, Sub-Lieutenant de Caqueray, Lieutenant Phillimore, R.N. (attached)
##Italian Admiral C. E. Morin, Commander Count Prasca, Lieutenant Lunghetti, Lieutenant Count Morano, Lieutenant Henderson. R.N. (attached)
##German Admiral his Royal Highness Prince Henry of Prussia, Captain Muller, Lieutenant von Spee, Sub-Lieutenant Wittman, Lieutenant Garforth, R.N. (attached)
##Japanese Admiral H.I.H. Prince Arizugawa, Captain Miura, Commander Tsuda, Lieutenant Stewart, R.N. (attached)
##Netherlands Admiral F. K. Englebrecht, Captain de Groot, Lieutenant Baron von Hardenbrock, Lieutenant Woolcombe, R.N. (attached)
##Norwegian Rear-Admiral von Krogh, Captain Muller, Lieutenant Petersen, Lieutenant Kerr Pearse, R.N. (attached)
##Portuguese Captain Barreto de Vascomellos, Captain de Cartillo, Lieutenant Trye, R.N. (attached)
##Russian Admiral Nicholas Skrydloff, Captain Domojiroff, Lieutenant Stetsenkoff, Lieutenant Twisleton Wykeham Fiennes, R.N. (attached)
##Spanish Admiral Don Segismundo Bermijo y Merelo, Captain Don Antonio Eulate y Fery, Lieutenant Don Juan Romero, Lieutenant Don Antonio Romero, Lieutenant Fair, R.N. (attached)
##Swedish Admiral A. F. H. Klintberg, Captain Ingelman, Commander Flack, Lieutenant Alton, R.N. (attached)
##United States Admiral J. N. Miller, Lieutenaut Richmond (attached)
##Captain de Mar E. Guerra
##Captain R. S. D. Cumins
#The Lord Lieutenant of Ireland and Countess Cadogan
#The Right Hon. the Speaker and Mrs. Gully, Miss Gully, and Miss Shelly Gully
#Cardinal Vaughan
#Right Hon. the Lord Mayor and Lady Mayoress, and Misses Faudel Phillips (2)
#The Gold Stick in Waiting, Silver Stick in Waiting, Silver Stick Adjutant in Waiting
#Officer Commanding 1st Life Guards and five officers
#Officer Commanding 2nd Life Guards and four officers
#Officer Commanding Royal Horse Guards and four officers
#Officer Commanding 2nd Dragoons and three officers
#Field Officer in Brigade Waiting, Adjutant in Brigade Waiting
#Commanding Officer Grenadier Guards
#Commanding Officer Coldstream Guards
#Commanding Officer Scots Guards, a Regimental Adjutant
#Commanding Officer 1st, 2nd, and 3rd Battalions Grenadier Guards and three officers of each Battalion
#Commanding Officer 1st and 2nd Battalions Coldstream Guards and three officers of each Battalion
#Commanding Officer 1st and 2nd Battalions of Scots Guards and three officers of each Battalion
#Commanding Officer Woolwich District and six officers
#Commanding Officer R.H.A. Home District and two officers
#Commanding Officer R.E. and four officers
#Commanding Officer 2nd Battalion Lincolnshire Regiment and three officers
#Commanding Officer Royal Marines (Chatham) and four officers
#Commanding Officer Royal Marines (Portsmouth) and two officers
#Four officers of the Honourable Corps of the Gentlemen at Arms
#Archbishops — Canterbury, York, Armagh, Ontario, Rupertsland
#Dukes and Duchesses
##The Duke and Duchess of [[Social Victorians/People/Argyll|Argyll]]
##The Duke and Duchess of [[Social Victorians/People/Abercorn|Abercorn]]
##The Duchess of De Baileu
##The Duke and Duchess of [[Social Victorians/People/Buccleuch|Buccleuch]]
##The Duchess of [[Social Victorians/People/Cleveland|Cleveland]]
##The Duke and Duchess of [[Social Victorians/People/Devonshire|Devonshire]]
##The Duchess of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Hamilton]]
##The Duke and Duchess of [[Social Victorians/People/Leeds|Leeds]]
##The Duke and Duchess of [[Social Victorians/People/Marlborough|Marlborough]]
##The Duke and Duchess of [[Social Victorians/People/Manchester|Manchester]]
##The Duke and Duchess of [[Social Victorians/People/Montrose|Montrose]]
##The Duke and Duchess of [[Social Victorians/People/Newcastle|Newcastle]]
##The Duke of [[Social Victorians/People/Norfolk|Norfolk]]
##The Duke of [[Social Victorians/People/Northumberland|Northumberland]]
##The Duke and Duchess of [[Social Victorians/People/Portland|Portland]]
##The Duke of [[Social Victorians/People/Richmond and Gordon|Richmond and Gordon]]
##The Duke and Duchess of [[Social Victorians/People/Roxburghe|Roxburghe]]
##The Duke and Duchess of [[Social Victorians/People/Somerset|Somerset]]
##The Duke and Duchess of [[Social Victorians/People/Sutherland|Sutherland]]
##The Duke and Duchess of St. Albans
##The Duke and Duchess of Wellington
##The Duchess of [[Social Victorians/People/Westminster|Westminster]]
#Marquises and Marchionesses
##The Marquis of Abergavenny
##The Marchioness of Ailesbury
##The Marquis and Marchioness of Ailsa
##The Marquis of Anglesey
##The Marquis and Marchioness of [[Social Victorians/People/Breadalbane|Breadalbane]]
##The Marchioness of [[Social Victorians/People/Marlborough#Marchioness of Blandford|Blandford]]
##The Marquis and Marchioness of Bristol
##The Marquis of [[Social Victorians/People/Camden|Camden]]
##The Marquis and Marchioness of Conyngham
##Dowager [Marchioness of] Conyngham
##The Marchioness of Cassar de Sai[n]
##The Marquis and Marchioness of Cholmondeley
##The Marquis of D'Auerstadt
##The Marquis and Marchioness [[Social Victorians/People/Stonor|D'Hautpoul]]
##The Marquis and Marchioness of Downshire
##Dowager [Marchioness of] Downshire
##The Marquis and Marchioness of [[Social Victorians/People/Hamilton Temple Blackwood|Dufferin and Ava]]
##The Marquis and Marchioness of [[Social Victorians/People/Exeter|Exeter]]
##The Marquis and Marchioness of Granby
##The Marchioness of [[Social Victorians/People/Florence Rawdon-Hastings Chetwynd|Hastings]]
##The Marquis and Marchioness of [[Social Victorians/People/Bective|Headfort]]
##The Marquis and Marchioness of Hertford
##The Marquis and Marchioness of Huntly
##The Marquis and Marchioness of [[Social Victorians/People/Abercorn#James Hamilton, Marquess of Hamilton|Hamilton]]
##The Marquis and Marchioness of [[Social Victorians/People/Lansdowne|Lansdowne]]
##The Marquis and Marchioness of Lothian
##Dowager (Marchioness of) [[Social Victorians/People/Londonderry|Londonderry]]
##The Marquis and Marchioness of [[Social Victorians/People/Londonderry|Londonderry]]
##The Marquis and Marchioness of [[Social Victorians/People/Ormonde|Ormonde]]
##The Marchioness of [[Social Victorians/People/Queensberry|Queensberry]]
##The Marquis and Marchioness of [[Social Victorians/People/Ripon|Ripon]]
##The Marquis and Marchioness of [[Social Victorians/People/Salisbury|Salisbury]]
##The Marquis and Marchioness of [[Social Victorians/People/Tweeddale|Tweeddale]]
##Dowager (Marchioness of) [[Social Victorians/People/Tweeddale|Tweeddale]]
##John Stewart-Murray, [[Social Victorians/People/Atholl|Marquess of Tullibardine]]
##Lawrence, [[Social Victorians/People/Zetland|Marquess of Zetland]] and Lilian, [[Social Victorians/People/Zetland|Marchioness of Zetland]]
#Earls and Countesses 
##Countess of Aberdeen and Dowager Countess of Aberdeen
##Earl and Countess of Albemarle and Dowager Countess of Albemarle
##Earl and Countess of Ancaster
##Earl and Countess of Amherst
##Earl of Ava
##Earl and Countess of Antrim
##Earl and Countess of Aylesford
##Earl and Countess of Annesley
##Earl and Countess of Airlie
##Earl and Countess of Arran
##Earl of Aberdeen
##Earl and Countess of Bandon
##Countess of Bantry
##Earl and Countess of Beauchamp
##Earl and Countess of Bathurst and Dowager Countess of Bathurst
##Countess of Bective
##Earl and Countess of Belmore
##Earl of Bradford
##Countess of Bremer
##Earl and Countess of Brownlow
##Earl and Countess of Buckinghamshire
##Earl of Burford
##Earl and Countess of Cairns
##Earl and Countess of Caledon
##Earl of Camperdown
##Earl of Cardigan
##Earl and Countess of Carnarvon and Dowager Countess of Carnarvon
##Earl of Carnwath
##Earl and Countess of Carrington
##Earl and Countess of Carysfort
##Earl and Countess of Castlestuart
##Earl and Countess of Cathcart
##Earl and Countess of Cavan
##Earl and Countess of Chesterfield
##Earl and Countess of Chichester
##Dowager Countess of Clancarty
##Countess of Clanwilliam
##Earl and Countess of Compton
##Countess of Cottenham
##Earl of Courtown
##Earl and Countess of Cowper
##Earl and Countess of Cranbrook
##Earl and Countess of Craven and Dowager Countess of Craven
##Earl and Countess of Crawford
##Earl of Crewe
##Earl and Countess of Cork and Orrery
##Earl and Countess of Coventry
##Countess of Cromartie and Dowager Countess of Cromartie
##Earl and Countess of Dalkeith
##Earl and Countess of Dartmouth
##Earl and Countess of De Grey
##Dowager Countess of De La Warr
##Earl and Countess of Denbigh
##Earl and Countess of Derby
##Earl and Countess of Donoughmore
##Earl and Countess of Drogheda
##Earl of Ducie
##Earl and Countess of Dudley and Dowager Countess of Dudley
##Earl and Countess of Dundonald
##Earl and Countess of Dunmore
##Earl and Countess of Dunraven
##Earl of Durham
##Earl and Countess of Eglinton and Winton
##Earl of Eldon
##Earl and Countess of Ellesinere
##Earl and Countess of Enniskillen
##Earl and Countess of Erne
##Earl and Countess of Errol
##Earl and Countess of Essex and Dowager Countess of Erroll
##Earl of Euston
##Earl and Countess of Feversham
##Earl and Countess of Fingall
##Earl of Fortescue
##Earl and Countess of Gainsborough
##Earl and Countess of Galloway
##Earl and Countess of Glasgow
##Countess of Gosford
##Earl and Countess of Granard
##Countess of Granville
##Earl and Countess of Grey
##Countess of Grosvenor
##Countess of Guilford
##Earl and Countess of Harewood and Dowager Countess of Harewood
##Earl and Countess of Harrington
##Earl and Countess of Hopetoun
##Earl and Countess of Huntingdon
##Earl and Countess of Harrowby
##Countess of Hohenau
##Countess of Howe
##Earl and Countess of Iddesleigh
##Earl and Countess of Jersey
##Earl and Countess of Kenmare
##Earl of Kerry
##Earl and Countess of Kilmorey
##Earl of Kimberley
##Earl and Countess of Kingston
##Earl of Kinnoull
##Josephine, Countess Kinsky
##Earl and Countess of Kintore
##Countess of Leitrim
##Earl and Countess of Lanesborough
##Countess of Lathom
##Earl and Countess of Lauderdale
##Countess of Leicester
##Earl and Countess of Leven and Melville
##Earl and Countess of Lichfield
##Earl and Countess of Limerick
##Earl and Countess of Lindsay
##Earl and Countess of Lisburne
##Earl and Countess of Listowel
##Earl and Countess of Londesborough
##Earl and Countess of Longford
##Earl and Countess of Lonsdale and Dowager Countess of Lonsdale
##Earl and Countess of Loudoun
##Earl and Countess of Lovelace
##Earl and Countess of Lucan
##Countess of Lytton
##Countess of Macclesfield
##Earl and Countess of Malmesbury and Dowager Countess of Malmesbury
##Earl and Countess of Mar
##Earl and Countess of Mar and Kellie and Dowager Countess of Mar and Kellie
##Earl and Countess of Mayo and Dowager Countess of Mayo
##Countess of Meath
##Countess of Metaxas
##Earl and Countess of Mexborough
##Earl and Countess of Minto
##Earl of De Montalt
##Earl and Countess of Morley
##Earl and Countess of Morton and Dowager Countess of Morton
##Earl of Nelson
##Earl and Countess of Norbury
##Earl of Northbrook
##Earl and Countess of Northesk and Dowager Countess of Northesk
##Earl and Countess of Onslow
##Earl of Orford
##Countess of Oxford
##Earl and Countess of Pembroke
##Countess of Percy
##Earl and Countess of Portarlington
##Earl and Countess of Portsmouth
##Earl and Countess of Powis
##Earl and Countess of Radnor
##Earl and Countess of Ravensworth
##Earl and Countess of Roden
##Earl and Countess of Romney
##Lawrence, [[Social Victorians/People/Zetland|Earl of Ronaldshay]]
##Earl of Rosebery
##Earl and Countess of Rosse
##Earl and Countess of Rosslyn and Dowager Countess of Rosslyn
##Earl of Sandwich
##Earl of Scarbrough
##Earl and Countess of Selborne
##Countess of Selkirk
##Countess of Shaftesbury
##Dowager Countess of Shrewsbury and Talbot
##Earl and Countess of Spencer
##Earl and Countess of Stamford
##Earl and Countess of Stanhope
##Earl and Countess of St. Germans
##Earl of Stradbroke
##Earl of Strafford
##Earl and Countess of Suffolk and Berkshire
##Earl and Countess of Temple (of Stowe)
##Earl and Countess of Verulam
##Earl and Countess of Waldegrave
##Earl and Countess of Warwick
##Earl and Countess of Westmeath
##Earl and Countess of Wharncliffe
##Elizabeth, Dowager Countess of Wilton and Isabella, Dowager Countess of Wilton
##Earl and Countess of Winchilsea and Nottingham
##Earl and Countess of Winterton
##Earl and Countess of Yarborough and Dowager Countess of Yarborough
#Viscounts&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 3c / Col. 4a) and Viscountesses
##Viscount and Viscountess of Boyne
##Viscountess of Cantelupe
##Viscount and Viscountess of Castlerosse
##Viscount and Viscountess of Chelsea
##Viscount and Viscountess of Chetwynd
##Viscountess of Chewton
##Viscount and Viscountess of Clifden
##Viscount and Viscountess of Cobham
##Viscount and Viscountess of Coke
##Viscount of Corry
##Viscount and Viscountess of Cranborne
##Viscount of Crichton
##Viscount and Viscountess of Cross
##Viscount of Curzon
##Viscount and Viscountess of Dalrymple
##Viscount and Viscountess of Deerhurst
##Viscount and Viscountess of De Vesci
##Viscount and Viscountess of Dillon
##Viscount of Doneraile
##Viscount and Viscountess of Duncannon
##Viscount of Dungarvan
##Viscount and Viscountess of Ebrington
##Viscount and Viscountess of Emlyn
##Viscount of Encombe
##Viscount and Viscountess of Exmouth
##Viscount and Viscountess of Falkland
##Viscount and Viscountess of Falmouth
##Viscount of Fitz Harris
##Viscount and Viscountess of Folkestone
##Viscount and Viscountess of Frankfort de Montmorency
##Viscount and Viscountess of Gage
##Viscount and Viscountess of Galway
##Viscount and Viscountess of Garnock
##Viscount and Viscountess of Gough
##Viscount of Gort
##Viscount and Viscountess of Halifax
##Viscount and Viscountess of Hardinge
##Viscount of Harrington
##Viscount and Viscountess of Hood
##Viscount and Viscountess of Kilcoursie
##Viscount and Viscountess of Knutsford
##Viscount and Viscountess of Lifford
##Viscount of Llandaff
##Viscount and Viscountess of Maitland
##Viscount and Viscountess of Marsham
##Viscount and Viscountess of Massereene and Ferrard
##Viscount and Viscountess of Melville
##Viscount and Viscountess of Midleton
##Viscount and Viscountess of Milton
##Viscount and Viscountess of Monck
##Viscount and Viscountess of Morpeth
##Dowager Viscountess of Mountmorres
##Viscount and Viscountess of Newark
##Viscount and Viscountess of Newport
##Viscount and Viscountess of Oxenbridge
##Viscount of Parker
##Viscount of Peel
##Viscount and Viscountess of Portman
##Viscount and Viscountess of Powerscourt
##Viscount and Viscountess of Raincliffe
##Viscountess of Sherbrooke
##Viscount of Sidmouth
##Viscount of St. Cyres
##Viscount of Southwell
##Viscount of Suirdale
##Viscount and Viscountess of Templetown
##Viscountess of Torrington
##Viscount and Viscountess of Trafalgar
##Viscount and Viscountess of Valentia
##Viscount of Valletort
##Viscount of Villiers
##Viscountess of Wolseley
#Bishops — Auckland, Barry, Bath and Wells, British Colombia, Chichester, Durham, Ely, Exeter, Gloucester and Bristol, Gibraltar, Hereford, London, Lichfield, Lincoln, Manchester, Newcastle, Norwich, Oxford, Peterborough, Rochester, Ripon, Stepney, Southwark, St. Albans, Salisbury, Sodor and Man, Southwell, Sydney, Sierra Leone, Worcester, Winchester, Wellington
#Baronesses — Burdett-Coutts, Macdonald
#Lords and Ladies&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 4b / Col. 5a) — 
##Lord and Lady Abercromby
##Lord and Lady Aberdare
##Lord Aberdour
##Lady Abinger
##Lady Alexandra Acheson
##Lady Adam
##Lady Adderley
##Lord and Lady Addington
##Lady Adye
##Lady Agnew
##Lady Alderson
##Lord and Lady Alington
##Lady Alison
##Lady Mildred Allsopp
##Lord and Lady Amherst of Hackney
##Lady Heathcoat Amory
##Lord and Lady Ampthill
##Lady Agnes Anderson
##Lady Bertha Anson
##Lady Arbuthnot
##Lady Alice Archer Houblon
##Lord Ardee
##Lord and Lady Ardilaun
##Lady Armstrong
##Lady Arnold
##Lady Arnott
##Lord and Lady Ashbourne
##Lord and Lady Ashburton and Dowager Ashburton
##Lord and Lady Ashcombe
##Lady Alice Ashley
##Lady Edith Ashley
##Lady Ashmead-Bartlett
##Lord and Lady Ashton
##Lord and Lady Ashtown
##Lady Florence Astley
##Lady Gertrude Astley-Corbett
##Lady Austin
##Lord Bagot
##Lady Bailey
##Lady Blanche Baillie
##Lady Baird
##Lady Baker
##Lord Balcarres
##Lord and Lady Balfour of Burleigh, Lady Nina Balfour and Lady Betty Balfour
##Lord Balvaird
##Lord Bangor
##Dowager Lady Barclay
##Lord and Lady Barnard
##Lady Florence Barnardiston
##Lady Constance Barne
##Lady Barran
##Lady Barrington
##Lord and Lady Basing
##Lord and Lady Bateman
##Lady Evelyn Bathurst
##Lord and Lady Battersea
##Lady Steuart Bayley
##Lady Violet Beauchamp
##Lord Osborne Beauclerk and Lady Beauclerk (2)
##Lady A. Beaumont
##Lady Bedford
##Lord and Lady Belhaven and Stenton and Dowager Belhaven and Stenton
##Lord and Lady Bellew and Dowager Bellew
##Lord and Lady Belper
##Lady Charles Beresford
##Lady William Beresford (Lilian Duchess of Marlborough)
##Lady Bergne
##Lord and Lady Bertie and Lady Elizabeth Bertie
##Lady Biddulph, Lady Elizabeth Biddulph and Lady Wilfreda Biddulph
##Lady Bigge
##Lord and Lady Bingham
##Lord and Lady Binning
##Lord Blackwood, Lord Basil Blackwood. Lady Hermione Blackwood and Lord Terence Blackwood
##Lady Bloomfield
##Lady Blythswood
##Lord and Lady Bolton
##Lady Maud Bootle-Wilbraham, Lady Bertha Bootle-Wilbraham and Lady Edith Bootle-Wilbraham
##Lord Borthwick
##Lady Margaret Boscawen
##Lord and Lady Boston
##Lady Boughey
##Lady Albreda Bourke and Lady Florence Bourke
##Lady Bowen
##Lady Bower
##Lady Muriel Boyle and Lady Boyle (2)
##Lady Mary Brabazon
##Lady Brackenbury
##Lady Braddon
##Lady Bramwell
##Lady Bramston
##Lord Brassey, Lady Idina Brassey and Lady Violet Brassey
##Lord and Lady Braye
##Lady Mary Bridgeman
##Lady Eleanor Brodie
##Lady Hilda Brodrick
##Lady De Capel Brooke and Dowager Brooke
##Lady Cunliffe Brooks
##Lord and Lady Brougham and Vaux
##Lord and Lady Ulick Browne, Lady Browne and Lady Crichton Browne
##Lady Brownlow
##Lord and Lady F. Brudenell-Bruce
##Lady Brunner
##Dowager Buchanan-Riddeil
##Lady Audrey Buller
##Lady Burdett
##Lord and Lady Burghclere
##Lord Burghley
##Lady Agnes Burne
##Lady Burrell
##Lord and Lady Burton
##Lady Butler and Lady Butler (2)
##Lord and Lady Arthur Butter
##Lady Buxton and Lady Victoria Buxton
##Lady Susan Byng
##Lord and Calthorpe
##Lady C. Cameron and Lady Margaret Cameron
##Lord and Lady Archibald Campbell and Lady A. Campbell
##Lord and Lady George Campbell
##Lady Campbell-Bannerman
##Lord and Lady Camoys
##Lord and Lady Carbery and Dowager Carbery
##Lady Carbutt
##Lady Cardon
##Lord and Lady Cardross
##Lord and Lady Carew
##Lady Carmichael
##Lord and Lady Carnegie
##Lord and Lady Castlemaine
##Lord and Lady Castletown
##Lady Eva Cathcart and Lady R. Cathcart
##Lady Frederick Cavendish, Lady Myra Cavendish, Lady Evelyn Cavendish and Lady Harriet Cavendish
##Lord Charles Cavendish-Bentinck, Lord and Lady Henry Cavendish-Bentinck, Lord William Cavendish-Bentinck, Lady Ottoline Cavendish-Bentinck
##Lord and Eustace Cecil, Lord Hugh Cecil, Lord and John Cecil, Lord and Edward Cecil, Lord and Lady Robert Cecil, Lord W. Cecil, Lady Gwendolen Cecil, Lady Florence Cecil, Lady William Cecil, Lady Louisa Cecil
##Lady Francis Cecil-Dallas
##Lady Chamberlain
##Lady Chelmsford
##Lord and Lady Chesham
##Lady Chetwode
##Lord Cheylesmore
##Lord and Lady Fitzwarine Chichester
##Lady Chitty
##Lady Cholmeley
##Lady Henry Cholmondeley
##Lady Clements (2)
##Lady Churchill, Lady Randolph Churchill, Dowager Churchill, Lady Spencer Churchill (2)
##Lord Edward Spencer-Churchill, Lady Alfred Spencer-Churchill
##Lord and Lady Churston
##Lord and Lady Clifford of Chudleigh
##Lady Marshal Clarke, Lady E. Clarke
##Lady Isabel Clayton
##Lord and Lady Clinton
##Lord and Lady Clonbrock
##Lord Cloncurry
##Lady Muriel Close
##Lady Evelyn Cobbold
##Lady Cochrane, Lady Gertrude Cochrane, Lady Adela Cochrane
##Lady Coddington
##Lady Mabel Coke
##Lord and Lady Colchester
##Lady Cole (2)
##Lady Colebrooke
##Lord and Lady Coleridge
##Lady Collins
##Lady Colomb
##Lady Colvile, Lady Colville
##Lord and Lady Colville of Culross
##Lady Jane Seymour Combe, Lady Constance Combe
##Lady Commerell
##Lord and Lady Alwyne Compton
##Lady Dowager Congleton
##Lord and Lady Connemara
##Lady Conyers
##Lady Blanche Conyngham
##Lady Cooper
##Lady Evelyn Cotterell
##Lord and Lady Cottesloe
##Lady Couch
##Lord and Lady Courtenay
##Lady Coventry (2)
##Lady Cowell
##Lady Helen Craven
##Lord and Lady Crawshaw
##Lady Evelyn Crichton, Lady Emma Crichton
##Lord Crofton
##Lady Cromer
##Lady Mary Crosse
##Lady Crossley
##Lady Mary Cuffe
##Lady Culme-Seymour
##Lady Cunliffe
##Lady Georgiana Curzon
##Lady Elizabeth Cust
##Lady Ida Dalzell
##Lady Mary Dashwood
##Lord and Lady Davey
##Lady Victoria Dawnay, Lady Evelyn Dawnay, Lady Adelaide Dawnay
##Lady Decies
##Lord and Lady De Freyne
##Lord and Lady De L’Isle and Dudley
##Lord De Manley
##Lady Mildred Denison, Lady Elinor Denison
##Lord Deramore
##Lord and Lady De Ramsey
##Lady Dering
##Lady De Ross
##Lord and Lady De Saumarez
##Lady Des Voeux
##Lady De Trafford, Lady Agnes De Trafford
##Lady De Winton
##Lord and Lady Digby
##Lady Dorchester
##Lady Dorington
##Lady Margaret Douglas, Lady Edith Douglas
##Lady H. Douglas-Hamilton
##Lady Dowell
##Lady Drummond, Lady Edith Drummond
##Lady Du Cane
##Lady Duckworth
##Lady Eva Dugdale
##Lord Dunally
##Lady Florence Duncombe, Lady Ulrica Duncombe, Lady Caroline Duncombe
##Lady Alice Dundas
##Lord and Lady Dunleath
##Lord Dunglass
##Lady Dunn
##Lord Dunsandle and Clanconal
##Lady Durand
##Lord Dynevor
##Lord Ebury
##Lady Edmonstone
##Lady Edwards, Lady J. B. Edwards, Lady Blanche Edwards
##Lady Ernestine Edgcumbe
##Lady Egerton (2)
##Lord Egerton of Tatton
##Lady Grey-Egerton
##Lord and Lady Elcho
##Lord and Lady Elibank
##Lady Ellenborough
##Lady Ellis
##Lord and Lady Elphinstone
##Lady Winifred Cary-Elwes
##Lady Engleheart
##Lord Erskine, Lady Erskine (2), Lady Horatia Erskine, Lady Erskine
##Lord and Lady Esher
##Lady Evans
##Lady Evelyn Ewart, Lady Mary Ewart
##Lady Evelyn Eyre
##Lady Fairbairn
##Lady Fairfax
##Lady Anne Fane, Lady Augusta Fane
##Lady Farquhar
##Lord and Lady Farrer
##Lady Fayrer
##Lady Louisa Feilding
##Lady Helen Munro Ferguson
##Lady Fergusson
##Lady Ffolkes
##Lady Finlay
##Lady Fisher
##Lady Dorothea Fitz-Clarence, Lady Maria Fitz-Clarence, Lady Dorothy Fitzclarence
##Lord and Lady Henry Fitz-Gerald, Lady B. Fitz Gerald, Lady M. FitzGerald, Lord Seymour Fitz-Gerald
##Lady Beatrix Fitzmaurice
##Lord and Lady F. FitzRoy, Lady C. Fitz-Roy
##Lady Mary Fitzwilliam
##Lady FitzWygram
##Lady Fletcher
##Lady Flower, Lady Flower
##Lord Foley, Lady Mary Foley
##Lady Gertrude Foljambe
##Lady Angela Forbes, Lady Forbes (2), Dowager Helen Forbes
##Lord and Lady Forester
##Lady Forrest
##Lady Susan Fortescue
##Lady Forwood
##Lady Foster
##Lady Fowler
##Lady Edith Franklin
##Lady Fremantle, Lady Fremantle
##Lady Frere
##Lady Fulton
##Lady Gardiner, Lady Lynedoch Gardiner
##Lord Garioch
##Lady Galton
##Lady Katharine Gathorne-Hardy
##Lady Garvagh
##Lord and Lady Gerard
##Lady Gilbey
##Lady Gillford
##Lady Susan Gilmour
##Lady Gipps
##Lord and Lady Glamis
##Lord and Lady Glenesk
##Lady Glyn, Lady Mary Carr Glyn
##Lady D'Arcy Godolphin-Osborne
##Lady Gordon
##Lady Margaret Ormsby Gore, Lady Constance Gore
##Lady Gore Langton (2)
##Lord Walter Gordon-Lennox, Lord Algernon Gordon-Lennox
##Lady Evelyn Goschen
##Lord R. S. Gower
##Lady Graham, Lady Margaret Graham, Lady Helen Graham
##Lady Charlotte Graham-Toler
##Lady Grant, Lady Florence Grant
##Lady Grant-Duff
##Lady Green
##Lord Greenock
##Lady Grenfell
##Lady Frances Gresley
##Lady Victoria Grey, Lady Grey
##Lady Jane Grey-Trefusis
##Lady Griffin
##Lady Helen Grimston
##Lord and Lady Arthur Grosvenor, Lady Grosvenor (2)
##Lady Gull
##Lady Haldon
##Lady Haliburton
##Lady Basil Hall
##Lady Halle
##Lord and Lady Halsbury
##Lord and Lady E. Hamilton, Lord F. Hamilton, Lady F. Douglas Hamilton, Lady Alexandra Hamilton, Lady Baillie Hamilton (2), Lady C. Hamilton, Lady Victoria Hamilton, Lady George Hamilton
##Lady Hanson
##Lady Harcourt
##Lady Cicely Hardy, Lady Hardy
##Lady Beatrice Hare
##Lord Harlech
##Lady Constance Harris, Lady Harris
##Lady Harrison, Lady Harriet Harrison
##Lady Hart
##Lady Emily Hart-Dyke
##Lady Dixon-Hartland
##Lady Hartopp
##Lord and Lady Hastings
##Lord and Lady Hatherton
##Lady Alice Havelock-Allan
##Lady Hawke
##Lord and Lady Hawkesbury
##Lady John Hay, Lady Hay
##Lady Blanche Haygarth
##Lady Hayter
##Lady Hely-Hutchinson (2)
##Lady Hemming
##Lord and Lady Heneage
##Lord and Lady Henley
##Lord Henniker
##Lady Beatrix Herbert, Lady Herbert (2)
##Lord and Lady Herries
##Lord and Lady Herschell
##Lord Francis Hervey, Lady Augustus Hervey
##Lady Hervey-Bathurst
##Lady Fermor Hesketh
##Lady Hibbert
##Lady Lucy Hicks-Beach
##Lord and Lady Arthur Hill, Lady Clement Hill, Lady Stock Hill
##Lord and Lady Hillingdon
##Lord and Lady Hindlip
##Lord and Lady Hobhouse
##Lady Norah Hodgson
##Lady Holdich
##Lady Mary Holland
##Lady Beatrix Douglas Home
##Lady Maria Hood
##Lady Hood of Avalon
##Lady Hooker
##Lady Mary Hope
##Lady Hoskins
##Lord and Lady Hotham
##Lord and Lady Hothfield
##Lady Houldsworth
##Lady Eleanor Howard, Lady Agnes Howard, Lady Howard (2), Lady Mabel Howard, Lady Rachel Howard
##Lord and Lady Howard of Glossop
##Lady Howarth
##Lady Mary Hozier
##Lady Florentia Hughes
##Lady Seager Hunt
##Lady Hunter
##Lord Hyde
##Lady Hylton
##Lord and Lady Inchiquin
##Lord Inverurie
##Lord and Lady Iveagh
##Lady Jackson
##Lord James of Hereford
##Lady Margaret Jenkins, Lady Jenkins
##Lady Jenner
##Lady Jephson
##Dowager Jessel, Lady Jessell
##Lady Jeune
##Lady Hill Johnes
##Lady Joicey
##Lady Alice Jolliffe
##Lady Burn Jones
##Lady Caroline Lister Kaye, Lady Beatrice Lister Kaye, Lady Lister Kaye
##Lady Isabella Keane
##Lady Keith-Falconer (2)
##Lord and Lady Kelvin
##Lady Kemball
##Lady Beatrice Kemp
##Lady Kennard
##Lady Kennaway
##Lady Aline Kennedy
##Lady Kennett-Barrington
##Lord Kenyon
##Lady Mabel Kenyon-Slaney
##Lord Kensington
##Lady Mary Stuart Keppel
##Lady Innes-Ker (2)
##Lady Kerr (2)
##Lord Kilmarnock
##Lady King
##Lady Florence King King
##Lady Emily Kingscote
##Lady Edith King-Tenison
##Lord and Lady Kinnaird
##Lady Kitson
##Lady Laking
##Lady Frances Lambart, Lady Ellen Lambart
##Lady Victoria Lambton
##Lady Adela Larking
##Lady Isabel Larnach
##Lady Mary Lascelles
##Lord and Lady Lawrence
##Lady Lawson
##Lord and Lady Leconfield
##Lady Elliott Lees, Lady Lees
##Lady Leese
##Lady Legard
##Lord and Lady Leigh
##Lady Henry Gordon-Lennox, Lady Walter Gordon-Lennox, Lady Algernon Gordon-Lennox, Lady Caroline Gordon-Lennox
##Lady Katharine Le Poer Trench
##Lady Constance Leslie
##Lady Susan Leslie-Melville
##Lady Lewis
##Lady Lilian Liddell
##Lady Lindley
##Lady Harriet Lindsay, Lady Jane Lindsay, Lady Jane Lindsay
##Lord and Lady Lingen
##Lord and Lady Lister
##Lady Gwendolen Little
##Lady Margaret Littleton
##Lord and Lady Llangattock
##Lady Llewelyn
##Lord and Lady Loch
##Lady Lockwood
##Lady Louise Loder
##Lady Catherine Loftus
##Lady Doreen Long
##Lady Longley
##Lady Albertha Lopes
##Lady Loraine
##Lord and Lady Lovat
##Lady Drury Lowe, Lady Lucy Drury Lowe
##Lady Lowry-Corry (2)
##Lady Mary Loyd
##Lady Lubbock
##Lord and Lady Lurgan and Dowager Lurgan
##Lady Lyall
##Lady Lyell
##Lady Mary Lygon
##Lady Lyons
##Lady Lysons
##Lady Lyttelton
##Lady Emily Lytton
##Lady MacCormac
##Lord and Lady Macdonald
##Lady Macgregor, Lady MacGregor, Lady Helen MacGregor
##Lady Mackenzie, Lady Mackenzie
##Lady Mackworth
##Lady Maclean
##Lord and Lady Macnaghten
##Lady Macpherson-Grant
##Lady Caroline Madden, Lady Madden
##Lady Louisa Magenis
##Lady Magheramorne, Dowager Magheramorne
##Lady Nora Maitland
##Lady Margaret Crichton-Maitland
##Lady Margaret Majendie
##Lord Cecil Manners, Lord Edward Manners, Lord Manners, Lady Victoria Manners, Lady Manners
##Lady Blundell Maple
##Lady Mappin
##Lady Marjoribanks
##Lady Markham
##Lady Marriott
##Lady Martin, Lady Martin
##Lady Evelyn Mason
##Lady Maude (2)
##Lady H. Maxwell, Lady Maxwell, Lady Maxwell, Lady Maxwell
##Lady Heron-Maxwell
##Lady M'Clintock
##Lady Evelyn M'Donnell
##Lady Meade (2)
##Lord and Lady Medway
##Lady Methuen
##Lady Meysey-Thompson
##Lord and Lady Middleton, Lady Middleton
##Lady Mary Milbanke
##Lady Miller
##Lady Milner
##Lady Clementina Mitford
##Lady Lady M'lver
##Lady Hilda M'Neile
##Lady Monckton
##Lord Moncreiff, Lady Scott Moncrieff
##Lady Moncreiffe
##Lord and Lady Monkswell
##Lady Monson
##Lord Charles Montagu, Lady Cecil Scott Montagu, Lady S. Montagu, Lady Agneta Montagu
##Lord Montagu of Beaulieu
##Lord and Lady Monteagle
##Lady Edith Montgomerie, Lady Sophia Montgomerie
##Lady Charlotte Montgomery
##Lady More-Molyneux
##Lord and Lady Moreton
##Lady Morgan
##Lord and Lady Morris
##Lady Blanche Morris
##Lady Mary Morrison
##Lady Moseley
##Lord and Lady Mostyn
##Lord and Lady Mowbray and Stourton, Dowager Mowbray and Stourton, Lady Mowbray
##Lord and Lady Muncaster
##Lady Anne Murray
##Lady Murray (2)
##Lady Georgiana Mure, Lady Georgiana Mure [sic]
##Lord and Lady Napier and Ettrick
##Lord and Lady Napier of Magdala and Dowager Napier of Magdala
##Lady Naylor-Leyland
##Lady Nelson
##Lord and Lady Henry Nevill
##Lord and Lady Newton
##Lord and Lady Newtown-Butler
##Lady Nicolson
##Lady Augusta Noel, Lady Agnes Noel
##Lady Norman
##Lord and Lady Norreys
##Lord and Lady North, Lady Muriel North
##Lady Northcote, Lady Northcote (2)
##Lord Norton
##Lady Elizabeth Nugent
##Lady O'Brien, Lady O'Brien [sic]
##Lady O'Hagan
##Lady Olpherts
##Lord and Lady O'Neill
##Lady Gwendoline O'Shee
##Princep [sic] Alice Packe
##Lord and Lady Berkeley Paget
##Lady Alfred Paget
##Lady Paget of Cranmore
##Lady Katherine Pakenham
##Lady Palgrave
##Lady Sophia Palmer, Lady Palmer
##Lady Evelyn Parker
##Lady Parratt
##Lady Maude Parry
##Lady Muriel Parsons
##Lord and Lady Pearson, Lady Pearson
##Lady Peel, Lady Georgiana Peel
##Lady Constance Childe-Pemberton
##Lord and Lady Penrhyn
##Lady Mary Pepys
##Lady Perceval
##Lady Percy (2)
##Lady Petre
##Dowager Lady Peyton
##Lady Phillimore
##Lady William Phipps
##Lord and Lady Pirbright
##Lord and Lady Playfair
##Lady Chichele Plowden
##Lady Anna Chandos-Pole
##Lady Pollock
##Lord and Lady Poltimore
##Lady Pontifex
##Lady Alice Portal
##Lady Powell, Lady Powell [sic]
##Lady Baden-Powell
##Lady Dickson-Poynder
##Lady Poynter
##Lord and Lady George Pratt
##Lady Priestley
##Lady Probyn
##Lady Eva Wyndham-Quin, Lady Wyndham-Quin (2)
##Lord and Lady Raglan, Dowager Raglan
##Lady Ramsay
##Lord and Lady Rathdonnell
##Lady Rathmore
##Lord and Lady Rayleigh, Dowager Rayleigh
##Lord and Lady Reay
##Lady Reid
##Lord and Lady Rendel
##Lord Rendlesham
##Lady Jane Repton
##Lord Revelstoke
##Lord and Lady Ribblesdale
##Lady Laura Ridding
##Lord and Lady Robartes
##Lady O. Roberts
##Lady Roberts of Kandahar
##Lady Robinson
##Lord and Lady Rodney
##Lord Romilly
##Lord and Lady Rookwood
##Lord and Lady Rossmore
##Lord Rowton
##Lady Roxburgh
##Lord and Lady Rothschild
##Lady Victoria Russell, Lady Arthur Russell, Lady G. Russell, Lady W. H. Russell, Lady Alexander Russell
##Lord and Lady Russell of Killowen
##Lord and Lady Ruthven
##Lady Jane Ryan
##Lady Mary Sackville
##Lady Salmon
##Lord and Lady Saltoun
##Lady Samuelson, Lady S. Samuel
##Lady Mary Saurin
##Lord and Lady Savile, Lady Marie Savile
##Lady Savory
##Lord George Scott, Lord Henry Scott, Lord Herbert Scott, Lady Sophie Scott, Lady Charles Scott, Lady Louisa Scott, Lady Scott (2)
##Lord and Lady Seaton
##Lord and Lady Settrington
##Lady Seymour, Lady Albert Seymour, Lady William Seymour, Lady Seymour (2)
##Lord and Lady Shand
##Lady Shaw
##Lady Constance Shaw-Lefevre
##Lady Octavia Shaw-Stewart, Lady Alice Shaw-Stewart
##Lady Mary Shelley
##Lord and Lady Sherborne
##Lady Shippard
##Lady Shute
##Lady Kay-Shuttleworth
##Lady Simeon
##Lady Simmons
##Lady Simpson of Windsor
##Lord and Lady Sinclair
##Lord and Lady Skelmersdale
##Lady Esther Smith, Lady Barbara Smith, Lady Smith, Lady Blanche Smith, Lady Sybil Smith, Lady Euan Smith, Lady D. Smith
##Lady Smyth
##Lady Catherine Somerset, Lady Geraldine Somerset, Lady Henry Somerset
##Lord and Lady Southampton, Dowager Southampton
##Lady Edward Spencer-Churchill
##Lady Margaret Spicer
##Lady Sprigg
##Lady Stafford
##Lord Stalbridge
##Lady Stanhope (2)
##Lord Stanmore
##Lord Stanley, Lady Alice Stanley, Lady Isobel Stanley
##Lady Stansfield
##Lord Stavordale
##Lady Stephenson
##Lady Stevenson
##Lady Helen Stewart, Lady Mary Stewart, Lady Mark Stewart, Lady Stewart, Lady Houston Stewart, Lady Stewart [sic], Lady Isabel Stewart
##Lady Stewart of Grantully
##Lady Edith St. Aubyn
##Lord and Lady St. Levan
##Lady St. Leonards
##Lord and Lady St. Oswald
##Lady Stone
##Lady Charlotte Stopford
##Lord and Lady Stratheden and Campbell
##Lady Mary Stuart-Richardson
##Lord Suffield
##Lady Sutherland
##Lady Evelyn Sutton, Lady Susan Sutton
##Lord and Lady Swansea
##Lady Swinnerton Dyer
##Lady Kathleen Swinnerton-Pilkington
##Lord and Lady E. Talbot, Lady Emma Talbot
##Lady Jane Taylor
##Lady Taylour (2)
##Lady Tatton Sykes
##Lord Herbert Vane-Tempest, Lord Henry Vane-Tempest
##Lord and Lady Templemore
##Lady Tennant
##Lord and Lady Tennyson
##Lady Tenterden
##Lord Tewkesbury
##Lord and Lady Teynham
##Lord and Lady Thring
##Lady E. Thornton
##Lady Thursby
##Lady Ulrica Thynne
##Lord and Lady Tollemache
##Lady Agnes Townshend
##Lady Mary Trefusis
##Lady Tredegar
##Lady Trevelyan, Lady Trevelyan [sic]
##Lord and Lady Trevor
##Lady Troubridge
##Lady Turner
##Lady Henrietta Turnor
##Lady Tuson
##Lord and Lady Tweedmouth
##Lady Tyler
##Lady Emily Van De Weyer
##Lady Jane Van Koughnet
##Lord and Lady Ventry
##Lady Villiers (2), Lady Edith Villiers
##Lady Howard Vincent, Lady Helen Vincent, Lady Vincent
##Lady Vivian, Lady Jane Vivian
##Lady Mary Waldegrave
##Lady F. F. Walker, Lady James Walker
##Lady Walrond
##Lady Clementine Walsh
##Lord Wandsworth
##Lady Wantage
##Lord Warksworth
##Lady Leucha Warner
##Lady Warrender
##Lord and Lady Watson
##Lady Cecilia Webb
##Lady Rose Weigall
##Lord Welby
##Lady Willes
##Lady Willis
##Lady Arthur Wellesley
##Lord and Lady Wenlock
##Lord and Lady Westbury and Dowager Westbury
##Lady Isabella Whitbread
##Lady White
##Lady Whitehead
##Lady Whiteway
##Lady Elizabeth Williamson
##Lady Williams-Wynn
##Lady Willoughby (2)
##Lord Willoughby de Broke
##Lord Willoughby de Eresby
##Lady Willshire
##Lady Wilson, Lady Sarah Gordon Wilson
##Lord and Lady Wimborne
##Lady Windeyer
##Lord and Lady Windsor
##Lady Winnington
##Lady Constance Wodehouse
##Lord and Lady Wolverton
##Lady Julia Wombwell
##Lady Wood, Lady Mary Wood
##Lady Woods
##Lord Wrottesley
##Lady Hugh Wyndham
##Lady Barbara Yeatman
##Lady Lilian Yorke
##Lord Zouche
#Right Honourables
##H. H. Asquith
##E. Ashley
##A. H. Dyke Acland
##J. Atkinson
##J. B. Balfour
##Sir G. Bowen
##G. W. Balfour
##Sir Hicks-Beach
##A. J. Balfour
##James Bryce
##Sir H. Campbell-Bannerman
##A. H. Smith-Barry
##E. Carson
##H. Chaplin
##Sir J. Chitty
##Jesse Collings
##Sir R. Couch
##G. N. Curzon
##J. Chamberlain
##L. Courtney
##Sir M. Grant-Duff
##A. Akers-Douglas
##Sir W. Hart Dyke
##Sir H. Elliot
##F. Foljambe
##Sir H. Fowler
##Sir A. B. Forwood
##Sir J. Fergusson
##Herbert Gladstone
##Sir J. Gorst
##G. J. Goschen
##W. E. Gladstone
##Sir G. Grey
##C. H. Hemphill
##Charles Seale-Hayne
##R. W. Hanbury
##Lord George Hamilton
##Staveley Hill
##Sir J. T. Hibbert
##Sir W. Harcourt
##lon Hamilton
##Sir Arthur Hayter
##Sir F. Jeune
##W. L. Jackson
##Sir John Kennaway
##G. Shaw-Lefevre
##W. Lidderdale
##Sir Massey Lopes
##James Lowther
##Sir J. Lubbock
##Sir H. Lopes
##Walter Long
##Sir N. Lindley
##J. W. Mellor
##Sir G. O. Morgan
##John Morley
##Arnold Morley
##Sir J. Mowbray
##A. J. Mundella
##J. H. Macdonald
##F. Max Müller
##Sir W. Marriott
##Graham Murray (the Lord Advocate)
##Sir E. Monson
##Sir P. O'Brien
##Sir A. Otway
##Sir F. Peel
##Sir R. Paget of Cranmore
##W. J. Pirrie
##J. P. Robertson
##Sir. J. Rigby
##C. T. Ritchie
##Sir S. H. Strong
##Sir B. Saunderson
##Sir J. Stansfeld
##Sir A. Smith
##C. R. Spencer
##Sir C. Kay-Shuttleworth
##Sir R. Temple
##Sir R. Thompson
##Sir E. Thornton
##Lord Henry Thynne
##Sir G. O. Trevelyan
##C. P. Villiers
##Sir Algernon West
##Sir C. L. Wyke
##C. B. Stuart-Wortley
##S. J. Way
#Honourables&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5a / Col. 5b) and Honourable Ladies&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5b / Col. 5c)
##Mrs. Acland
##Mrs. Alexander
##H. Allsopp, Mrs. Allsopp, George Allsopp
##Mrs. Anstruther
##Mrs. Armytage
##[Hon. Lady] Vere Annesley
##Mrs. Bagot, Mrs. Bagot [sic 2x]
##Mrs. Baillie of Dochfour
##Mrs. Balfour
##[Hon.] Coplestone and [Hon.] Mrs. Bampfylde
##John Baring, Susan Baring, Lilian Baring
##Mrs. Barker
##Mrs. Barlow
##Eric Barrington, Mrs. Barrington
##Mrs. Hamar Bass
##Misses Bateman-Hanbury (2)
##Allen B. Bathurst
##Mrs. Benyon
##[Hon. Lady] Beresford
##[Hon.] R. Chetwynd
##Arthur Chichester
##Lady Biddulph
##C. E. Bingham, Mrs. Bingham, Albert Bingham, Mrs. Bingham [sic x2]
##Lady Birkbeck
##Ivo Bligh, Mrs. Bligh
##Diana Sclater-Booth
##O. Borthwick
##J. Boscawen
##Henry Bourke, Mrs. H. Bourke, Charles Bourke, Terence Bourke, Mrs. T. Bourke, Algernon Bourke, Mrs. A. Bourke, Mrs. E. R. Bourke
##Charles Brand, Arthur Brand, Mrs. Brand, Mrs. T. Brand
##T. Brassey, Mrs. A. Brassey
##Mrs. Stapleton Bretherton
##Reginald Brett, Mrs. Brett
##Mrs. F. Bridgeman, Misses Bridgeman (2)
##Mrs. Britten
##W. St. John Brodrick, Albinia Brodrick
##Emmeline Brownlow
##Mrs. T. C. Bruce, Misses Bruce (2)
##Misses M'Clintock Bunbury (2)
##Mary Byng
##T. J. Byrnes
##Arthur Cadogan, Mrs. A. Cadogan, Mrs. C. Cadogan, Ethel Cadogan
##Mrs. Gough-Calthorpe, Rachel (Gough) Calthorpe, Misses Gough Calthorpe (2)
##Mrs. Candy
##G. H. Campbell, K. Campbell, Hugh Campbell, Mrs. H. Campbell, Mrs. Ronald Campbell, Misses Campbell (2), Mrs. J. B. Campbell, Mildred Campbell
##Mrs. Carington
##Mrs. Carpenter
##Emily Cathcart
##W. Cavendish, Mrs. W. Cavendish, Mrs. Cavendish
##Eleonora Chetwynd, Mrs. R. Chetwynd
##Mrs. A. Chichester, Hilda Chichester
##Mrs. Clowes
##T. H. Cochrane
##Audrey Coleridge
##George Colville
##Mrs. Corbett
##Mrs. H. Corry
##Caroline Courtenay
##Henry Coventry
##Osbert Craven
##Misses Cross
##Mrs. P. Crutchley
##Henry Cubitt, Mrs. Cubitt
##Hamilton Cuffe, Mrs. Otway Cuffe
##Lady Cunningham
##Montagu Curzon, Darea Curzon, Mrs. Curzon
##Hew Dalrymple
##John Dawnay, Eustace Dawnay, W. Dawnay, Mrs. Dawnay (2)
##Misses de Montmorency (2)
##Mrs. H. Dennison
##R. C. Devereux, Mrs. R. C. Devereux
##Mrs. Digby
##Conrad Dillon, Mrs. C. Dillon, Edith Dillon
##Misses Douglas-Pennant (2)
##A. Hay Drummond, Mrs. Hay Drummond, Frances Drummond, Mrs. M. Drummond
##Hubert V. Duncombe, Cecil Duncombe, Mrs. C. Duncombe
##C. T. Dundas, Mrs. C. T. Dundas, W. Dundas, Mrs. W. Dundas, Mrs. John Dundas
##Lady Du Cane
##Herbert Eaton, Mrs. H. Eaton
##F. Egerton, Mrs. A. F. Egerton, Lady Grey Egerton, Tatton Egerton, Mrs. T. Egerton
##Arthur Elliot, Mrs. Arthur Elliot, Lady Elliot, Mrs. Eliot
##Lilian Elphinstone
##Mrs. Ellis
##Muriel Erskine
##H. Escombe, Mrs. Escombe
##Mrs. Evans
##Mrs. C. Keith-Falconer
##Sir S. Ponsonby Fane
##Mrs. W. Farquhar
##Ailwyn Fellowes, Mrs. A. Fellowes
##Mrs. Ferguson of Pitfour
##Everard Fielding
##N. Fitzgerald, Mrs. N. Fitzgerald, Mrs. Fitzgerald, , Mrs. F. G. FitzGerald, Lady FitzGerald
##R. Fitzwilliam, W. H. Fitzwilliam
##Mary Forester
##Sir John Forrest
##Mrs. W. H. Forster
##Mrs. Lionel Fortescue
##Sir C. Fremantle, Mary Fremantle
##Sir Malcolm Fraser, Misses Fraser (2)
##Mrs. Charles Keith-Fraser
##Violet Gibson
##Evelyn Giffard
##Mrs. Henry Gladstone
##Lady Godley
##George Ormsby Gore
##F. Leveson-Gower
##Mrs. Gough
##Mrs. Alaric Grant
##Ronald Greville, Mrs. R. Greville, Louis Greville, Mrs. L. Greville, Sidney Greville, Mrs. A. Greville, Mrs. A. H. F. Greville
##Robert Grosvenor, Algernon Grosvenor, Mrs. A. Grosvenor, Maud Grosvenor, Elizabeth Grosvenor
##Lady Hamilton Gordon, [Hon. Lady] Nevil Gordon
##Misses Guest (2)
##Geoffrey Browne Guthrie
##Mrs. Gye
##Mrs. A. Haig
##Mrs. Halford
##, Misses Hamilton (2)
##Mrs. North Dalrymple-Hamilton
##Mrs. Hobart Hampden
##Mrs. Assheton Harbord, Mrs. C. Harbord, Judith Harbord, Bridget Harbord, Mrs. Harbord
##C. Hardinge, Mrs. C. Hardinge, A. Hardinge
##A. E. Gathorne-Hardy, Nina Gathorne-Hardy
##Misses Hawke (2)
##C. G. Hay
##Misses Heneage (2)
##Helen Henniker, Mrs. Henniker
##Robert Herbert, Sir Robert Herbert, Mrs. R. Herbert, Mrs. Herbert
##A. Holland Hibbert, Mrs. A. Holland Hibbert
##Lady Higginson
##Mrs. Hill
##Lionel Holland, Sydney Holland
##Grosvenor Hood, Dorothy Hood
##Lady Acland-Hood
##Fanny Hood of Avalon
##Mrs. Curzon Howe
##[Hon.] Evelyn Hubbard, Mrs. E. Hubbard, Alice Hubbard
##Mary Hughes
##Mrs. Meynell Ingram
##G. Jolliffe, Sydney H. Jolliffe, Mrs. Jolliffe
##Lady Johnston
##G. Keppel, Mrs. Keppel, Derek Keppel, Mrs. William Keppel
##Mrs. Alfred Ker
##Constance Kerr
##Mrs. Kingscote
##C. C. Kingston
##Lady Knollys
##Bertha Lambart
##F. W. Lambton, Mrs. Lambton
##Mary Lascelles
##Charles Laurence, Herbert Laurence
##Wilfrid Laurier
##Mrs. Lawley
##Mrs. C. Lawrence, Misses Lawrence (2), Mrs. H. Lawrence
##Mrs. Legge
##T. W. Legh, Mrs. Legh, Sybil Legh
##F. D. Leigh, Mrs. F. D. Leigh, E. Chandos Leigh, Mrs. E. C. Leigh, Cordelia Leigh
##C. Hanbury Lennox, Mrs. Hanbury Lennox
##G. W. Leslie
##R. l’Estrange
##Atholl Liddell, Mrs. A. Liddell
##Mrs. H. Gore-Lindsay
##Reginald Lister
##Henry Littleton, Misses Littleton (2)
##Misses Loch (2)
##William Lowther, Mrs. W. Lowther, L. Lowther, Mrs. L. Lowther
##Mrs. E. H. Loyd
##Mrs. Lumley
##Alfred Lyttelton, Mrs. A. Lyttelton, Misses Lyttelton (2), Mrs. Lyttelton
##Flora Macdonald, Lady Macdonald
##Mrs. Mackinnon
##Mrs. Maclagan
##Mrs. Magniac
##Mrs. Maguire
##W. Massey-Mainwaring, Mrs. Massey-Mainwaring
##Mrs. Fuller-Maitland
##Aline Majendie
##Misses Henniker Major (2)
##Mrs. Mallet
##Archibald Marjoribanks
##Misses Constable Maxwell (2)
##Mrs. M'Calmont
##Schomberg M'Donnell
##Charles Mills, Violet Mills, Mrs. Mills
##Mrs. Percy Mitford
##Maud de Moleyns
##Mrs. C. Molyneux
##Annette Monck, Mrs. Monck
##Violet Monckton
##Mrs. Monson
##John Scott Montagu
##[Hon.] Evelyn Moore
##R. Moreton, Mrs. R. Moreton
##Mrs. Mostyn, Misses Mostyn (2)
##Mrs. G. H. Murray, Alice Murray
##Lady Musgrave
##[Hon. Lady] Napier, Emilia Napier, Mrs. Scott Napier
##Mrs. Neeld
##Sir Hugh Nelson
##[Hon.] R. Nevill
##Mrs. Newdigate
##Sir H. S. Northcote
##Misses O'Brien (2)
##Mary O'Hagan
##Mrs. Okeover
##Mrs. Oliphant
##R. Terence O'Neill, Henrietta O'Neill
##Misses Palk (2)
##Cecil Parker, R. Parker, F. Parker, Mrs. F. Parker, Mrs. Parker
##Mabel Parnell
##[Hon.] C. B. Parsons, Mrs. Parsons
##Mrs. W. Paton
##[Hon.] Sydney Peel, Misses Peel (2)
##Mrs. Anderson Pelham
##E. S. Douglas-Pennant, Mrs. E. S. Douglas-Pennant
##Mrs. Heber Percy
##Albert Petre, Mrs. A. Petre
##Harriet Phipps
##Mrs. Pirie
##Thomas Playford
##Horace C. Plunkett
##[Hon.] Ashley Ponsonby, Mrs. Ponsonby, Misses Ponsonby (2)
##H. Orde Powlett, Mrs. Orde-Powlett, Myra Orde-Powlett
##E. W. B. Portman, Mrs. Portman, Mary Portman
##Mrs. Pretyman
##C. Ramsay, Mrs. C. Ramsay
##G. H. Reid
##Misses Rendel (2)
##Misses Rice (2)
##Lady White Ridley
##Mrs. Ritchie
##F. Roberts, Mrs. Phillips Roberts
##Misses Roberts (of Kandahar) (2)
##J. M. Rolls, Eleanor Rolls
##W. Rothschild, Evelina Rothschild
##W. Rowley, Mrs W. Rowley, Lady Thelluson Rowley
##A. Russell, Misses Russell (2)
##Gustavus Hamilton-Russell, Misses Hamilton Russell (2)
##the Master of Ruthven, Mrs. Ruthven
##Mrs. J. D. Ryder
##Sir Saul Samuel
##A. Saumarez, Mrs. A. Saumarez
##Mrs. E. J. Saunderson
##J. Maxwell Scott, Mrs. Maxwell Scott
##R. J. Seddon
##Mary Sidney
##Lady Simeon
##Misses Skeffington (2)
##Sir Donald Smith, Mrs. A. H. Smith, [Hon.] W. F. D. Smith
##Granville Somerset, Mrs. G. Somerset, Arthur Somerset, Mrs. A. Somerset, R. Somerset, Violet Somerset
##Mrs. C. R. Spencer
##Sir J. Gordon Sprigg
##Lyulph Stanley, F. C. Stanley, George Stanley, Mrs. E. J. Stanley, Mrs. Stanley, Mrs. V. A. Stanley, Maude Stanley
##Lady Cowell-Stepney
##Randolph Stewart, Mrs. R. Stewart, FitzRoy Stewart, Mrs. Stewart
##Mabel St. Aubyn
##Misses St. Clair (2)
##Mrs. Stirling
##Horatia Stopford
##[Hon. Lady] Alison Stourton
##Mrs. Strutt, Misses Strutt (2)
##Hilda Sugden
##Alfred Talbot, Mrs. Talbot, Mrs. R. A. J. Talbot
##Sir D. Tennant
##S. R. Thayer
##Misses Thellusson (2)
##Edward Thesiger, Mrs. E. Thesiger, Frederick Thesiger, Mrs. F. Thesiger, Mary Thesiger
##Lady Thorold
##Katharine Thring
##Misses Tollemache (2)
##R. Marsham-Townshend, Mrs. Marsham-Townshend
##Alice Hanbury-Tracy
##Charles Grey Trefusis, Misses Trefusis (2)
##Mrs. Trelawny
##Mrs Tremayne
##Mrs. W. le Poer Trench
##Charles Trevor
##George Hill-Trevor, Marcus Hill-Trevor, Mrs. Hill-Trevor, Misses Hill-Trevor (2)
##Mrs. C. W. Trotter
##Lady Tryon
##Rosamond Tufton
##Sir G. Turner
##Rev. L. Tyrwhitt
##Misses Tyssen Amherst (2)
##Misses Vereker (2)
##R. Greville-Verney, Mrs. R. G. Verney, Misses Verney (2)
##F. Villiers, Mrs. F. Villiers
##Misses Vivian (2)
##Arthur Walsh
##Mrs. P. E. Warburton
##Robert Ward, Mrs. Dudley-Ward
##Mrs. West
##Mrs. Whateley
##Sir W. Whiteway
##F. Bootle-Wilbraham
##Ella Williamson
##Tatton Willoughby
##Lady Wilson
##[Hon.] Armine Wodehouse, Mrs. Wodehouse
##Frances Wolseley
##F. Wood, Misses Wood (2)
##Mrs. G. Wrottesley, Evelyn Wrottesley
##Percy Wyndham, Mrs. P. Wyndham, Misses Wyndham (2)
##Maud Wynn
##Lois Yarde-Buller
##Alex. G. Yorke, Mrs. J. Yorke, Mrs. E. C. Yorke
#Sirs&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5c–6a)
##Augustus Adderley
##Edwin Arnold
##John Austin
##George Arthur
##John Heathcoat-Amory
##A. Armstrong
##Andrew Agnew
##Frederick Abel
##Henry Acland
##A. Arnold
##Alexander Arbuthnot
##John Barran
##G. Bower
##J. W. Bonser
##J. Crichton-Browne
##Joseph Bailey
##E. Ashmead-Bartlett
##Henry Barkly
##R. Beauchamp
##Raymond Burrell
##Charles Barrington
##David Baird
##Arthur Birch
##Edward Birkbeck
##W. Cunliffe Brooks
##A. de Capel Brooke
##Courtenay Boyle
##F. Burton
##F. Buxton
##Steuart Bayley
##John Bramston
##John Baker
##H. Bullard
##J. T. Brunner
##H. Bellingham
##Henry Bergne
##Thomas Boughey
##F. J. Bramwell
##E. Burne-Jones
##James Blyth
##Seymour Blane
##Henry Chamberlain
##Roderick Cameron
##Hugh Cholmeley
##John Conroy
##Edward Clarke
##C. Cameron
##E. Carbutt
##W. Coddington
##Marshal Clarke
##Reginald Cathcart
##Savile Crossley
##Edward Colebrooke
##Reginald Cust
##Charles Crosthwaite
##John Colomb
##Daniel Cooper
##F. Astley-Corbett
##Donald Currie
##Henry Cunningham
##Robert Cunliffe
##Henry Cotterell
##T. D. Gibson Carmichael
##F. Curden,
##George Dallas
##James Drummond
##Mortimer Durand
##G. Des Vieux
##Henry Dering
##J. N. Dick
##Dyce Duckworth
##T. Swinnerton Dyer
##E. Hastings Doyle
##John Dorington
##William Dunn
##Humphrey de Trafford
##Charles Dalrymple
##G. Dashwood
##Gardner
##Engleheart
##Francis Evans
##A. Edmonstone
##Whittaker Ellis
##W. H. Flower
##Horace Farquhar
##Joseph Fayrer
##H. Fletcher
##William Ffolkes
##William Fraser
##Bartle Frere
##Gerald Seymour Fitz-Gerald
##Robert Finlay
##B. Walter Foster
##Gerald FitzGerald
##R. FitzGerald
##Maurice FitzGerald
##Forrest Fulton
##William Flower
##Andrew Fairbairn
##John Gilbert
##E. T. Gourley
##Edward Grey
##W. Gull
##Walter Gilbey
##Lepel Griffin
##G. Macpherson-Grant
##Reginald Graham
##Philip Grey Egerton
##Douglas Galton
##R. Glyn
##Arthur Godley
##Charles Grant
##R. Gresley
##Alexander Acland-Hood
##T. G. Fermor Hesketh
##Arthur Haliburton
##Brydges Henniker
##F. Dixon-Hartland
##R. Hanson
##Alfred Hickman
##W. Houldsworth
##Henry Howorth
##F. Seager Hunt
##Charles Hall
##E. W. Hamilton
##Reginald Hardy
##Clement Hill
##Basil Hall
##Joseph Hooker
##Charles Hunter
##Charles Hartopp
##Victor Houlton
##Augustus Hemming
##Henry Irving
##Frederic Johnstone
##W. Jenner
##J. Jenkins
##James Joicey
##Charles Jessell
##Harry Johnston
##Edward Jenkinson
##James Hill Johnes
##John Jackson
##H. Seymour King
##James Kitson
##J. Lister-Kaye
##V. Kennett-Barrington
##George Kekewich
##John Leslie
##Thomas Dick Lander
##T. Villiers Lister
##James Linton
##Charles Lees
##Charles Legard
##Thomas Lea
##Wilfrid Lawson
##Elliott Lees
##A. C. Lyall
##J. T. D. Llewelyn
##Joseph Leese
##Leonard Lyell
##F. Laking
##Godfrey Lushington
##F. Lockwood
##Henry Longley
##George Lewis
##F. Milner
##Herbert Maxwell
##Francis Montefiore
##Graham Montgomery
##Robert Moncreiffe
##Musgrave
##Colin Scott Moncrieff
##Francis Mowatt
##Evan MacGregor
##J. G. Miller
##F. D. Maclean
##J. Blundell Maple
##Allan Mackenzie
##Lewis M'lver
##F. Mappin
##Theodore Martin
##Samuel Montagu
##William MacCormac
##Hubert Miller
##Lewis Morris
##Clements Markham
##A. C. Mackenzie
##John Monckton
##J. Stirling-Maxwell
##J. Heron Maxwell
##Kenneth Matheson
##J. S. Montefiore
##Acquin Martin
##W. Maxwell
##Oswald Moseley
##Arthur Nicolson
##Terence O'Brien
##Reginald Ogilvy
##Herbert Oakeley
##Hush Owen
##G. G. Petre
##Walter Parratt
##Frederick Pollock
##Herbert Perrott
##Douglas Powell
##Weetman Pearson
##Joseph Pease
##Francis S. Powell
##Reginald Palgrave
##W. Priestley
##E. G. Poynter
##G. S. Baden-Powell
##Charles Pontifex
##J. Dickson-Poynder
##James Paget
##C. M. Palmer
##C. Lennox Peel
##James B. Peile
##Westby Perceval
##Charles Pigott
##John Puleston
##W. Plowden
##Richard Quain
##George Russell
##C. Lister Ryan
##W. H. Russell
##J. Ramsay
##Owen Roberts
##R. T. Reid
##Charles Robinson
##J. Thellusson Rowley
##James Reid
##C. Euan-Smith
##J. Barrington Simeon
##J. B. Stone
##M. Shaw-Stewart
##Edward Sieveking
##T. H. Sanderson
##Augustus K. Stephenson
##Thomas Sutherland
##Mark Stewart
##Andrew Scoble
##Joseph Savory
##Douglas Straight
##Charles Shelley
##S. Shippard
##E. Sassoon
##A. Condie Stephen
##E. Sullivan
##Arthur Sullivan
##S. Scott
##H. Simpson
##E. Stafford
##Ernest Satow
##Tatton Sykes
##John Tyler
##Charles Tennant
##John Tenniel
##J. Thorold
##John Thursby
##Thomas Troubridge
##Charles Turner
##H. Meysey-Thompson
##W. Vincent
##Edgar Vincent
##Arthur Vicars
##W. Williams-Wynn
##James Walker
##R. Webster
##George Wombwell
##C. Rivers Wilson
##W. H. Wills
##Donald Mackenzie Wallace
##George Warrender
##F. Winnington
##James Whitehead
##Arthur Willshire
##Henry Wood
##Hugh Wyndham
##W. White
##Sidney Waterlow
##Hedworth Williamson
##Jacob Wilson
##W. Windeyer
##Albert Woods (Garter)
##Allen Young
#Chairman of County Council (Dr. Collins)
#Counts and Countesses
##Count Cassini
##Count and Countess De Ganay
##Count Gurowski
##Count Hohenau
##Count Theodor Bolesta Koziebrodski
##Count Leon Mniszeek
##Count and Countess Potocki
##Count and Countess Raben
#Barons and Baronesses
##Baroness Emile Beaumont d'Erlanger
##Baroness De Brienen
##Baron De Onethau and Baroness D’Onethan [sic]
##Baron and Baroness Alphonse de Rothschild
##Baron Ferdinand Rothschild
##Baron and Baroness Schröder
##Baron and Baroness von Deichmann
##Baron von Heeckeren van Wassenaer
##Baroness von Hügel, Baroness Gertrud von Hügel [sic]
##Baron and Baroness Campbell von Laurentz
##Baroness Wilhelm von Rothschild
#Rev. the Moderator of the General Assembly of the Church of Scotland
#Deans — Christ Church, St. Paul's, Westminster, Windsor
#The Provost of Eton
#Master of Trinity (Mr. Butler)
#The Sub-Dean of the Chapels Royal
#Canons — Blundell, Dalton, Duckworth, Fleming, Hervey, Teignmouth Shore, Wilberforce
#Dr. Adler (Chief Rabbi)
#Dr. M'Cormick
#Chaplain of the Fleet
#Chaplain General
#Reverend Doctors — Edmund Warre, C. J. Welldon
#Reverends — Prebendary Hawkshaw, Albert Baillie, W. H. Bliss, M. Ebrington Bisset, Lord W. Cecil, Lord Charles Fitzroy, J. H. Ellison, H. Haweis, W. R. Jolly, G. J. Martin, Newton Mant, Marquis of Normanby, A. Robins. W. Gunion Rutherford, Clement Smith, Montagu Villiers
#Doctors — Lennox Browne, J. V. Bridge, Barlow, Robert Farquharson, J. F. Fox, Surgeon-Major Kilkelly, John Lowe, C. H. H. Parry, G. V. Poore, Dorrien Smith, S. Wilks
#Messieurs&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 6b–7a),  Mesdames (4, Col. 7a–b) and Misses&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7c – 5, Col. 1a)
##Mme Abdy
##Mr C. T. Dyke-Acland, Mme A. H. Dyke Acland, Mme Dyke Acland
##Mme Adair
##Misses Adam (2)
##Mr and Mme Adeane
##Misses Adye [?] (2)
##Mme Agar
##Mr Hamilton Aidé
##Mr John Aird, Misses Aird (2)
##Miss Akers-Douglas
##Mr Edward Alderson
##Mr George Alexander, Mme Alexander, Miss Alexander
##Miss Alison
##Mr and Mme Allhusen
##Mme Alma-Tadema
##Mr W. Ambrose
##Miss Heathcoat-Amory
##Mr R. Anderson, Miss Florence Anderson
##Mr E. H. Anson
##Mr H. T. Anstruther, Miss Rosomond Anstruther
##Mme Antrobus
##Mr Arbuthnot, Miss Arbuthnott [sic]
##Miss Archer-Houblon
##Mme Argles
##Mme Arkwright, Miss Arkwright
##Misses Armytage (2)
##Miss Arnott
##Mr and Mme Ascroft, Miss Ascroft
##Mr Arthur Ash
##Mr A. Asher
##Mme Ashton
##Mme Asquith
##Mr Astor, Mr W. Astor
##Mr B. F. Astley
##Mme Evelyn Atherley
##Mr and Mme Alfred Austin, Misses Austin (2)
##Mr and Mrs C. H. Babington
##Mr and Mrs Bagge
##Mrs Charles Bagot, Mrs J. F. Bagot, Miss Alice Bagot
##Mr James Bailey, Mrs J. Bailey, Mrs Bailey, Misses Bailey (2)
##Mrs Duncan Baillie, Misses Duncan Baillie (2)
##Mr Baillie of Dochfour
##Mr and Mrs W. A. Baillie-Hamilton
##Mr E. Bainbridge
##Mr and Mrs H. R. Baird, Mr and Mrs J. G. A. Baird, Misses Baird (2)
##Mr and Mrs Baldwin
##Mr and Mrs E. Balfour, Mr and Mrs Charles Balfour, Miss Balfour
##Mr and Mrs Banbury, Miss Banbury
##Mr and Mrs S. B. Bancroft [actor &quot;Bancroft and his wife accepted with becoming grace the congratulations with which they were well-nigh overwhelmed&quot;&lt;ref name=&quot;:3&quot; /&gt; (5, Col. 6b)]
##Bandanaratke [?]
##Mrs Bankes
##Mr Banks
##Mr and Mrs Walter Baring, Miss Baring
##Miss Barker
##Mr J. Emmott Barlow, Mrs Barlow, Mrs Barlow [sic 2x]
##Misses Barnardiston (2)
##Miss Barne
##Mr and Mrs F. G. Barnes, Mr and Mrs Barnes, Misses Barnes (2)
##Miss Barran (2)
##Mr and Mrs J. Wolfe Barry, Mr and Mrs F. Tress Barry, Mrs A. Barry
##Misses Bartlett (2)
##Mr and Mrs D. P. Barton, Mr and Mrs Barton
##Mr Hamar Bass
##Mrs Bates, Miss Bates
##Mr and Mrs H. Bathurst, Misses Bathurst (2)
##Mr and Mrs Baxendale, Miss Baxendale
##Miss Mariot [?] Bayley
##Mr and Mrs W. W. Beach, Miss Beach
##Misses Hicks-Beach (2)
##Mr R. M. Beachcroft
##Mr and Mrs Wentworth Beaumont, Mr Wentworth B. Beaumont, Mrs Beaumont, Miss Hilda Beaumont
##Mr and Mrs Rupert Beckett, Mr E. W. Beckett
##Mr and Mrs Beer
##Mr and Mrs F. F. Begg
##Mr  Charles Bell, Mr and Mrs Bell, Misses Bell (2)
##Miss Bellingham
##Mr and Mrs R. Benson, Mr and Mrs Benson
##Miss Berens
##Mr and Mrs Beresford, Miss Beresford
##Miss Berkeley, Misses Berkeley (2)
##Mr and Mrs Bertier, Miss Bertier
##Mr and Mrs Cosmo Bevan, Mr and Mrs F. Bevan, Miss Bevan
##Mr M. M. Bhownaggree
##Mr and Mrs F. Bibby
##Mr Leonard Biddulph, Mr Biddulph, Mr Victor Biddulph, Mr M. Biddulph, Mrs H. M. Biddulph, Misses Biddulph (2), Miss Biddulph, Miss Freda Biddulph
##Mr and Mrs Bigham
##Mr Bigwood
##Mrs C. Bill, Miss Bill
##Miss Birch
##Mrs Birch-Reynardson, Misses Birch-Reynardson (2)
##Mr A. Birrell, Mrs Birrell
##Mr and Mrs Bischoffsheim
##Mrs Ebrington Bissett
##Misses Blackwood (2)
##Mr and Mrs R. G. Blennerhassett
##Mrs W. H. Bliss
##Mrs Blundell, Miss Blundell
##Misses Blyth (2)
##Mr and Mrs Bolitho, Miss Bolitho
##Mr H. C. O. Bonsor, Mrs Bonsor, Miss Bonsor
##Mrs W. Borsel
##Mrs Griffith-Boscawen
##Mr and Mrs Boulnois
##Miss Bourke
##Mr W. R. Bousfield
##Mrs Bowden-Smith, Misses Bowden-Smith (2)
##Miss Bowen (2)
##Mr T. G. Bowles, Mrs Bowles
##Mr Edmund R. Boyle
##Miss Mabel Brackenbury
##Mrs Bradley, Miss Bradley
##Miss Beryl Bradford
##Miss Braddon
##Miss Bramwell
##Mr H. L. C. Brassey, Mrs H. A. Brassey, Misses Brassey (2), Misses Brassey (2) [sic 2x]
##Mr Stapleton Bretherton, Misses Stapleton Bretherton (2), Mr F. Stapleton Bretherton
##Mrs Bridge
##Mr G. Bridgman, Mr and Mrs C. G. O. Bridgeman
##Mr Brigg
##Mrs Brocklehurst
##Misses Brodie (2)
##Mr and Mrs Brookfield, Miss Brookfield
##Miss Bromley-Davenport
##Miss Brooke
##Miss Rhoda Broughton
##Mr and Mrs A. H. Brown, Miss Brown
##Mrs Browne, Misses Browne (2), Misses Browne (2) [sic 2x]
##Mrs Brownrigg, Miss Brownrigg
##Mr A. O. Bruce, Mrs A. C. Bruce [sic], Misses Bruce (2)
##Miss Brunner
##Mrs Bryce
##Mr Brymer
##Mr and Mrs Buchanan
##Mrs C. E. Buckle
##Mr Bucknill
##Miss Budgett
##Miss Mary Bulteel
##Miss Burdett
##Mr and Mrs Burges, Misses Burges (2)
##Mrs C. K. Burn
##Mr and Mrs F. C Burnand
##Miss Evelyne Burne
##Mr and Mrs W. Burns, Miss Burns
##Misses Burrell (2)
##Mr J. G. Butcher
##Mrs Butler, Mrs Butler, Miss Butler
##Mr Sydney Buxton, Mrs S. Buxton, Misses Buxton (2)
##Mr P. H. Calderon
##Mrs Calley
##Mrs Archibald Calvert, Miss Calvert
##Mr Cameron, Miss Cameron, Misses Cameron (2)
##Mr and Mrs J. D. Campbell, Mr J. A. Campbell, Miss J. A. Campbell, Mrs F. Campbell, Mrs W. Campbell, Mrs Hastings Campbell, Mrs W. Campbell [sic 2x], Mrs F. L. Campbell, Mrs D. B. O. Campbell, Miss Lilah Campbell, Miss Campbell, Miss Ronald Campbell, Misses Campbell (2)
##Miss Grace de Capell-Brooke
##Miss Carden
##Miss Carleton
##Mr and Mrs W. W. Carlile, Miss Carlisle
##Mrs Rivett Carnac
##Mrs Carnegy
##Mrs Boyd Carpenter, Misses Boyd Carpenter (2)
##Mrs Carson
##Mr and Mrs D'Oyly Carte
##Mrs Carter
##Mrs Castance
##Mr R. K. Causton, Mrs Causton, Miss Causton
##Mrs Cavaye
##Mr and Mrs C. Tyrall Cavendish, Mr Victor Cavendish, Mr Henry Cavendish, Mr Cavendish, Mrs Cavendish
##Mr and Mrs F. Cavendish-Bentinck, Mr Cavendish-Bentinck, Mrs W. G. Cavendish-Bentinck
##Mr F. Cawley
##Mr and Mrs Cayzer, Miss Cayzer
##Mr and Mrs W. M. Cazalet
##Mr F. Cazenove
##Mr Evelyn Cecil, Miss Cecil
##Mrs Chaine
##Mrs Chaloner
##Mr Austen Chamberlain, Mrs Chamberlain, Misses Chamberlain (2)
##Misses Chaning (2)
##Mr and Mrs Channing
##Mr and Mrs Cecil Chaplin, Misses Chaplin (2), Miss Edith Chaplin, Miss Chaplin
##Mrs Chapman
##Misses Chetwode (2)
##Mrs W. Chetwynd, Miss Chetwynd (2)
##Mr Childe-Pemberton
##Miss Chitty
##Miss Leila Crichton
##Miss Cholmeley (2)
##Miss Cholmondeley
##Miss Chrichton-Maitland
##Mrs H. Churchill
##Miss Spencer Churchill
##Mr J. D. Clark, Mr and Mrs Atkinson Clark, Mr Clark, Mrs B. F. Clark, Mrs G. D. Clark, Stanley Clark, Miss Clark
##Mr Purdon Clarke, Mr Ernest Clarke, Miss Clarke, Miss Stanley Clarke
##Mrs Clerk
##Mr and Mrs Henry Pelham Clinton
##Mrs Clive, Misses Clive (2)
##Mrs Close
##Mr Clough
##Mr Clowes, Misses Clowes (2)
##Mr Cobbold
##Mr T. B. Cochrane, Miss Cochrane
##Mr and Mrs W. A. Cockerell, Miss Cockerell, Miss Cockerell [sic 2x]
##Mr and Mrs D. Coghill
##Mr B. Cohen
##Mr Wentworth Cole
##Miss Colomb
##Mr and Mrs Colston
##Miss Colville
##Mr Richard Combe
##Miss Commerell, Miss Commerell [sic 2x]
##Mr and Mrs Compton
##Mr and Mrs Consett, Miss Vera Consett
##Mr and Mrs F. L. Cook, Mr Ward Cook, Miss Cook
##Mr and Mrs Kinloch Cooke, Mr Cooke, Mr and Mrs C. Kinloch Cooke
##Mr and Mrs Daniel Cooper, Mrs E. H. Cooper, Misses Cooper (2), Miss Cooper
##Mr and Mrs Cameron Corbett, Miss Corbett
##Mr and Mrs V. Seymour Corkran, Miss Corkran
##Mr and Mrs F. S. W. Cornwallis
##Mr and Mrs Cory
##Mrs Armar Corry, Mrs Clifford Corry, Miss Corry
##Mr J. R. G. Cotterell, Miss Cotterell (2)
##Mrs Stapleton Coton
##Mr and Mrs George Courroux
##Mrs Courtney
##Mr Burdett-Coutts
##Mrs Coventry
##Miss Cowell
##Miss Cowell-Stepney
##Mr and Mrs R. Cox, Mrs Cox, Miss Cox
##Mrs Crabbe, Misses Crabbe (2)
##Mrs Craik
##Mr and Mrs Crawshay
##Mrs Creignton, Miss Lucia Creighton
##Mr C. A. Cripps, Mr and Mrs Wilfrid Cripps
##Mr and Mrs Critchett
##Mr and Mrs Croombie
##Mrs A. B. Crosbie
##Mr and Mrs Shepherd Cross, Mr A. Cross, Miss Crosse
##Mr and Mrs Cruddas, Misses Cruddas (2)
##Mr and Mrs Percy Crutchley, Misses Crutchley (2)
##Miss Cuffe
##Miss Culme-Seymour
##Mrs Cuninghame
##Miss Cunliffe
##Mrs Dick-Cunynghame
##Mrs Curzon
##Misses Cust (2)
##Miss Custance
##Mrs Dalbiac
##Miss Gladys Dalgety [?]
##Mr C. B. Dalison
##Miss Dalrymple
##Mrs Dalton
##Mrs Denis Daly
##Mr and Mrs Darling
##Miss Dashwood
##Mr W. Bromley-Davenport
##Miss Davey
##Mr and Mrs Louis Davidson, Mrs Randall Davidson
##Mr W. Rees Davies, Mr Ben Davies, Mr and Mrs Vaughan Davies
##Mrs Davis
##Miss Dawnay (2)
##Mrs de Arcos
##Misses De Brienen (2)
##[Miss] La Baronne de Friesen
##Mrs R. C. de Grey Vyner
##[Miss] La Baronne Sirtema de Grovestins [?]
##Mr and Mrs J. de la Cour
##Mr and Mrs Edwin de Lisle
##Mr W. E. Denison
##Mrs Denny
##Miss De Perpigna
##Mrs de Salis
##Mr de Soria
##Mr De Trafford, Miss De Trafford
##Mr Deverell, Miss Deverell
##Mr and Mrs W. de Winton, Miss De Winton
##Mr and Mrs Gerard Dicconson
##Mr and Mrs Dicken
##Mr and Mrs C. S. Dickson, Mrs Dickson
##Mr J. K. Digby, Kenelm E. Digby, Mrs Digby, Misses Digby (2), Miss Digby
##Mr and Mrs J. Diggle
##Mr Lee Dillon, Misses Dillon (2)
##Mr and Mrs Coningsby Disraeli, Mr and Mrs R. Disraeli, Miss Disraeli
##Mrs Domvile, Miss Domvile
##Mr Greville Douglas, Mrs A. L. Douglas, Misses Douglas (2)
##Mrs Akers-Douglas
##Miss Dowell
##Mr and Mrs Doxford, Miss Doxford
##Mrs Geoffrey Drage
##Mr A. Drummond, Mr and Mrs G. Drummond, Mrs A. Hay Drummond, Mrs Lawrence Drummond, Mrs Drummond, Miss Edith Drummond,  Misses Drummond (2), Miss Mary Drummond, Miss Adelizs [?] Drummond, Misses Drummond (2) [sic 2x]
##Misses Du Cane (2)
##Miss Du Chair
##Mr W. H. Dudley-Ward, Miss Sybil Dudley-Ward
##Mr F. Dugdale
##Misses Duncombe (2)
##Mrs Dundas, Miss May Dundas
##Miss Dunn
##Mrs Dunne, Miss Marion Dunne
##Mr Du Plat Taylor, Mrs G. Du Plat Taylor
##Mrs Durnford
##Mr and Mrs Thiselton Dyer
##Mrs East, Misses East (2)
##Mr F. Eaton
##Mr R. Edgcumb
##Mrs Edis, Misses Edis (2)
##Mr Bevan Edwards, Miss Bevan Edwards (2), Mr C. C. Edwards, Mrs Edwards
##Mrs Egerton, Miss Egerton (2), Miss Egerton
##Miss Grey Egerton
##Mr and Mrs M. Eliot, Misses Eliot (2)
##Miss Ellaby
##Mrs Ellicott, Miss Ellicott
##Mr and Mrs F. Elliot, Mr T. H. Elliott, Miss Gertrude Elliot
##Mr T. E. Ellis, Miss Ellis (2), Miss Evelyn Ellis
##Mrs Ellison, Miss Ellison
##Misses Elphinstone (2)
##Mr Cary-Elwes
##Mr Erskine, Miss Rachel Erskine
##Mr Maurice Euphrussi
##Mr W. H. Evans, Misses Evans (2)
##Mr H. P. Ewart, Mrs C. B. Ewart
##Mr Eyre
##Mr Cecil Fane, Mr G. H. Fane, Mr Fane
##Mr Dyafer Fakhry
##Misses Keith Falconer (2)
##Mrs Fane
##Mrs Fanshawe, Miss Fanshawe
##Mr and Mrs Fardell, Misses Fardell (2)
##Mr and Mrs Farmer, Mrs Lancelot Farmer, Miss Farmer
##Mrs Farnham
##Mr Alfred Farquhar, Mr W. Farquhar, Mr and Mrs E. Farquhar, Mrs G. M. Farquhar
##Mr J. N. Farquharson, Miss Amelia Farquharson, Miss Henrietta Farquharson
##Mr and Mrs Farquharson of Invercauld, Misses Farquharson of Invercauld (2)
##Misses Feilding (2)
##Mrs Fellowes
##Mrs Fenn
##Mrs Fenwick, Misses Fenwick (2)
##Mr and Mrs Johnson-Ferguson
##Mr Munro-Ferguson
##Misses Ferguson of Pitfour (2)
##Miss Fergusson
##Miss Dorothy Ffolkes
##Mrs Field
##Mr and Mrs Fielden, Misses Fielden (2)
##Mrs G. H. Finch, Mrs Wynne Finch, Misses Finch (2)
##Mr and Mrs Firbank
##Mr Herbert Fisher, Mr and Mrs Hayes Fisher, Misses Fisher (2)
##Mr and Mrs Fison, Miss Fison
##Miss FitzClarence (2)
##Mrs FitzGeorge, Miss Olga FitzGeorge
##Mr Fitzgerald, Mr F. G. Fitzgerald, Miss Fitz Gerald
##Mr and Mrs Almeric Fitzroy, Miss Ethel Fitz-Roy
##Mrs R. Fitzwilliam, Misses Fitzwilliam (2)
##Mr Flannery
##Mr E. Flower, Miss Flower, Miss Flower [sic 2x]
##Mrs Floyd
##Mrs H. Fludyer
##Mr H. St. George Foley
##Mrs Barrington Foote
##Mr J. S. Forbes, Mr Forbes
##Mr John Ford
##Mr H. W. Forster
##Mr and Mrs Arnold-Forster
##Mr and Mrs Bevill Fortescue
##Misses Forwood (2)
##Mr W. S. Foster, Mrs W. H. Foster, Mrs H. S. Foster, Miss Foster
##Misses Fowler (2)
##Mr Franklin
##Mrs Houston French
##Misses Frere (2)
##Mr L. Fry
##Mrs Fullerton, Misses Fullerton (2)
##Mr Gadson
##Mr Wilhelm Ganz
##Miss Gardiner, Miss Gardiner [sic 2x]
##Mrs Gardner
##Mr and Mrs Garfit [?]
##Mrs Gathorne-Hardy, Miss Gathorne-Hardy
##Mr Hamilton Gatliff
##Mr and Mrs Scott Gatty
##Mr and Mrs Sydney Gedge
##Mr Geoffrey Drage [sic; does this belong here?]
##Mr F. W. Gibbs, Misses Gibbs (2)
##Mr and Mrs Walter Gibson
##Miss Gilbey
##Mr and Mrs Tyrell Giles
##Mr W. Gillett
##Mr and Mrs Gilliat, Misses Gilliat (2)
##Mr Henry Gladstone, Mrs Gladstone, Miss Helen Gladstone
##Miss Glyn
##Misses Godley (2)
##Mrs Godson
##Mr and Mrs Goelet, Miss Goelet
##Mr Charles Gold, Miss Gold
##Mr G. P. Goldney
##Mr and Mrs S. Hoffnung Goldnung Goldsmid
##Mrs A. Goldsmid, Miss Goldsmid
##Mr Otto Goldsmidt
##Mrs Goldsworthy
##Mrs Goodden, Miss Gurrney Goodden
##Mrs Goodenough
##Mr and Mrs John Gordon, Mr and Mrs J. E. Gordon, Mrs Gordon, Mrs G. G. Gordon, Mrs S. Gordon, Mrs Gordon [sic 2x], Miss Hamilton Gordon, Misses Gordon (2)
##Mr and Mrs Frank Gore, V. Gore, Mr and Mrs S. W. Gore, Mrs Gore, Miss Gore
##Mr and Mrs Goschen, Misses Goschen (2)
##Mr and Mrs A. Gosling, Miss Gosling
##Mr and Mrs F. R. Gosset
##Misses Gough-Calthorpe (2)
##Mr E. A. Goulding
##Mr G. Leveson-Gower
##Mr F. Graham, Mr Graham, Mr H. R. Graham, Mr and Mrs C. C. Graham
##Mrs Grant, Miss Grant
##Miss Victona Grant-Duff
##Mr and Mrs Henry Graves, Miss Graves
##Mr Ernest Gray
##Mrs Green
##Mr H. D. Greene, Mr W. R. Greene
##Mrs Gregory, Miss Gregory
##Mr and Mrs W. H. Grenfell, Mrs H. Grenfell, Miss Maud Grenfell
##Mr J. A. Gretton
##Mr Howard of Greystoke
##Mr Grifflth-Boscawen
##Mr and Mrs W. H. Kendal Grimston
##Mr George Grossmith [&quot;George Grossmith was not a little lionised by titled ladies&quot;&lt;ref name=&quot;:3&quot;&gt;“The Queen’s Garden Party. Buckingham Palace Grounds. A Brilliant Scene. The Queen’s Cup of Tea.” ''Daily News'' (London) 29 June 1897, Tuesday: 5 [of 10], Col. 6a [of 7] – 6, Col. 2a. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970629/021/0005. Print pp. 5–6.&lt;/ref&gt; (5, Col. 6b)]
##Mr Montagu Guest
##Mrs Gunter, Misses Gunter (2)
##Mr Gurdon
##Mrs Gurney
##Mrs Guy-Pym
##Mr and Mrs Gye
##Mr and Mrs Carl Haag, Miss Carl Haag
##The Munshi Abdul Hafiz Karim
##Mr and Mrs Haggard
##Miss Haig
##Mr R. B. Haldane
##Mr Halford, Misses Halford (2)
##Mr and Mrs Lewis Hall, Mrs Hall, Miss Hall, Miss (Lewis) Hall
##Mr and Mrs Thomas Halsey, Misses Halsey (2)
##Mr Francis Hamilton, Mrs R. W. Hamilton, Mrs Ian Hamilton, Misses Hamilton (2), Miss Hamilton
##Mrs Hammet
##Mr and Mrs Hanbury, Miss Dora Hanbury, Mrs Hanbury
##Miss V. Hanson
##Mr L. V. Harcourt
##Mr and Mrs Hardcastle, Misses Hardcastle (2)
##Mr and Mrs Hardy, Misses Hardy (2)
##Mr Cozens-Hardy
##Mr T. Hare, Mr Augustus Hare, Mr and Mrs John Hare, Mrs Marcus Hare, Mrs Marcus Hare [sic 2x], Miss Hare, Misses Hare (2), Misses Hare (2) [sic 2x]
##Mrs Harford
##Mrs Hargreaves-Rogers
##Mr C. Harrison, Miss Harrison
##Miss Hart
##Misses Hart-Dyke (2)
##Mr and Mrs Hartmann
##Mr George Harwood, Misses Harwood (2)
##Mr Hatch
##Mrs Hatton
##Mrs Haweis
##Mr and Mrs Claude Hay, Misses Hay (2), Misses Hay (2) [sic 2x]
##Mrs Arthur Heath, Mrs Heath
##Miss Louisa Heathcote
##Mr J. Henniker Heaton
##Misses Hemming (2)
##Mr Philip Henriques
##Mrs Heneage, Miss Heneage
##Mrs Henderson
##Miss (Brydges) Henniker
##Mrs Philip Henriques
##Mrs Herbert, Miss Herbert
##Mr and Mrs Hermon-Hodge
##Miss Heron Maxwell (2)
##Mr G. T. Hertslet
##Mrs Hervey, Miss Hervey
##Misses Hervey-Bathurst (2)
##Mr and Mrs Heseltine, Miss Heseltine
##Miss Hickman
##Mr H. Higgins, Mr Cecil Higgins, Mrs Higgins
##Mrs Platt-Higgins
##Miss Gladys Higginson
##Mrs Hildyard
##Mrs Staveley Hill, Miss Hill
##Misses (Stock) Hill (2)
##Mrs Hills
##Mrs Hippisley
##Mr and Mrs E. Brodie Hoare, Misses (Brodie) Hoare (2), Mr G. Hoare, Mrs S. Hoare, Misses Hoare (2)
##Mr and Mrs H. Hobhouse
##Mr R. K. Hodgson
##Mr and Mrs C. D. Hohler
##Mr R. R. Holmes, Mrs Holmes
##Mr R. Hallett Holt
##Mr Maurice Holzman
##Misses Hood (2)
##Miss Margaret Acland Hood
##Miss Hooker
##Mr and Mrs E. Hope, Mr and Mrs Adrian Hope, Misses Adrian Hope (2), Mr and Mrs James Hope, Mr Hope, Miss Mary Hope, Miss Hope
##Mr and Mrs Beresford-Hope, Miss Agnes Beresford Hope
##Mr and Mrs W. H. Hornby, Misses Hornby (2)
##Mr and Mrs Horner
##Mr and Mrs Hornyold [sic]
##Mr J. C. Horsley
##Miss Jean Hotham
##Misses Houldsworth (2)
##Mr R. P. Houston
##Mr E. S. Howard, Mr and Mrs A. C. Howard, Mr Joseph and Mrs J. Howard, Mr and Mrs H. Howard, Mrs Howard, Misses Howard (2), Miss Howard, Misses Howard (2) [sic 2x]
##Mr J. Hozier
##Mr and Mrs G. B. Hudson
##Hughes, Misses Hughes (2)
##Mr and Mrs A. C. Humphreys-Owen
##Mr and Mrs Hungerford
##Miss M. Carew Hunt
##Mrs W. G. G. Hutchinson, Miss Hutchinson
##Mr and Mrs G. M. Hutton, Mr John Hutton, Mr A. E. Hutton, Mrs G. Hutton
##Mrs Inglefield
##Mr and Mrs Wootton Isaacson
##Mrs Jackson, Miss Grace Jackson
##Mr Jacobs
##Mrs Jacoby
##Mr and Mrs W. James, Mr Arthur and Mrs A. James, Miss Helena James
##Mrs J. E. Jameson, Misses Jameson (2)
##Mr and Mrs Jebb
##Mr and Mrs A. F. Jeffreys
##Mr E. R. Jenkins, Missess Jenkins (2)
##Mrs Jenkinson
##Miss Jenner
##Mr and Mrs H. C. Jervoise, Miss Jervoise
##Mrs Jessel, Miss Jessel
##Mrs Cotton-Jodrell, Miss Cotton-Jodrell
##Mr and Mrs J. H. Johnstone, Mrs G. Johnstone, Miss Johnstone, Miss S. L. Johnstone
##Mrs Joicey, Miss Joicey
##Misses Jolliffe (2)
##Mr and Mrs Atherley-Jones
##Mr and Mrs Brynmor-Jones
##Mrs Inigo Jones
##Mr Philip Burne Jones
##Mrs Pryce Jones
##Mr Henry Joslin
##Mr and Mrs Kearley
##Mrs Keeley
##Misses Keith-Falconer (2)
##Miss Kemball
##Mr George Kemp
##Mr C. Kempe
##Mr and Mrs A. Kennard, Mrs Hegan Kennard, Miss Kennard, Misses Kennard (2)
##Misses Kennaway (2)
##Mrs Kennedy, Miss Kennedy
##Mrs Kennion [?]
##Mrs Kennison
##Mr and Mrs W. Kenny, Miss Ethel Kenny
##Mr J. Kenyon
##Mrs Colin Keppel, Miss Keppel
##Misses Ker (2)
##Misses Kerr (2), Miss Nona Kerr
##Mrs Kilkelly
##Mr and Mrs Kimber, Miss Kimber
##Mr King King, Miss King King
##Mr Nigel Kingscote, Mr T. Kingscote
##Mrs Kingston
##Mrs Kitching
##Misses Kitson (2)
##Mr Lees Knowles and Mrs Knowles
##Mr Knowles
##Mr and Mrs Kuhe
##Mrs A. P. Lake
##Misses Lambart (2)
##Miss Aline Lambton
##Mr Landon
##Mrs Lane
##Mrs Langenbach
##Miss Larking
##Mrs Lascelles
##Mr W. F. Laurence
##Mr Edwin Laurence
##Misses Laurie (2)
##Mrs Laurier
##Mr and Mrs E. Law
##Mrs E. Lawrence, Misses Lawrence (2)
##Mrs Lawrie
##Mr J. Grant Lawson, Miss J. Lawson
##Mr and Mrs Lecky
##Mrs Hanning Lee, Misses Hanning Lee (2)
##Miss C. Lees
##Miss Leese
##Miss Violet Leigh
##Mr and Mrs S. Leighton, Misses Leighton (2)
##Mrs Leslie
##Mr L’Estrange, Miss l’Estrange
##Mr Letchworth
##Mrs Lewis, Misses Lewis (2)
##Mrs Naylor Leyland
##Mrs Liddell, Miss Liddell, Misses Liddell (2)
##Mrs Lidderdale, Misses Lidderdale (2)
##Misses Lindley (2)
##Mr Henry Gore Lindsay, Miss Gore Lindsay, Mr H. B. Lindsay, Mr W. A. Lindsay, Miss Lindsay, Miss Lindsay [sic 2x]
##Mr Leonard Lindsey
##Misses Linton (2)
##Miss Lister
##Mr Cecil Lister-Kaye
##Miss Llewelyn
##Mr E. Lloyd, Mrs Lloyd, Misses Lloyd (2)
##Miss Alice Loch, Miss Emily Loch
##Mrs Lockhart
##Mrs Lockwood, Miss Lockwood
##Mr Loder
##Miss Loftus
##Mr Heathcote Long and Mrs Long
##Mr H. T. Lopes, Misses Lopes (2)
##Mr Drury Lowe and Mrs Lowe, Miss Drury Lowe
##Mr and Mrs J. W. Lowther, Miss Aimee Lowther
##Mr E. H. Loyd, Mr and Mrs A. K. Loyd
##Mr and Mrs H Lubbock, Miss Lubbock
##Mr Reginald Lucas, Mrs Lucas, Mrs F. A. Lucas
##Mrs Lucas-Shadwell, Miss Lucas-Shadwell
##Mrs Luck
##Mr and Mrs Fairfax Lucy
##Mr H. Luttrell, Mr W. C. F. Luttrell, Mrs Luttrell, Miss Luttrell
##Miss Lyall
##Misses Lyell (2)
##Mr and Mrs Lyon
##Miss Lyson
##Misses Lyte (2)
##Mr W. G. E. Macartney
##Mr and Mrs  J. C. Macdona
##Miss Macdonald
##Mr Alpin Macgregor, Misses MacGregor(2), Miss Macgregor
##Mr and Mrs Muir Mackenzie, Miss Margaret Muir MacKenzie, Miss Mackenzie
##Mr Mackinnon
##Misses Mackworth (2)
##Mr and Mrs J. Maclean
##Mr and Mrs J. W. Maclure, Miss Maclure
##Mr and Mrs Frederick Macmillan
##Miss Macnaghten
##Miss Macpherson-Grant
##Mr Madden, Miss Madden, Misses Madden (2)
##Misses Magniac (2)
##Mr R. Maguire
##Mr Fuller Maitland
##Mr lan Z. Malcolm
##Mrs Malet, Miss Malet
##Mr B. Mallet
##Mr and Mrs G. Manners
##Mrs Newton Mant
##Mr and Mrs Marjoribanks, Mrs Majoribanks
##Mr T. C. March
##Mrs Markham, Misses Markham (2)
##Mr and Mrs H. H. Marks
##Mrs Marshall
##Mr and Mrs W. A. M’Arthur
##Mr Martin, Mrs R. B. Martin
##Miss Martyn
##Mr Mason
##Miss Massey-Mainwaring
##Mr C. Maud
##Mr and Mrs F. W. Maude, Mrs C. Maude, Miss Constance Maude
##Mrs Maurice
##Misses Maxwell (2)
##Mr and Mrs Maxwell-Lyte
##Mrs May
##Mr H. L. B. MCalmont, Mrs J. M'Calmont
##Miss M'Clintock
##Mrs J. M'Donald
##Mrs Meeking, Miss Meeking
##Mrs Mellor, Misses Mellor (2)
##Mr and Mrs Beresford Melville
##Mr and Mrs T. G. Menzies, Miss Menzies
##Mr and Mrs M'Ewan
##Mr and Mrs P. C. Milbank, Miss May Milbank
##Mr F. Bingham Mildmay, Mr and Mrs Bingham Mildmay, Miss Beatrice Mildmay
##Mr and Mrs Arundel St. John Mildmay, Miss St. John Mildmay
##Mrs Napier Miles
##Mrs Millett
##Mr and Mrs A. Milman, Miss Lena Milman, Misses Milman (2)
##Mr and Mrs Milvain
##Mrs Victor Milward, Miss Milward
##Mr A. B. F. Mitford, Miss Mitford
##Mr and Mrs M’Laren
##Mrs M'Neile
##Mr and Mrs C. M’Neill, Mrs M’Neill
##Mrs W. C. F. Molyneux
##Mrs G. Moncrieff
##Mr Monk, Misses Monk (2)
##Mr E. P. Monckton
##Mr Ronald Moncrieffe
##Miss Cicely Monson
##Mr V. Montagu, Misses Montagu (2)
##Mrs Montefiore
##Mr Montgomerie
##Mrs Montgomery, Miss Montgomery
##Mr Moon
##Miss Mary Moore
##Mrs Moorhouse
##Mrs Manvers Moorson
##Mr R. J. More
##Miss More-Molyneux
##Miss Evelyn Moreton
##Mr Charles Morley
##Mr and Mrs Morrell
##Mrs Ashurst Morris, Misses Morris (2), Miss Ethel Morris
##Mr Hugh Morrison
##Misses Moseley (2)
##Mrs Mostyn
##Mr and Mrs Mount, Misses Mount (2)
##Misses Mowatt (2)
##Miss Mowbray
##Mrs Max Muller
##Miss Mundella
##Mr and Mrs Campbell Munro, Miss Campbell Munro
##Mr and Mrs Muntz, Miss Muntz
##Mr and Mrs Murdoch, Miss Murdoch
##Mr and Mrs W. J. Mure, Mr Mure, Miss Mure
##Mr C. J. Murray, Mr G. H. Murray, Mrs Graham Murray, Miss Graham Murray, Mrs J. Murray, Mrs Wyndham Murray, Miss Murray
##Mr W. H. Myers
##Mr M. Myther
##Misses Nelson (2)
##Misses Nevill (2), Miss Nevill
##Mrs F. Neville, Miss Neville
##Mrs Nevul [?]
##Mr F. A. Newdigate
##Mrs Newhouse
##Mrs H. F. Nicholson
##Mr and Mrs Nicol, Miss Nicol
##Mr G. and Mrs Noel, Misses Noel (2)
##Mrs Nugent
##Mr T. W. Nussey
##Mrs Oakley, Miss Oakley
##Misses O’Brien (2)
##Mr J. C. O'Dowd
##Miss Ogilvy
##Miss V. A. Okeover
##Mrs H. H. Oldham
##Mr and Mrs M. Oldroyd
##Mr and Mrs Arthur Oliphant
##Misses Olpherts [?] (2)
##Mr and Mrs Oppenheim, Miss Linda Oppenheim
##Mr and Mrs Charles Orde
##Mr C. L. Orr-Ewing
##Miss Alina O'Shee
##Mr and Mrs R. A. Oswald, Mr and Mrs Oswald
##Miss Phoebe Otway
##Miss Humphreys Owen
##Mr Hussey Packe, Misses Packe (2)
##Mrs A. Paget, Mrs Paget, Miss Alice Paget, Miss Paget
##Misses Paget of Cranmore (2)
##Mrs Pakenham, Mrs Pakenham
##Miss Palgrave
##Mrs Dampier Palmer, Miss Palmer
##Mr Paoli
##Miss Parker
##Mr E.  and Mrs Parkes
##Mrs H. Parr
##Miss Parry
##Mr Paton
##Mr and Mrs J. L. Pattison, Miss Pattison
##Mr J. Balfour Paul
##Mr J. M. Paulton
##Mr Walter Peace, Miss Peace
##Mrs Peacocke [?]
##Mr Godfrey and Mrs G. Pearse
##Mr Joseph and Mrs J. Pease, Mr Arthur and Mrs A. Pease, Mr A. E. Pease, Miss Pease, Miss Pease
##Mr A. Peckover, Misses Peekover [?]
##Mr Archibald Peel, Mr Algernon Peel, Mrs A. Peel, Misses Peel (2), Miss Cecilia Peel
##Mrs Aldrich Pelham, Miss Anderson Pelham
##Misses Pelly (2)
##Mr J. and Mrs Pender
##Mr J. Penn, Miss Penn
##Mr Pennefather
##Mr Heber Percy
##Miss Pereira
##Mr and Mrs Perks
##Mrs Perowne, Miss Perowne
##Mr Henry Petre
##Mrs Peyton, Miss Peyton
##Mrs N. G. Philips
##Mr and Mrs Phellps, Miss Pollock Phellps
##Mr and Mrs Lort Phillips, Miss [?] Phillips
##Mr B. Faudel-Phillips, Mr L. Faudel-Phillips
##Mrs Phillpotts
##Mr and Mrs Constantine Phipps, Mr Charles Phipps, Miss Phipps, Mr and Mrs Wilton Phipps, Miss Wilton Phipps
##Mrs Pipon
##Mrs Pirie [?]
##Mrs Pitman
##Mrs Fox Pitt
##Mr Platt-Higgins
##Mrs Poe
##Miss Pole, Miss Chandos Pole
##Mr and Mrs Pollock
##Mr and Mrs John Ponsonby, Miss Julia Ponsonby, Miss Ponsonby
##Mr and Mrs Wyndham Portal
##Mrs F. Post, Miss Post
##Mr and Mrs Powell, Miss Baden Powell, Miss Powell (2)
##Mrs Powlett, Miss Powlett, Miss Orde Powlett
##Mr Herbert Praed
##Mr and Mrs Price, Mr and Mrs Montagu Price
##Miss Priestley
##Mr H. W. and Mrs Primrose
##Mrs Upton Prior
##Mr and Mrs Leslie Probyn, Miss Charlotte Probyn
##Mr Roland and Mrs R. Protheroe
##Mr Provand
##Mr A. V. Pryor
##Mr Guy Pym
##Miss Quain
##Mr and Mrs Quilter, Misses Quilter (2)
##Mr Henry and Mrs H. Raikes, Mrs Raikes, Miss Lucy Raikes
##Mr Pandeli Ralli
##Mr and Mrs Alexander Ramsay, Miss Ramsay
##Mr and Mrs J. Rankin, Miss Rankin
##Mr Read
##Mr and Mrs Rebow, Miss Rebow
##Mr G. A. Redford and Mrs Redford
##Miss K. Reiss
##Mr and Mrs J. Rennell Rodd, Mrs Rennell Rodd
##Mr and Mrs Renshaw
##Mr J. A. Rentoul
##Mr Repton
##Mrs I[?]. Ricardo, Misses Ricardo (2), Miss Ricardo
##Mrs Rice, Miss Beatrix Rice
##Mr H. C. Richards
##Mr T. Richardson, Mr and Mrs Richardson
##Mrs Riddel
##Miss Rigby
##Alderman and Sheriff Ritchie, Misses Ritchie (2)
##Mr A. T. Phillips Roberts
##Mr Forbes Robertson, Mr Edmund Robertson, Mrs Robertson, Miss Sibyl Robertson
##Mrs Robins, Miss Robins
##Mr Brooke Robinson, Mrs Robinson
##Miss Frances Rod
##Sheriff Hargreaves Rogers
##Mrs Rolfe
##[[Social Victorians/People/Fanny Ronalds|Mrs Ronalds]]
##Mrs James Ronand
##Mrs Carl Rosa
##Mrs Harcourt Rose
##Mr and Mrs Leopold Rothschild, Mr Alfred Rothschild
##Mr James Round, Misses Round (2)
##Mr Bowen Rowlands
##Mr and Mrs Royds, Misses Royds (2)
##Mrs Arnold Royle [? Royce?], Mrs G. Royle
##Mr Hugo von Ruffer
##Mr G. W. E. Russell, Mr H. J. H. Russell, Mr T. W. RusseII, Mrs J. C. Russell, Mrs F. Russell, Miss Russell, Misses Russell (2), Misses Russell (2)
##Misses Russell of Killowen (2)
##Mrs W. W. Russon[?]
##Mr John Rutherford and Mrs Rutherford
##Miss Jane Ryan
##Mr G. L. Ryder, Mr J. D. Ryder
##Mrs Salmon [?]
##Mrs Salmond, Misses Salmond (2)
##Mr and Mrs Salting
##Mr and Mrs H. S. Samuel
##Mr Albert and Mrs A. Sandeman, Mrs Sandeman, Misses Sandeman (2)
##Miss [Sanderson?]
##Mrs Sandford
##Mr and Mrs Sant, Miss Sant
##Misses Sar[?] (2)
##Misses Sartorius (2)
##Mr and Mrs R. Sassoon, Mr and Mrs Arthur Sassoon, Misses Sassoon (2)
##Miss Saumarez Smith
##Miss Truda Saunderson
##Miss Saurin
##Mrs Graves Sawle
##Mr and Mrs Scaramanga
##Mr Leo Schuster
##Mr P. L. Sclater, Miss P. L. Sclater
##Mrs Scobell
##Mr J. Murray Scott, Mrs Scott, Miss Maxwell Scott
##Mrs Seddon, Misses Seddon (2)
##Mr and Mrs C. H. Seely
##Mr and Mrs Senhouse
##Mrs Sergison [?]
##Mr and Mrs H. Seton-Karr
##Mrs Settle, Mrs Settle, 
##Miss Lily Severn
##Mr Horace and Mrs H. Seymour, Mrs L. Seymour, Miss M. Seymour, Misses Seymour (2), Miss Mabel Seymour
##Mr Lucas Shadwell
##Mr W. E. T. Sharpe
##Mr H. H. Shaw, Mr C. E. Shaw, Mr and Mrs Shaw, Miss Shaw
##Mr Michael Shaw-Stewart, Miss Shaw-Stewart
##Miss Shelley
##Mr and Mrs Shelley-Bontein
##Mrs Edgar Shephard
##Miss [Sheppart?]
##Mrs Brinsley Sheridan
##Miss Maud S[?]hey
##Mrs Teignmouth [?] Shore
##Miss Shute
##Misses Kay Shuttleworth [?] (2)
##Mr W. Sidebottom, Mr and Mrs T. H. Sidebottom
##Mr and Mrs Louis Sinclair
##Mr and Mrs T. Skewes-Cox
##Mrs Bridgman Simpson
##Mr and Mrs Skefflngton Smyth
##Mrs Slade, Mrs Frederick Slade
##Mrs P. L. Slater
##Mrs Hawley Smart
##Mr and Mrs Abel Smith, Miss Abel Smith, Mr and Mrs J. P. Smith, Mr A. H. Smith, Mr and Mrs H. C. Smith, Mr and Mrs T. Smith, Mr Smith, Mr G. D. Smith, Mr and Mrs Dudley Smith, Miss Dudley Smith, Mrs Graham Smith, Mrs C. Smith, Miss Smith, Miss Dorrien Smith, Smith (2), Miss [?]-Smith, Miss Smith (Clement), Miss Rachel Smith
##Mrs Smith-Barry
##Mr Philip Somers-Cocks
##Mr H. Somerset
##Mr Augustus Spalding
##Mr and Mrs E. B. Sparke, Miss Sparke
##Miss Sparkes
##Miss Ruby Spencer Churchill
##Mr and Mrs A. Spicer
##Misses Sprigg (2)
##Miss Stafford
##Mr and Mrs H. M. Stanley, Mr E. J. Stanley, Miss Magaret Stanley, Miss [Stanley?], Miss Evelyn Stanley
##Mrs Starkie
##Miss Evelyn Starling[?]
##Mrs St. Clair
##Mr and Mrs Leslie Stephen, Miss [?] Stephen [could this have been Virginia Woolf?]
##Misses Stephenson (2)
##Mrs Sterling, Miss R. Sterling
##Mr and Mrs J. Stern, Misses Stern (2)
##Mr and Mrs Stevenson, Mrs Stevenson, Miss Stevenson
##Mr and Mrs Steward
##Mr C. J. Stewart, Mrs A. C. Stewart, Mrs Stewart, Miss Stewart, Misses Stewart (2), Miss Nita Houston Stewart, Miss Hilda Stewart
##Mr Stibbert
##Mr and Mrs J. H. Stock
##Miss Dora Stone
##Mrs Stopford, Misses Stopford (2)
##Mr and Mrs Eames Storey
##Mr and Mrs E. Strachey
##Mr and Mrs J. Sturgis
##Mrs Napier Sturt
##Mrs Sullivan, Miss [Sullivan?]
##Mrs Surtees
##Mr F. Sutton
##Mrs Swaine, Miss Swaine
##Mr and Mrs J. A. Swettenham
##Miss Swinburne
##Mr Christopher Sykes
##Mrs R. F. Synge, Mrs Synge
##Mr Alma-Tadema
##Miss Satyendra Bala Tagore
##Miss Mary Talbot, Miss Talbot
##Mr and Mrs Tarleton
##Mr and Mrs F. Taylor, Mr John Taylor, Mrs J. W. Taylor, Mrs Taylor, Mrs Brook Taylor, Miss Taylor, Miss Ella Taylor, Misses Taylor (2)
##Mrs Temple, Miss Temple
##Mr H. J. Tennant
##Miss Ellen Terry
##Mr and Mrs Montagu Tharp
##Miss Thesiger
##Mr Abel Thomas, Mrs H. Thomas, Miss Ethel Thomas
##Mrs Thomson, Mrs Anstruther Thomson, Mrs C. F. Anstruther Thomson, Misses Thomson (2)
##Mr Montagu and Mrs M. Thorold
##Mr P. Thornton, Misses Thornton (2)
##Miss Thorold
##Mrs Thwaites
##Mr and Mrs J. C. Thynne, Mr and Mrs C. E. Thynne, Mr F. J. Thynne, Mrs R. T. Thynne, Miss Rachel Thynne, Miss Thynne, Misses Thynne (2), Miss Thynne
##Mr and Mrs Edward Tighe
##Mrs Tillard
##Mrs Tillbrook
##Mr W. H. Wilson-Todd
##Mr and Mrs H. Graham Toler
##Mr H. F. Tollemache
##Mr W. E. M. Tomlinson, Miss Tomlinson
##Mr A. M. Torrance
##Mr and Mrs Tosti
##Mr and Mrs Christopher Tower
##Mr and Mrs Beerbohm Tree
##Miss Adela[?] Trefusis
##Mr H. D. Trelawny, Miss Trelawny [?]
##Mr Tremayne, Misses Tremayne (2)
##Mr A. J. R. Trendell
##Mr and Mrs C. E. Tritton, Misses Tritton (2)
##Mr and Mrs C. W. Trotter
##Mr and Mrs Tudway
##Mr and Mrs Dan Tupper, Mrs Tupper, Miss G. le M. Tupper
##Miss Turner
##Mr Algernon Turnor
##Miss E. Tuson
##Mr and Mrs A. Ure
##Mr T. Usborne
##Mr and Mrs T. Usher, Miss Usher
##Mr and Mrs Val Prinsep
##Miss Van [der Byl?]
##Mr Van De Weyer, Misses Van de Weyer (2)
##Mr L. Van Loon
##Mr and Mrs Chas van Raalte
##Mrs Vance
##Mrs Edmund Vaughan
##Mr and Mrs Venning, Miss Venning
##Mr and Mrs Hope Vere
##Mrs Verschoyle, Miss Verschoyle
##Mr and Mrs F. E. Villiers, Mrs E. Villiers, Mrs Villiers, Miss Dorothy Villiers, Miss Freda Villiers, Miss Villiers
##Mr Graham Vivian, Mrs R. Vivian, Misses [Vivian?] (2)
##Mr and Mrs Von André
##Miss Hilda von Deichmann
##Mr R. C. de Grey Vyner
##Mr Charles Waldstein
##Mr F. Walker, Miss F. Walker, Miss Smart Walker, Miss Walker
##Mrs Wallis
##Mr and Mrs Spencer Walpole, Miss Maud Walpole
##Misses Walrond (2)
##Mr and Mrs Walter
##Mr and Mrs Walton
##Mr Wanklyn
##Mr Piers Egerton Warburton
##Mr and Mrs J. Humphrey Ward, Mrs C. [?] E. Ward, Miss Ward
##Mr and Mrs Lee Warner, Mrs Warner
##Mr A. F. Warr
##Mrs Warre
##Misses Warren (2)
##Misses Warrender (2)
##Mrs Watson, Miss Watson
##Mrs S. J. Way
##Mr Thomas Wayman
##Mr Godfrey Webb
##Mr and Mrs R. G. Webster, Misses Webster (2)
##Mr Weigall, Miss Rachel Weigall
##Mr and Mrs John Welby, Mr and Mrs J. Welby
##Mr W. West
##Mr and Mrs Cornwallis West, Miss Cornwallis West
##Mr and Mrs Sackville West
##Miss Evelyn Wellesley
##Mrs Wells
##Mrs F. Charteris Wemyss
##Mrs Weywan
##Mrs Wharton
##Mr Whateley
##Mrs Whatley
##Mrs Whatman, Misses Whatman (2)
##Misses Whitehead (2)
##Mr and Mrs G. Whiteley, Mrs Herbert Whiteley, Mrs H. Whiteley
##Mr Whitbread
##Mrs Wickham
##Mrs Wilberforce, Misses Wilberforce (2), Miss Wilberforce
##Miss Wilbraham
##Mr and Mrs Hwfa Williams, Mr Powell Williams, Mrs Williams, Mrs Ellis Williams, Miss Williams
##Misses Wills (2)
##Mrs Eardley Wilmot, Miss Eardley-Wilmot
##Mr and Mrs A. S. Wilson, Mr G. Fleetwood Wilson, Miss Fleetwood Wilson, Mr F. W. Wilson, Mr J. W. Wilson, Mr A. Wilson, Mr and Mrs John Wilson, Mr and Mrs C. H. Wilson, Mrs Wilson, Miss [?hend?] Wilson, Miss Wilson
##Mr and Mrs Wingfleld
##Mr Wodehouse, Mrs E. F. Wodehouse
##Mr S. Wombwell, Miss Wombwell
##Mrs Wood, Mrs Charles Wood, Misses Wood (2)
##Miss de la Wood[?]
##Mr W. Woodall
##Mrs J. Woodford, Miss Wood[ford?]
##Mr H. C. Woods, M.D.
##Misses [Workham?] (2)
##Mrs Stuart Wortley
##Mrs Wray
##Mr Wylie
##Mrs Williams Wynn
##Mr and Mrs Wynne
##Mr D’Arcy Wyvill, Misses [Wyvill?] (2)
##Mr Wyndham
##Miss Yeatman
##Mr and Mrs Yerburgh
##Mr H. Yorke
##Messieurs , , , , , , , , , , , , , , , , , Deputy Inspector-General, Charles Wyndham
#Mesdames&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7a–b) — , , , , , , , , Mrs Charles Wyndham
#Misses&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7c – 5, Col. 1a) —  , , , , , , , , , [?], Muriel [?], [?], [?], I. C. (2), , , Wynd[ham?] (2), [W?Ieyer?] (2), [?] (2)
#Admirals of the Fleet [initial large caps, rest sm caps] — Earl of Clanwilliam, Lord John [Hay?], the Hon. Sir H. Keppel
#Admirals — H. G. Andoe, C. E. Buckle, Sir F. Bedford, Britten, the Hon. W. Carpenter, H. F. Cleveland, Sir H. Chads, Close, [?], Carr, E. J. Church, Sir W. Dowell, R. G. Douglas, A. L. [?], C. E. Domvile, A. T. Dale, D’Eyncourt, Field, Sir A. [Farquhar?], Fitzgerald, Fellowes, Fanshawe, Sir H. Fairfax, Sir [?] Fisher, C. J. Fane, Fullerton, the Hon. Sir E. Fremantle, [?] FitzGeorge, Woods Pasha, Sir W. Hunt-Grubbe, Sir Anthony [?] Hoskin, Lord Hood of Avalon, Sir Leopold Heath, Sir [?] [F.?] Hotham, Sir Algernon Heneage, R. H. Hamond, the Right Hon. Sir [J.?] Hay, St. G. C. D’Arcy Irvine, Jones, Kennedy, Sir A. [?s], A. P. Lake, R. M. Lloyd, Sir L. Loraine, A. H. Markham, [Sir?] R. More-Molyneux, Sir F. L. M'Clintock, Sir R. Macdonald, [the?] Hon. V. Montagu, Nicholson, Noel, Marquis of Northampton, Sir E. Ommaney [?], Sir Augustus Phillimore, A. T. Powlett, [?], [?. ?.] Rowley, Sir F. Richards, Lord Charles Scott, [? St.? John?], W. H. C. St. Clair, Bowden Smith, Sulivan, E. H. Sey[mour?], H. Stephenson, Sir Nowell Salmon, Sir W. Houston [Stewart?], Sir M. [Cuhne?]-Seymour, E. W. Turnour, E. W. Van[?] Wharton, Sir G. Willes, the Hon. W. J. Ward
#Captain, R.N. — W. A. D. Acland, C. J. Barlow, F. R. Board[?], H. Bainbridge, Hon. T. Brand, Bickford, Lord Charles [B?ford?], B. F. Clark, Colville, Carter, Hon. S. Cecil Colville, [?ford?], A. G. Douglas, Sir C. Domville, Hon. A. Hay Dru[?], [?] [W.?] [?] Gordon, Hammet, Hon. Curzon Howe, Hender[?], [?] Ingles, Jellicoe, Jephson, Johnstone, Jeffreys, H. C. [?], Hon. A. Littleton, Hon. Hedworth Lambton, Moore, May, [? Net?], Poe, Pipon [?], Aldrich Pelham, Alfred Paget, [Bi.idcl?], Rolleston, John Sinclair, Bridgeman Simpson, [?], Van Koughnet [?], Burges Watson, Eardley-Wilmot, [?ham, Winsloe, Hon. J. Yorke
#[Lieutenants???] — Anson, G. R. Bethell, Blair, Bayley, Cave[?], [?] Cave,Hon. Cecil Cadogan, de Salis, Fraser, Floyd, Hon. [?] [F?], Alaric Grant, Morgan, Moore, Marescaux, [?] Stuart, Tupper, Wells, Williams, G. J. S. Warrender
#[Lieutenants?] R.N. — Alton, Murray Aynsley, Boyle, Bather, [?], [R. F.?] Boyle, Chaytor, Sir Charles Cust, G. W. Davy, [?] Wyndham-Fiennes, Fair, Godfrey Faussett, Garforth, [L?]ord Clifford, Hopkinson, Henderson, Keyes, Keppel, [?] Lloyd, Majendie, Mitchell, Morant, Kerr-Pearse, [?] Richmond, Rae, Stewart, Hon. Victor Stanley, [?] [Calta?]-Seymoar, Trye, Thring, Hon. Cyril Ward, W[?], R. E. Wemyss, Woolcombe
#[Captain?] Trinity House, Sir J. Sydney Webbe
#[Field?] Marshall — Sir F. P. Haines, Sir Lintorn Simmons, Sir [?] Stewart, Lord Roberts of Kandahar, Viscount Wolseley
#[Generals?] —Sir J. Ardagh, Sir A. Alison, Sir H. J. Alderson, [?n] Annesley, J. Alleyne, Sir J. M. Adye, Sir C. G. [Arbuth?]not, Sir H. Havelock-Allan, R. Bateson, Sir W. F. [B?er, Sir H. Brackenbury, H. M. Bengough, the Right Hon. [?] Buller, Sir Owen Tador-Burne, H. J. Buchanan, Sir C. H. [Brown?low], Sir S. Browne, Sir M. Biddulph, Viscount Bridport, [?. O.?] Barnard, E. F. Chapman, Lord Clarina, C. F. Clery, the Hon. S. Gough-Calthorpe, E. H. Clive, Godfrey Clerk, Lord [Ch?]sford, the Hon. Sir Andrew Clarke, Sir E. Du Cane, Crutchley [?], Lord de Ros, Sir John Donelly, J. H. Dunne, Sir Martin Dillon, Sir Collingwood Dickson, Sir H. de Bathe, Davis, Sir F. de Winton, Sir T. Dennehy, Sir H. Ewart, Sir J. B. Edwards, C. B. Ewart, Cecil East, Arthur French, Sir T. Fitz-Wygram, the Hon. Sir P. Feilding, Sir T. E. Gallwey, Sir T. Goldsmid, Sir R. Gipps, Sir R. Grant, Sir F. W. Grenfell, Coleridge Grove, Goldsworthy, J. J. H. Gordon, Sir E. A. Holdich, Sir E. W. Higginson, Sir R. J. Hay, Sir R. Harrison, Julian Hall, Earl Howe, the Hon. W. Home, J. Jameson, Sir Arnold Kemball, Kelly-Kenay, Lord Mark Kerr, F. T. Lloyd, Sir D. Lysons, Sir Drury Lowe, G. Luck, J. W. Laurie, F. Marshall, the Hon. R. Monck, Crichton Maitland, Sir J. M'Neill, Montgomery, the Hon. S. Mostyn, G. Moncrieff, E. Markham, Sir W. A. Mackinnon, Bryan Milman [?], H. M’Calmont [?], M'Donnell, W. C. F. Molyneux, Lord [Methuen?], J. F. Maurice, Sir F. Middleton, O. H. Nicolls, Sir E. [?] Newdegate, Sir H. N[orman?], Sir W. Olpherts, F. Peyton [?], G. [?] Upton Prior, T. H. Pakenham, G. W. T. Rich, Lord [?der] Russell, Robinson, Rowlands, J. C. Russell, F. [Russell?], A. C. Stewart, Sir Henry Smyth, Sterling, Sir C. [?] Shute, N. Stevenson, Swaine, Lord William Seymour, [?] [Sahmond?], Sir Frederick Stephenson, Sir John Stokes, Sir R. [?], Sir H. B. Tuson, the Hon. R. A. J. Talbot, G. le M. [Tupper?], Taylor, Hon. C. Thesiger, R. T. Thynne, Upperton, [?]H. Utterson, Sir J. Watson, Sir C. W. Wilson, Sir F. F. Walker, Sir Evelyn Wood, Sir C. Warren, Albert Williams, the Hon. G. Wrottesley, Sir G. H. Willis, Sir H. Wilmot
#Colonels — Armytage, Arkwright, Pat Boyle, Burges, the Hon. [?] Byng, H. B. H. Blundell, M. S. Brownrigg, Sir E. Bradford, Sir A. [Blyge? Bigge?], the Hon. F. Bridgeman, Brassey, Lord William Beresford, St. John Barne, N. Barnardiston, Lord Blythswood, [?] Cunynghame, F. H. Custance, Clayton, Sir Henry Colville, [?] Carnac [?], Cavaye, Seymour Corkran, the Hon. Charles [?], W. Campbell, Chaloner, Archibald Calvert, the Hon. [?] Campbell, the Hon. Wenman C. Coke, the Hon. W. [?ton], the Hon. Sir W. Colville, Chaine, A. B. Crosbie, [T.?] [R?] Crosse, Lord Edward Pelham Clinton, the Hon. Henry [C?hton], E. H. Cooper, the Hon. H. Corry, John Clerk, Lord Dorchestcr, C. R. Dease, the Hon, Lewis Dawnay, [the?] Hon. H. Denison, Denny, Dalbiac, A. Davidson, the Hon. Cathbert Edwards, the Right Hon. Sir F. Edwards, [?son], R. Edis, the Hon. Charles Edgecumbe, Aubone Fife, [?], Wynne Finch, Ferguson of Pitfour, Forster, Lancelot [?r] H. Frudyer, Barrington Foote, Goldsmid, Gore, Grenfell, [?n], C. G. Gordon, R. Gunter, Alan Gardner, Hon. G. Gough, [?] [?iton], the Hon. A. Hood, the Earl of Home, Lord Claud [Hamilton?], Harford, Herbert, the Earl of Haddington, Haygarth, G. Hatton [?], Hillyard, Arthur Haig, Sir E. Stock Hill, R. Hennell, Archer Houblon [?], the Hon. Cospatrick Home, the Hon. C. Gathorne-Hardy, Johnstone, Cotton-Jodrell, Hegan, [H?nard], Sir N. Kingscote, H. A. Lascelles, the Hon. Heneage [L?], Hanning Lee, F. A. Lucas, the Hon. H. Lyttelton, Lockwood, L. V. Loyd, C. W. Long, Ronald Lane, Lucas, J. Leslie, the Hon. Caryl [?]Molyneux, John Murray, Sir A. W. Mackworth, J. M'Calmont [?], Milward, the Hon. F. C. Morgan, J. J. Mellor, Meeking, Manvers [?], Moorsom, H. Malet, the Earl of Mount Edgecumbe, the [Earl?] of March, Wyndham Murray, Sir V. Majendie, the Hon. G. [Napper?], H. H. Oldham, L. J. Oliphant, A. Paget, Dampier Palmer, [Earl?] Percy, George Paget, C. D. Patterson, Arthur Peel, [Birch?] [Richardson?], the Hon. F. W. Stopford, Sir W. G. Stirling, E J. [Sanderson?], T. M. Sandys, H. Smith, J. F. Sandeman, Renyon-[Surrey?], C. E Stewart, E. H. Sartorius, the Hon. Walter [Stewart?], L. Seymour, Settle, Stevenson, Starkie, C. H. Seafe, the Hon. Sir W. P. Talbot, J. Du Plat[?] Taylor, H. Thomas, A. W. [T?], the Hon. W. Ie Poer Trench, H. P. Vance, Sir C. E. Howard Vincent, M.P.; R. Vivian, A. P. Vivian, E. Villiers, the Duke of Westminster, the Earl of Wemyss, Lord Wantage, Ward, [Waring?], [Earle?] Welby, Lord Arthur Wellesley, Robert Williams, the Hon. H. L. Wood, Sir W. H. Walroud, F. Smart Walker, A. [Williams?] Wynn, Wardrop
#Majors — Anne, Atherley, Ashton, F. H. Bowles, the Hon. [?] R. Bourke, Carnegy, H. Candy, Close, the Hon. F. Colborue, the Hon. Wenman Coke, Lawrence Drummond, Alfred [Edgecombe?], G. Egerton, E. H. Elliot, the Hon. A. Henniker, J. [H?a?h], the Hon. Assheton Harbord, the Hon. North Dalrymple [Hamilton?], Jameson, Pryce Jones, Larnach, the Hon. Osbert [Lumley?], C. Little, Marindin, the Hon. J. Scott Napier, Wyndham Quin, F. C Rasch, the Hon. A. Sidney, the Hon. J. T. St. Aubyn, Sir Edgar Sebright, Stirling, T. E. M. Swinnington-Parkington, [?.] M. Temple, Tillbrook, Anstruther Thomson, [E.?] [L.?] Woodhouse, and the Marquis of Winchester
#Captains — O. Ames, J. Acland, Alan Boisragon, Bates, H. M. [Biddulph?], the Hon. Baring, Butler, the Hon. J. Byng, the Hon. [N.?] Yarde-Butler, E. W. Blunt, J. F. Bagot, the Hon. W. Bagot, Seymour Combe, W. Chetwynd, Dundas, Denis Daly, Cecil Drummond, M. Drummond, Ellison, Houston French, Gye, R. G. [Gilmour?], P. Green, W. G. Grice-Hutchinson, Ahmed Hussain, G. [L.?] Holford, Jessel, the Hon. W. Lambton, the Hon. G. H. [L?], Sir H. Naylor-Leyland, G. Lister, Matthews, A. D. Miller, [?],M. M'Neill, C. Norton, Phillpotts, N. G. Philips, Prety[man?], Duncan Pirie, Pitman, Fox Pitt, Petre, Harcourt Rose, [W.?] [J.?] Stopford, Sir Eyre Shaw, H. G. D. Shute, Spicer, the Hon. [?.] St. Aubyn, Sutton, Tillard, Webbe, Wray, and Gordon [Watson?]
#Lieutenants — Baun, A. Cowell, the Hon. E. C. Lennox, F. Ponsonby, J. Ponsonby, Vandeleur, the Hon. C. Willoughby, and the Hon. C. S. H. D. Willoughby
===Entertainment===
&quot;The Bands of the 1st Life Guards, Grenadier Guards, and Royal Artillery played a selection of music during the afternoon.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)

==Anthology==

====Quote Intro====
&lt;quote&gt;&lt;/quote&gt; ()

== Notes and Questions ==
# 

==References==
*
&lt;references /&gt;</text>
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  <page>
    <title>Linear algebra (Osnabrück 2024-2025)/Part I/Forum</title>
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== Quantum me ==


[[Special:Contributions/41.210.146.80|41.210.146.80]] ([[User talk:41.210.146.80|discuss]]) 19:13, 19 December 2024 (UTC)</text>
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        <username>Bocardodarapti</username>
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    <title>Boolf prop/3-ary</title>
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      <text bytes="17320" sha1="gxe6319c75peb5rgzjorlupio168yjf" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692651</id>
      <parentid>2692647</parentid>
      <timestamp>2024-12-19T20:03:22Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692651</origin>
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      <text bytes="17715" sha1="mg6yy442ja4xkel216yru66ags34oi8" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
      <sha1>mg6yy442ja4xkel216yru66ags34oi8</sha1>
    </revision>
    <revision>
      <id>2692652</id>
      <parentid>2692651</parentid>
      <timestamp>2024-12-19T20:05:51Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
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      <text bytes="17972" sha1="0a7ixeuq8czwqustjhqzdgu6khdad08" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692658</id>
      <parentid>2692652</parentid>
      <timestamp>2024-12-19T20:13:34Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692658</origin>
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      <text bytes="18229" sha1="tug772vxfvxpawrldcuva0r5pwlu1dj" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
      <sha1>tug772vxfvxpawrldcuva0r5pwlu1dj</sha1>
    </revision>
    <revision>
      <id>2692679</id>
      <parentid>2692658</parentid>
      <timestamp>2024-12-19T20:34:16Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692679</origin>
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      <text bytes="19787" sha1="1lchhrcbv7982t7d22wipm5jwuzexnc" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692705</id>
      <parentid>2692679</parentid>
      <timestamp>2024-12-19T21:02:00Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <text bytes="20888" sha1="i1a2wuxespphr5o7pcsvkffm1ij1jhb" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
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|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
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|-
|class=&quot;number-of-blocks&quot;| 46
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    <revision>
      <id>2692709</id>
      <parentid>2692705</parentid>
      <timestamp>2024-12-19T21:25:06Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <text bytes="21163" sha1="c2j7hmhsl6kqjh6hs5qf0465vdcndfj" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692714</id>
      <parentid>2692709</parentid>
      <timestamp>2024-12-19T21:53:46Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <text bytes="21503" sha1="is3v7kwt3xvwspai4cewbunfdg78coj" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
      <sha1>is3v7kwt3xvwspai4cewbunfdg78coj</sha1>
    </revision>
    <revision>
      <id>2692718</id>
      <parentid>2692714</parentid>
      <timestamp>2024-12-19T21:59:57Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692718</origin>
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      <text bytes="21909" sha1="7gt1nof4w0vjfi67pblki7s9l2ep75a" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 30
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 8, 8, 14, 16, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super family|super family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692723</id>
      <parentid>2692718</parentid>
      <timestamp>2024-12-19T22:20:54Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
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      <text bytes="22987" sha1="qq8r4mbwoolhw0sh0gkffuwbn25ub0s" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 14
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super clan|super clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 18
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 8, 6, 16, 4, 32, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra family|ultra family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 30
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 8, 8, 14, 16, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super family|super family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692725</id>
      <parentid>2692723</parentid>
      <timestamp>2024-12-19T22:21:53Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 8, 2, 16, 2, 32, 3, 40, 1, 64, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra clan|ultra clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 14
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super clan|super clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 18
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 8, 6, 16, 4, 32, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra family|ultra family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 30
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 8, 8, 14, 16, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super family|super family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692735</id>
      <parentid>2692725</parentid>
      <timestamp>2024-12-19T22:43:49Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor weight|quaestor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor weight|praetor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron index weight|patron index weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 8, 2, 16, 2, 32, 3, 40, 1, 64, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra clan|ultra clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 14
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super clan|super clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 18
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 8, 6, 16, 4, 32, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra family|ultra family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 30
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 8, 8, 14, 16, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super family|super family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692740</id>
      <parentid>2692735</parentid>
      <timestamp>2024-12-19T23:01:13Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <text bytes="25181" sha1="ci3zryx7pqprff50g65olok7u86alqv" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect weight|prefect weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor weight|quaestor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor weight|praetor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron index weight|patron index weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 8, 2, 16, 2, 32, 3, 40, 1, 64, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra clan|ultra clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 14
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super clan|super clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 18
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 8, 6, 16, 4, 32, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra family|ultra family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 30
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 8, 8, 14, 16, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super family|super family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692741</id>
      <parentid>2692740</parentid>
      <timestamp>2024-12-19T23:10:36Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692741</origin>
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      <text bytes="25442" sha1="jg9fzrz0zp85j8q8dspt5k986d5rlcj" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect weight|prefect weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guardian|guardian]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor weight|quaestor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor weight|praetor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron index weight|patron index weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 8, 2, 16, 2, 32, 3, 40, 1, 64, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra clan|ultra clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 14
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 6, 2, 8, 2, 16, 2, 24, 4, 48, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super clan|super clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor|praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;praetor sword&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 3|nameless 3]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 4|nameless 4]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 18
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 8, 6, 16, 4, 32, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra family|ultra family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 4, 12, 12, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/squad|squad]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[10, 16, 24, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guild|guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 22
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 1, 4, 2, 6, 2, 8, 5, 12, 4, 24, 6]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/clan|clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 30
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 8, 8, 14, 16, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/super family|super family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/principality|principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 8, 3, 8, 4, 8, 6, 8, 12, 12]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 46
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 2, 7, 4, 14, 8, 23]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;7&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;23&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/family|family]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 64]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 80
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 16, 3, 48, 6, 16]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;48&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/faction|faction]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
      <sha1>jg9fzrz0zp85j8q8dspt5k986d5rlcj</sha1>
    </revision>
    <revision>
      <id>2692745</id>
      <parentid>2692741</parentid>
      <timestamp>2024-12-20T00:08:28Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692745</origin>
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      <format>text/x-wiki</format>
      <text bytes="25725" sha1="94uh7hxm6w4d7ajb02y1sk2q1lptul7" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_noble|is noble]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_linear|is linear]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[24, 1, 232, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;232&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_dense|is dense]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[57, 1, 199, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;57&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;199&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_honest|is honest]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[62, 1, 194, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;62&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;194&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blotless|is blotless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_dominion|great quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 1, 192, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;192&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#great_quaestor_sword_dominion|great quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[66, 1, 190, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;66&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;190&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_bloatless|is bloatless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[96, 1, 160, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;160&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_blightless|is blightless]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[97, 1, 159, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;97&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;159&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_male|is male]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_acute|is acute]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odd|is odd]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_odious|is odious]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_ugly|is ugly]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_rough|is rough]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_sharp|is sharp]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 2
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[128, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/binary#is_solid|is solid]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 1, 6, 1, 30, 1, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;30&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/valency|valency]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 2, 12, 1, 240, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;240&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/adicity|adicity]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword dominion|quaestor sword dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron dominion|great patron dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron principality|great patron principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul weight|consul weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 2, 96, 2]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect weight|prefect weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor|lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/lictor sword|lictor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor shield|praetor shield]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/reverse lictor|reverse lictor]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 5|nameless 5]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight quadrant|weight quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/guardian|guardian]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron index consul|patron index consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor weight|quaestor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/praetor weight|praetor weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 2, 64, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron index weight|patron index weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[2, 4, 10, 3, 218, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;218&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/atomvals|atomvals]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[8, 2, 24, 2, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/company|company]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor dominion|quaestor dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[16, 4, 48, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron dominion|patron dominion]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron principality&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron king index and quadrant&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor sword|quaestor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/leveled praetor sword|leveled praetor sword]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/octant|octant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 8
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[32, 8]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/consul|consul]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 9
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[1, 2, 8, 2, 28, 2, 56, 2, 70, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;28&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;56&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;70&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/weight|weight]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 8, 2, 16, 2, 32, 3, 40, 1, 64, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/ultra clan|ultra clan]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
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[[Category:Boolf prop/3-ary| ]]</text>
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  <page>
    <title>Complex Analysis/Chain</title>
    <ns>0</ns>
    <id>317282</id>
    <revision>
      <id>2692599</id>
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      <timestamp>2024-12-19T13:10:38Z</timestamp>
      <contributor>
        <ip>139.14.137.206</ip>
      </contributor>
      <comment>/* Definition - Chain */</comment>
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      <text bytes="3771" sha1="eq2v8no8ulnp86qoe8224ubrc28nmfb" xml:space="preserve">A chain is a formal linear combination of[[Complex Analysis/Trace|Trace of Curve]], we have

== Definition - Chain == 
Let &lt;math&gt;G \subseteq \mathbb C&lt;/math&gt;, let &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and let &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; be curves in &lt;math&gt;G&lt;/math&gt; and &lt;math&gt;n_i\in \mathbb Z&lt;/math&gt;. Then the [[formal linear combination]] &lt;math&gt;\sum_{i=1}^n n_i\gamma_i&lt;/math&gt; is called a chain in &lt;math&gt;\mathbb C&lt;/math&gt;. The set of all chains in &lt;math&gt;G&lt;/math&gt;, which is naturally an abelian group, is denoted by &lt;math&gt;C(G)&lt;/math&gt;.

== Definition - Trace of a Chain ==
The ''trace'' of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the traces of the individual curves &lt;math&gt;\gamma_i&lt;/math&gt;, i.e.

&lt;center&gt;&lt;math&gt; \mathrm{Trace}(\Gamma) := \bigcup_{i=1}^n \mathrm{Trace}(\gamma_i) &lt;/math&gt;&lt;/center&gt;
==Cycle== 
A chain &lt;math&gt;\Gamma = \sum_{i=1}^n n_i \gamma_i \in C(G)&lt;/math&gt; with &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; is called a cycle if each point of &lt;math&gt;G&lt;/math&gt; occurs equally often as the starting and ending point of curves in &lt;math&gt;G&lt;/math&gt;, i.e., if

&lt;center&gt;&lt;math&gt; \sum_{i=1}^n n_i |\{i: \gamma_i(a_i) = z \}| = \sum_{i=1}^n n_i|\{i: \gamma_i(b_i) = z\}| &lt;/math&gt;&lt;/center&gt;
holds for every &lt;math&gt;z \in G&lt;/math&gt;.
===Interior and Exterior Region=== 
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb C&lt;/math&gt;, with the help of the [[w:en:winding number|winding number]] one can consider a decomposition of &lt;math&gt;\mathbb C&lt;/math&gt; into three parts determined by &lt;math&gt;\Gamma&lt;/math&gt;, namely:

*The image set of &lt;math&gt;\mathrm{Trace}(\Gamma)&lt;/math&gt;

*The ''exterior region'', those points that are not traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; A_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) = 0}&lt;/math&gt;&lt;/center&gt;

*The ''interior region'' consists of those points that are traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; I_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) \ne 0}&lt;/math&gt;&lt;/center&gt;
== Page Information ==
You can display this page as 
'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Chain&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Chain&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''  were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' 
and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Chain https://en.wikiversity.org/wiki/Complex%20Analysis/Chain]
--&gt;
* [https://en.wikiversity.org/wiki/Complex%20Analysis/Chain This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Kette|Kurs:Funktionentheorie/Kette]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Kette&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;08:35&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Kette]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
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    </revision>
    <revision>
      <id>2692604</id>
      <parentid>2692599</parentid>
      <timestamp>2024-12-19T13:41:05Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <comment>/* Interior and Exterior Region */</comment>
      <origin>2692604</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3752" sha1="oxm83l9h6zjavc9osjgxsd2rbt3e29x" xml:space="preserve">A chain is a formal linear combination of[[Complex Analysis/Trace|Trace of Curve]], we have

== Definition - Chain == 
Let &lt;math&gt;G \subseteq \mathbb C&lt;/math&gt;, let &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and let &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; be curves in &lt;math&gt;G&lt;/math&gt; and &lt;math&gt;n_i\in \mathbb Z&lt;/math&gt;. Then the [[formal linear combination]] &lt;math&gt;\sum_{i=1}^n n_i\gamma_i&lt;/math&gt; is called a chain in &lt;math&gt;\mathbb C&lt;/math&gt;. The set of all chains in &lt;math&gt;G&lt;/math&gt;, which is naturally an abelian group, is denoted by &lt;math&gt;C(G)&lt;/math&gt;.

== Definition - Trace of a Chain ==
The ''trace'' of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the traces of the individual curves &lt;math&gt;\gamma_i&lt;/math&gt;, i.e.

&lt;center&gt;&lt;math&gt; \mathrm{Trace}(\Gamma) := \bigcup_{i=1}^n \mathrm{Trace}(\gamma_i) &lt;/math&gt;&lt;/center&gt;
==Cycle== 
A chain &lt;math&gt;\Gamma = \sum_{i=1}^n n_i \gamma_i \in C(G)&lt;/math&gt; with &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; is called a cycle if each point of &lt;math&gt;G&lt;/math&gt; occurs equally often as the starting and ending point of curves in &lt;math&gt;G&lt;/math&gt;, i.e., if

&lt;center&gt;&lt;math&gt; \sum_{i=1}^n n_i |\{i: \gamma_i(a_i) = z \}| = \sum_{i=1}^n n_i|\{i: \gamma_i(b_i) = z\}| &lt;/math&gt;&lt;/center&gt;
holds for every &lt;math&gt;z \in G&lt;/math&gt;.
===Interior and Exterior Region=== 
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb C&lt;/math&gt;, with the help of the [[winding number]] one can consider a decomposition of &lt;math&gt;\mathbb C&lt;/math&gt; into three parts determined by &lt;math&gt;\Gamma&lt;/math&gt;, namely:

*The image set of &lt;math&gt;\mathrm{Trace}(\Gamma)&lt;/math&gt;

*The ''exterior region'', those points that are not traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; A_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) = 0}&lt;/math&gt;&lt;/center&gt;

*The ''interior region'' consists of those points that are traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; I_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) \ne 0}&lt;/math&gt;&lt;/center&gt;

== Page Information ==
You can display this page as 
'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Chain&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Chain&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''  were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' 
and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Chain https://en.wikiversity.org/wiki/Complex%20Analysis/Chain]
--&gt;
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&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Kette&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;08:35&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Kette]]
&lt;/noinclude&gt;

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      <text bytes="9670" sha1="hs3qjxjzgk2o3kezmyj2ifok0942faf" xml:space="preserve">== Introduction ==
'''Chain''' and '''cycle''' are mathematical objects studied in [[Complex Analysis|Complex Analysis]] but also appear as special cases in [[w:en:Algebraic topology|Algebraic topology]]. A chain generalizes a [[w:en:Path (topology)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.

== Definitions ==
=== Chain ===
A chain on a set &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt; is defined as a finite integer linear combination of paths &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt;:
&lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}&lt;/math&gt;.
&lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; are generally continuous [[w:en:Path (topology)|curves]] in &lt;math&gt;G&lt;/math&gt;.

=== Integration over a chain ===
Let &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt; be integrable, and let &lt;math&gt;\Gamma&lt;/math&gt; be a chain of piecewise continuously differentiable paths (paths of integration) &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; in &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;. The integral over the chain &lt;math&gt;\Gamma&lt;/math&gt; is defined by:
:&lt;math&gt;\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz&lt;/math&gt;

=== Definition: Cycle ===
'''Version 1:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt;, where every point &lt;math&gt;a \in \mathbb{C}&lt;/math&gt; appears as the starting point as many times as it appears as the endpoint of the curves &lt;math&gt;\gamma_i&lt;/math&gt;, taking multiplicities &lt;math&gt;n_i&lt;/math&gt; into account.

'''Version 2:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; consisting of closed paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt;.

=== Connection Between Version 1 and Version 2 ===
Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; can be transformed into a chain &lt;math&gt;\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i&lt;/math&gt; of closed paths &lt;math&gt;\hat{\gamma}_1, \ldots, \hat{\gamma}_m&lt;/math&gt;.
If the paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt; are piecewise continuously differentiable, then the closed paths &lt;math&gt;\hat{\gamma}1, \ldots, \hat{\gamma}m&lt;/math&gt; are also continuously differentiable. For all holomorphic functions &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt;, it holds that:
&lt;math&gt;\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz&lt;/math&gt;.

=== Trace of a path ===
The '''trace''' of a path &lt;math&gt;\gamma : [a,b] \to G&lt;/math&gt; is defined as:
&lt;math&gt;\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }&lt;/math&gt;.

=== Trace of a cycle/chain ===
The trace of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the [[w:de:Image (mathematics)|images]] of its individual curves, i.e.:
&lt;math&gt;\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)&lt;/math&gt;.
If &lt;math&gt;\operatorname{Trace}(\Gamma) \subset \mathbb{C}&lt;/math&gt; is a subset of &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;, then &lt;math&gt;\Gamma&lt;/math&gt; is called a cycle '''in''' &lt;math&gt;G&lt;/math&gt; if and only if the trace &lt;math&gt;\operatorname{Trace}(\Gamma) \subseteq G&lt;/math&gt; lies in &lt;math&gt;G&lt;/math&gt;.

=== Winding number ===
The '''[[w:en:Winding number(Mathematics)|winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For &lt;math&gt;z \not\in \operatorname{Trace}(\Gamma)&lt;/math&gt;, it is given by:
&lt;math&gt;n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}&lt;/math&gt;.

=== Interior points of a cycle ===
The '''interior''' of a cycle consists of all points for which the winding number is non-zero:
&lt;math&gt;\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}&lt;/math&gt;.

=== Exterior points of a cycle ===
Analogously, the '''exterior''' is the set of points for which the winding number is zero:
&lt;math&gt;\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}&lt;/math&gt;.

=== zero-homologous cycle ===
A cycle is called '''null-homologous''' for a set &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if the interior &lt;math&gt;\operatorname{Int}(\Gamma)&lt;/math&gt; lies entirely within &lt;math&gt;G&lt;/math&gt;. This is equivalent to the winding number vanishing for all points in &lt;math&gt;\mathbb{C} \setminus G&lt;/math&gt;.

=== Homologous cycles ===
Two cycles &lt;math&gt;\Gamma_1&lt;/math&gt;, &lt;math&gt;\Gamma_2&lt;/math&gt; are called '''homologous''' in &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if their formal difference &lt;math&gt;\Gamma_1-\Gamma_2&lt;/math&gt; is null-homologous in &lt;math&gt;G&lt;/math&gt;.

== Integral Theorems ==
Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]], the [[w:en:Cauchy's integral formula|Cauchy's integral formula]], and the [[w:en:Residue theorem|Residue theorem]] can be proven for cycles.

== Relation to Homology Theory ==
To indicate that chains and cycles are special cases of objects in [[w:en:Homology (mathematics)|Homology (mathematics)]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.&lt;ref&gt;[[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20&lt;/ref&gt;. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.&lt;ref&gt;{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}&lt;/ref&gt;. Additionally, note that the plural of cycle is &quot;cycles,&quot; while the plural of Zykel is &quot;Zykel&quot; in German.

=== Embedding in Homology Theory ===
The terms chain and cycle are special cases of [[w:en:Mathematical object|objects]] in [[w:en:Topology (mathematics)|topology]]. In [[w:en:Algebraic topology|algebraic topology]], one considers [[w:en:Chain complex|complexes of p-chains]] and constructs [[w:en:Homology group|homology groups]] from them. These groups are [[w:en:Invariant (mathematics)|invariants]] in topology. A very important [[w:en:Homology theory|homology theory]] is that of [[w:en:Singular homology|singular homology groups]].

=== 1-Chain of the Singular Complex ===
A chain, as defined here, is a 1-chain of the [[w:de:Singular complex|singular complex]], which is a specific chain complex. The operator defined in the section on cycles, &lt;math&gt;\partial \colon C_1(X) \to \operatorname{Div}(X)&lt;/math&gt;, is the first [[w:de:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator &lt;math&gt;\partial&lt;/math&gt;, is a 1-[[w:de:Chain complex|cycle]] in the sense of the singular complex.

=== Algebraic Topology ===
In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:de:Singular homology|singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning.

== See also ==

*[[w:en:Global Cauchy Integral Theorem]]

*[[w:en:Stokes' theorem|Stokes' theorem]]

*[[w:en:Smooth function|smooth function]]

== References ==

{{Literature
| Author=Wolfgang Fischer, Ingo Lieb
| Title=Complex Analysis
| Edition=8th
| Publisher=Vieweg
| Location=Braunschweig
| Year=2003
| ISBN=3-528-77247-6
}}

[[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20


== Notes ==
&lt;references /&gt;

[[Category:Complex analysis]]

== Page Information ==
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'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/cycle&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=cycle&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

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* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Zyklus&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;08:50&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

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[[de:Kurs:Funktionentheorie/Zyklus]]
&lt;/noinclude&gt;

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      <text bytes="9670" sha1="0e5mo2vr4zzp3p3qjkft1uikqdk1gyj" xml:space="preserve">== Introduction ==
'''Chain''' and '''cycle''' are mathematical objects studied in [[Complex Analysis|Complex Analysis]] but also appear as special cases in [[w:en:Algebraic topology|Algebraic topology]]. A chain generalizes a [[w:en:Path (topology)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.

== Definitions ==
=== Chain ===
A chain on a set &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt; is defined as a finite integer linear combination of paths &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt;:
&lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}&lt;/math&gt;.
&lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; are generally continuous [[w:en:Path (topology)|curves]] in &lt;math&gt;G&lt;/math&gt;.

=== Integration over a chain ===
Let &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt; be integrable, and let &lt;math&gt;\Gamma&lt;/math&gt; be a chain of piecewise continuously differentiable paths (paths of integration) &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; in &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;. The integral over the chain &lt;math&gt;\Gamma&lt;/math&gt; is defined by:
:&lt;math&gt;\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz&lt;/math&gt;

=== Definition: Cycle ===
'''Version 1:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt;, where every point &lt;math&gt;a \in \mathbb{C}&lt;/math&gt; appears as the starting point as many times as it appears as the endpoint of the curves &lt;math&gt;\gamma_i&lt;/math&gt;, taking multiplicities &lt;math&gt;n_i&lt;/math&gt; into account.

'''Version 2:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; consisting of closed paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt;.

=== Connection Between Version 1 and Version 2 ===
Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; can be transformed into a chain &lt;math&gt;\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i&lt;/math&gt; of closed paths &lt;math&gt;\hat{\gamma}_1, \ldots, \hat{\gamma}_m&lt;/math&gt;.
If the paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt; are piecewise continuously differentiable, then the closed paths &lt;math&gt;\hat{\gamma}1, \ldots, \hat{\gamma}m&lt;/math&gt; are also continuously differentiable. For all holomorphic functions &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt;, it holds that:
&lt;math&gt;\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz&lt;/math&gt;.

=== Trace of a path ===
The '''trace''' of a path &lt;math&gt;\gamma : [a,b] \to G&lt;/math&gt; is defined as:
&lt;math&gt;\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }&lt;/math&gt;.

=== Trace of a cycle/chain ===
The trace of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the [[w:de:Image (mathematics)|images]] of its individual curves, i.e.:
&lt;math&gt;\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)&lt;/math&gt;.
If &lt;math&gt;\operatorname{Trace}(\Gamma) \subset \mathbb{C}&lt;/math&gt; is a subset of &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;, then &lt;math&gt;\Gamma&lt;/math&gt; is called a cycle '''in''' &lt;math&gt;G&lt;/math&gt; if and only if the trace &lt;math&gt;\operatorname{Trace}(\Gamma) \subseteq G&lt;/math&gt; lies in &lt;math&gt;G&lt;/math&gt;.

=== Winding number ===
The '''[[w:en:Winding number(Mathematics)|Winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For &lt;math&gt;z \not\in \operatorname{Trace}(\Gamma)&lt;/math&gt;, it is given by:
&lt;math&gt;n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}&lt;/math&gt;.

=== Interior points of a cycle ===
The '''interior''' of a cycle consists of all points for which the winding number is non-zero:
&lt;math&gt;\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}&lt;/math&gt;.

=== Exterior points of a cycle ===
Analogously, the '''exterior''' is the set of points for which the winding number is zero:
&lt;math&gt;\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}&lt;/math&gt;.

=== zero-homologous cycle ===
A cycle is called '''null-homologous''' for a set &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if the interior &lt;math&gt;\operatorname{Int}(\Gamma)&lt;/math&gt; lies entirely within &lt;math&gt;G&lt;/math&gt;. This is equivalent to the winding number vanishing for all points in &lt;math&gt;\mathbb{C} \setminus G&lt;/math&gt;.

=== Homologous cycles ===
Two cycles &lt;math&gt;\Gamma_1&lt;/math&gt;, &lt;math&gt;\Gamma_2&lt;/math&gt; are called '''homologous''' in &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if their formal difference &lt;math&gt;\Gamma_1-\Gamma_2&lt;/math&gt; is null-homologous in &lt;math&gt;G&lt;/math&gt;.

== Integral Theorems ==
Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]], the [[w:en:Cauchy's integral formula|Cauchy's integral formula]], and the [[w:en:Residue theorem|Residue theorem]] can be proven for cycles.

== Relation to Homology Theory ==
To indicate that chains and cycles are special cases of objects in [[w:en:Homology (mathematics)|Homology (mathematics)]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.&lt;ref&gt;[[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20&lt;/ref&gt;. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.&lt;ref&gt;{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}&lt;/ref&gt;. Additionally, note that the plural of cycle is &quot;cycles,&quot; while the plural of Zykel is &quot;Zykel&quot; in German.

=== Embedding in Homology Theory ===
The terms chain and cycle are special cases of [[w:en:Mathematical object|objects]] in [[w:en:Topology (mathematics)|topology]]. In [[w:en:Algebraic topology|algebraic topology]], one considers [[w:en:Chain complex|complexes of p-chains]] and constructs [[w:en:Homology group|homology groups]] from them. These groups are [[w:en:Invariant (mathematics)|invariants]] in topology. A very important [[w:en:Homology theory|homology theory]] is that of [[w:en:Singular homology|singular homology groups]].

=== 1-Chain of the Singular Complex ===
A chain, as defined here, is a 1-chain of the [[w:de:Singular complex|singular complex]], which is a specific chain complex. The operator defined in the section on cycles, &lt;math&gt;\partial \colon C_1(X) \to \operatorname{Div}(X)&lt;/math&gt;, is the first [[w:de:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator &lt;math&gt;\partial&lt;/math&gt;, is a 1-[[w:de:Chain complex|cycle]] in the sense of the singular complex.

=== Algebraic Topology ===
In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:de:Singular homology|singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning.

== See also ==

*[[w:en:Global Cauchy Integral Theorem]]

*[[w:en:Stokes' theorem|Stokes' theorem]]

*[[w:en:Smooth function|smooth function]]

== References ==

{{Literature
| Author=Wolfgang Fischer, Ingo Lieb
| Title=Complex Analysis
| Edition=8th
| Publisher=Vieweg
| Location=Braunschweig
| Year=2003
| ISBN=3-528-77247-6
}}

[[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20


== Notes ==
&lt;references /&gt;

[[Category:Complex analysis]]

== Page Information ==
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* Date: 12/17/2024

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&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Zyklus]]
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      <text bytes="9657" sha1="b71pr4ext4jtv6k9vi5zxubjvajnkpv" xml:space="preserve">== Introduction ==
'''Chain''' and '''cycle''' are mathematical objects studied in [[Complex Analysis|Complex Analysis]] but also appear as special cases in [[w:en:Algebraic topology|Algebraic topology]]. A chain generalizes a [[w:en:Path (topology)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.

== Definitions ==
=== Chain ===
A chain on a set &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt; is defined as a finite integer linear combination of paths &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt;:
&lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}&lt;/math&gt;.
&lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; are generally continuous [[w:en:Path (topology)|curves]] in &lt;math&gt;G&lt;/math&gt;.

=== Integration over a chain ===
Let &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt; be integrable, and let &lt;math&gt;\Gamma&lt;/math&gt; be a chain of piecewise continuously differentiable paths (paths of integration) &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; in &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;. The integral over the chain &lt;math&gt;\Gamma&lt;/math&gt; is defined by:
:&lt;math&gt;\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz&lt;/math&gt;

=== Definition: Cycle ===
'''Version 1:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt;, where every point &lt;math&gt;a \in \mathbb{C}&lt;/math&gt; appears as the starting point as many times as it appears as the endpoint of the curves &lt;math&gt;\gamma_i&lt;/math&gt;, taking multiplicities &lt;math&gt;n_i&lt;/math&gt; into account.

'''Version 2:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; consisting of closed paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt;.

=== Connection Between Version 1 and Version 2 ===
Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; can be transformed into a chain &lt;math&gt;\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i&lt;/math&gt; of closed paths &lt;math&gt;\hat{\gamma}_1, \ldots, \hat{\gamma}_m&lt;/math&gt;.
If the paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt; are piecewise continuously differentiable, then the closed paths &lt;math&gt;\hat{\gamma}1, \ldots, \hat{\gamma}m&lt;/math&gt; are also continuously differentiable. For all holomorphic functions &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt;, it holds that:
&lt;math&gt;\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz&lt;/math&gt;.

=== Trace of a path ===
The '''trace''' of a path &lt;math&gt;\gamma : [a,b] \to G&lt;/math&gt; is defined as:
&lt;math&gt;\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }&lt;/math&gt;.

=== Trace of a cycle/chain ===
The trace of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the [[w:de:Image (mathematics)|images]] of its individual curves, i.e.:
&lt;math&gt;\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)&lt;/math&gt;.
If &lt;math&gt;\operatorname{Trace}(\Gamma) \subset \mathbb{C}&lt;/math&gt; is a subset of &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;, then &lt;math&gt;\Gamma&lt;/math&gt; is called a cycle '''in''' &lt;math&gt;G&lt;/math&gt; if and only if the trace &lt;math&gt;\operatorname{Trace}(\Gamma) \subseteq G&lt;/math&gt; lies in &lt;math&gt;G&lt;/math&gt;.

=== Winding number ===
The '''[[w:en:Winding number|Winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For &lt;math&gt;z \not\in \operatorname{Trace}(\Gamma)&lt;/math&gt;, it is given by:
&lt;math&gt;n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}&lt;/math&gt;.

=== Interior points of a cycle ===
The '''interior''' of a cycle consists of all points for which the winding number is non-zero:
&lt;math&gt;\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}&lt;/math&gt;.

=== Exterior points of a cycle ===
Analogously, the '''exterior''' is the set of points for which the winding number is zero:
&lt;math&gt;\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}&lt;/math&gt;.

=== zero-homologous cycle ===
A cycle is called '''null-homologous''' for a set &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if the interior &lt;math&gt;\operatorname{Int}(\Gamma)&lt;/math&gt; lies entirely within &lt;math&gt;G&lt;/math&gt;. This is equivalent to the winding number vanishing for all points in &lt;math&gt;\mathbb{C} \setminus G&lt;/math&gt;.

=== Homologous cycles ===
Two cycles &lt;math&gt;\Gamma_1&lt;/math&gt;, &lt;math&gt;\Gamma_2&lt;/math&gt; are called '''homologous''' in &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if their formal difference &lt;math&gt;\Gamma_1-\Gamma_2&lt;/math&gt; is null-homologous in &lt;math&gt;G&lt;/math&gt;.

== Integral Theorems ==
Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]], the [[w:en:Cauchy's integral formula|Cauchy's integral formula]], and the [[w:en:Residue theorem|Residue theorem]] can be proven for cycles.

== Relation to Homology Theory ==
To indicate that chains and cycles are special cases of objects in [[w:en:Homology (mathematics)|Homology (mathematics)]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.&lt;ref&gt;[[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20&lt;/ref&gt;. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.&lt;ref&gt;{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}&lt;/ref&gt;. Additionally, note that the plural of cycle is &quot;cycles,&quot; while the plural of Zykel is &quot;Zykel&quot; in German.

=== Embedding in Homology Theory ===
The terms chain and cycle are special cases of [[w:en:Mathematical object|objects]] in [[w:en:Topology (mathematics)|topology]]. In [[w:en:Algebraic topology|algebraic topology]], one considers [[w:en:Chain complex|complexes of p-chains]] and constructs [[w:en:Homology group|homology groups]] from them. These groups are [[w:en:Invariant (mathematics)|invariants]] in topology. A very important [[w:en:Homology theory|homology theory]] is that of [[w:en:Singular homology|singular homology groups]].

=== 1-Chain of the Singular Complex ===
A chain, as defined here, is a 1-chain of the [[w:de:Singular complex|singular complex]], which is a specific chain complex. The operator defined in the section on cycles, &lt;math&gt;\partial \colon C_1(X) \to \operatorname{Div}(X)&lt;/math&gt;, is the first [[w:de:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator &lt;math&gt;\partial&lt;/math&gt;, is a 1-[[w:de:Chain complex|cycle]] in the sense of the singular complex.

=== Algebraic Topology ===
In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:de:Singular homology|singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning.

== See also ==

*[[w:en:Global Cauchy Integral Theorem]]

*[[w:en:Stokes' theorem|Stokes' theorem]]

*[[w:en:Smooth function|smooth function]]

== References ==

{{Literature
| Author=Wolfgang Fischer, Ingo Lieb
| Title=Complex Analysis
| Edition=8th
| Publisher=Vieweg
| Location=Braunschweig
| Year=2003
| ISBN=3-528-77247-6
}}

[[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20


== Notes ==
&lt;references /&gt;

[[Category:Complex analysis]]

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      <text bytes="9714" sha1="pgyf7fq1f3ssiy0jyjes2hn7wjzw547" xml:space="preserve">== Introduction ==
'''Chain''' and '''cycle''' are mathematical objects studied in [[Complex Analysis|Complex Analysis]] but also appear as special cases in [[w:en:Algebraic topology|Algebraic topology]]. A chain generalizes a [[w:en:Path (topology)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.

== Definitions ==
=== Chain ===
A chain on a set &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt; is defined as a finite integer linear combination of paths &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt;:
&lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}&lt;/math&gt;.
&lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; are generally continuous [[w:en:Path (topology)|curves]] in &lt;math&gt;G&lt;/math&gt;.

=== Integration over a chain ===
Let &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt; be integrable, and let &lt;math&gt;\Gamma&lt;/math&gt; be a chain of piecewise continuously differentiable paths (paths of integration) &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; in &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;. The integral over the chain &lt;math&gt;\Gamma&lt;/math&gt; is defined by:
:&lt;math&gt;\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz&lt;/math&gt;

=== Definition: Cycle ===
'''Version 1:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt;, where every point &lt;math&gt;a \in \mathbb{C}&lt;/math&gt; appears as the starting point as many times as it appears as the endpoint of the curves &lt;math&gt;\gamma_i&lt;/math&gt;, taking multiplicities &lt;math&gt;n_i&lt;/math&gt; into account.

'''Version 2:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; consisting of closed paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt;.

=== Connection Between Version 1 and Version 2 ===
Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; can be transformed into a chain &lt;math&gt;\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i&lt;/math&gt; of closed paths &lt;math&gt;\hat{\gamma}_1, \ldots, \hat{\gamma}_m&lt;/math&gt;.
If the paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt; are piecewise continuously differentiable, then the closed paths &lt;math&gt;\hat{\gamma}1, \ldots, \hat{\gamma}m&lt;/math&gt; are also continuously differentiable. For all holomorphic functions &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt;, it holds that:
&lt;math&gt;\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz&lt;/math&gt;.

=== Trace of a path ===
The '''trace''' of a path &lt;math&gt;\gamma : [a,b] \to G&lt;/math&gt; is defined as:
&lt;math&gt;\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }&lt;/math&gt;.

=== Trace of a cycle/chain ===
The trace of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the [[w:de:Image (mathematics)|images]] of its individual curves, i.e.:
&lt;math&gt;\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)&lt;/math&gt;.
If &lt;math&gt;\operatorname{Trace}(\Gamma) \subset \mathbb{C}&lt;/math&gt; is a subset of &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;, then &lt;math&gt;\Gamma&lt;/math&gt; is called a cycle '''in''' &lt;math&gt;G&lt;/math&gt; if and only if the trace &lt;math&gt;\operatorname{Trace}(\Gamma) \subseteq G&lt;/math&gt; lies in &lt;math&gt;G&lt;/math&gt;.

=== Winding number ===
The '''[[w:en:Winding number|Winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For &lt;math&gt;z \not\in \operatorname{Trace}(\Gamma)&lt;/math&gt;, it is given by:
&lt;math&gt;n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}&lt;/math&gt;.

=== Interior points of a cycle ===
The '''interior''' of a cycle consists of all points for which the winding number is non-zero:
&lt;math&gt;\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}&lt;/math&gt;.

=== Exterior points of a cycle ===
Analogously, the '''exterior''' is the set of points for which the winding number is zero:
&lt;math&gt;\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}&lt;/math&gt;.

=== zero-homologous cycle ===
A cycle is called '''null-homologous''' for a set &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if the interior &lt;math&gt;\operatorname{Int}(\Gamma)&lt;/math&gt; lies entirely within &lt;math&gt;G&lt;/math&gt;. This is equivalent to the winding number vanishing for all points in &lt;math&gt;\mathbb{C} \setminus G&lt;/math&gt;.

=== Homologous cycles ===
Two cycles &lt;math&gt;\Gamma_1&lt;/math&gt;, &lt;math&gt;\Gamma_2&lt;/math&gt; are called '''homologous''' in &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if their formal difference &lt;math&gt;\Gamma_1-\Gamma_2&lt;/math&gt; is null-homologous in &lt;math&gt;G&lt;/math&gt;.

== Integral Theorems ==
Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]], the [[w:en:Cauchy's integral formula|Cauchy's integral formula]], and the [[w:en:Residue theorem|Residue theorem]] can be proven for cycles.

== Relation to Homology Theory ==
To indicate that chains and cycles are special cases of objects in [[w:en:Homology (mathematics)|Homology (mathematics)]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.&lt;ref&gt;[[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20&lt;/ref&gt;. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.&lt;ref&gt;{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}&lt;/ref&gt;. Additionally, note that the plural of cycle is &quot;cycles,&quot; while the plural of Zykel is &quot;Zykel&quot; in German.

=== Embedding in Homology Theory ===
The terms chain and cycle are special cases of [[w:en:Mathematical object|Mathematical object]] in [[w:en:Topology (mathematics)|topology]]. In [[w:en:Algebraic topology|Algebraic topology]], one considers [[w:en:Chain complex|complexes of p-chains]] and constructs [[w:en:Homology (mathematics)|Homology (mathematics)]] from them. These groups are [[w:en:Invariant (mathematics)|Invariant (mathematics)]] in topology. A very important [[w:en:Homology (mathematics)|Homology (mathematics)]] is that of [[w:en:Singular homology|Singular homology]].

=== 1-Chain of the Singular Complex ===
A chain, as defined here, is a 1-chain of the [[w:en:Singular homology|Singular homology]], which is a specific chain complex. The operator defined in the section on cycles, &lt;math&gt;\partial \colon C_1(X) \to \operatorname{Div}(X)&lt;/math&gt;, is the first [[w:en:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator &lt;math&gt;\partial&lt;/math&gt;, is a 1-[[w:en:Chain complex|Chain complex]] in the sense of the singular complex.

=== Algebraic Topology ===
In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:en:Singular homology|Singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning.

== See also ==

*[[w:en:Global Cauchy Integral Theorem]]

*[[w:en:Stokes' theorem|Stokes' theorem]]

*[[w:en:Smooth function|smooth function]]

== References ==

{{Literature
| Author=Wolfgang Fischer, Ingo Lieb
| Title=Complex Analysis
| Edition=8th
| Publisher=Vieweg
| Location=Braunschweig
| Year=2003
| ISBN=3-528-77247-6
}}

[[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20


== Notes ==
&lt;references /&gt;

[[Category:Complex analysis]]

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==57 / 199==

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[[Category:Boolf prop/3-ary| ]]</text>
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[[../December_20th_2024|December 20th 2024]]</text>
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[[../December_19th_2024_1|December 19th 2024 1]]

[[../December_20th_2024|December 20th 2024]]</text>
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      <comment>Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]].</comment>
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  <page>
    <title>Formal linear combination</title>
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      <comment>New resource with &quot;In many areas of mathematics, especially in [[Algebra]] and its applications, such as [[Complex Analysis]] and [[Topology]], it is often helpful to consider &quot;linear combinations&quot; of elements of a set that does not have a module structure. In this article, we aim to briefly explain how these linear combinations can be mathematically represented as actual linear combinations.  Let &lt;math&gt;M&lt;/math&gt; be a set and &lt;math&gt;R&lt;/math&gt; a ring (in many classical cases, &lt;math&gt;R&lt;/math&gt; is...&quot;</comment>
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      <text bytes="4810" sha1="niqsohk1ttty1258d3dj8pndha5fj8a" xml:space="preserve">In many areas of mathematics, especially in [[Algebra]] and its applications, such as [[Complex Analysis]] and [[Topology]], it is often helpful to consider &quot;linear combinations&quot; of elements of a set that does not have a module structure. In this article, we aim to briefly explain how these linear combinations can be mathematically represented as actual linear combinations.

Let &lt;math&gt;M&lt;/math&gt; be a set and &lt;math&gt;R&lt;/math&gt; a ring (in many classical cases, &lt;math&gt;R&lt;/math&gt; is a field or the ring &lt;math&gt;\mathbb Z&lt;/math&gt; of integers).

==Classical Definition==
A ''formal linear combination'' of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt; is a sum of the form &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;, where &lt;math&gt;r_i \in R&lt;/math&gt; and &lt;math&gt;m_i \in M&lt;/math&gt;, with the &lt;math&gt;m_i&lt;/math&gt; being pairwise distinct. Two sums of this form are considered equal if the same elements of &lt;math&gt;M&lt;/math&gt; appear with the same coefficients (with a coefficient of &lt;math&gt;0&lt;/math&gt; meaning that the element does not appear). The set of these sums forms, with the operations

*&lt;math&gt;\sum_{i=1}^n r_i m_i + \sum_{i=1}^n s_i m_i := \sum_{i=1}^n (r_i + s_i)m_i&lt;/math&gt; (any two such sums can be written so that they contain the same elements of &lt;math&gt;M&lt;/math&gt;; if necessary, terms of the form &lt;math&gt;0 \cdot m&lt;/math&gt; with &lt;math&gt;m \in M&lt;/math&gt; can be added), with &lt;math&gt;r_i, s_i \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,

*&lt;math&gt;r \cdot \sum_{i=1}^n r_i m_i = \sum_{i=1}^n (rr_i)m_i&lt;/math&gt; with &lt;math&gt;r_i, r \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,
a natural &lt;math&gt;R&lt;/math&gt;-module (in the case of a field, this is a vector space; in the case of integers, an abelian group).
This &lt;math&gt;R&lt;/math&gt;-module is also called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;''.


The definition is somewhat unsatisfactory in that the elements of the free &lt;math&gt;R&lt;/math&gt;-module are undefined sums &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;; note that we can only define the operations once we have the elements, but we need the operations to describe the elements. One could argue that the sums are merely &quot;formal,&quot; but this is mathematically unsatisfactory since it is unclear what a formal sum actually is. The main idea of the precise definition is to assign the formal sums an exact meaning.

==Preliminary Considerations for the Exact Definition==
How can we assign the sum &lt;math&gt;s = \sum_{i=1}^n r_i m_i&lt;/math&gt; a mathematically precise object with the same meaning? The idea is to think about what we actually need from the sums. These are the pairs &lt;math&gt;(m_i, r_i)&lt;/math&gt;. Remembering that the &lt;math&gt;m_i&lt;/math&gt; were pairwise distinct, we see that we are dealing with a map &lt;math&gt;m_i \mapsto r_i&lt;/math&gt;. Extending this map to all of &lt;math&gt;R&lt;/math&gt; by assigning &lt;math&gt;0&lt;/math&gt; to other elements gives us a map &lt;math&gt;\phi_s \colon M \to R&lt;/math&gt; with the property that its so-called ''support'' &lt;math&gt;\mathrm{supp}(\phi_s) := \phi_s^{-1}[R \setminus {0}]&lt;/math&gt;, i.e., the set of elements not mapped to zero, is finite. Conversely, if &lt;math&gt;\phi \colon M \to R&lt;/math&gt; is a map with finite support &lt;math&gt;\mathrm{supp}(\phi) = {m_1, \ldots, m_n}&lt;/math&gt;, we can assign it the formal sum &lt;math&gt;s_\phi := \sum_{i=1}^n \phi(m_i) m_i&lt;/math&gt;. One can see that these constructions are inverses of each other, so formal sums correspond exactly to maps with finite support. We can thus define:

==Definition==
A ''formal linear combination of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt;'' is a map &lt;math&gt;\phi \colon M \to R&lt;/math&gt; with

&lt;center&gt;&lt;math&gt;|\mathrm{supp}(\phi)| &lt; \infty &lt;/math&gt;&lt;/center&gt;  
The set of all such maps is denoted by &lt;math&gt;R^{(M)}&lt;/math&gt; and is an &lt;math&gt;R&lt;/math&gt;-submodule of &lt;math&gt;R^M&lt;/math&gt;. It is called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;.''
===Notation===
Let &lt;math&gt;\phi \in R^{(M)}&lt;/math&gt;. Since sums are more convenient to work with than maps (as the sum compactly encodes all the information), we want to write &lt;math&gt;\phi&lt;/math&gt; as a sum. For &lt;math&gt;m \in M&lt;/math&gt;, consider the map &lt;math&gt;\phi_m \colon M \to R&lt;/math&gt;, which maps &lt;math&gt;m&lt;/math&gt; to &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;M \setminus {m}&lt;/math&gt; to &lt;math&gt;0&lt;/math&gt;. If &lt;math&gt;{m_1, \ldots, m_n}&lt;/math&gt; is a listing of the support of &lt;math&gt;\phi&lt;/math&gt; with pairwise distinct &lt;math&gt;m_i&lt;/math&gt;, then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i) \phi_{m_i}&lt;/math&gt; (note that this is an actual sum in the &lt;math&gt;R&lt;/math&gt;-module &lt;math&gt;R^{(M)}&lt;/math&gt; with pointwise operations). The notation is often further simplified by identifying the elements &lt;math&gt;m \in M&lt;/math&gt; with the map &lt;math&gt;\phi_m&lt;/math&gt;, so &lt;math&gt;\phi_m&lt;/math&gt; is simply called &lt;math&gt;m&lt;/math&gt;. Then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i)m_i&lt;/math&gt;, allowing us to work with convenient sums while ensuring that everything rests on a solid mathematical foundation.</text>
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      <text bytes="6887" sha1="gkzpgl952k61hanykcvq1gu73kg16s7" xml:space="preserve">In many areas of mathematics, especially in [[Algebra]] and its applications, such as [[Complex Analysis]] and [[Topology]], it is often helpful to consider &quot;linear combinations&quot; of elements of a set that does not have a module structure. In this article, we aim to briefly explain how these linear combinations can be mathematically represented as actual linear combinations.

Let &lt;math&gt;M&lt;/math&gt; be a set and &lt;math&gt;R&lt;/math&gt; a ring (in many classical cases, &lt;math&gt;R&lt;/math&gt; is a field or the ring &lt;math&gt;\mathbb Z&lt;/math&gt; of integers).

==Classical Definition==
A ''formal linear combination'' of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt; is a sum of the form &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;, where &lt;math&gt;r_i \in R&lt;/math&gt; and &lt;math&gt;m_i \in M&lt;/math&gt;, with the &lt;math&gt;m_i&lt;/math&gt; being pairwise distinct. Two sums of this form are considered equal if the same elements of &lt;math&gt;M&lt;/math&gt; appear with the same coefficients (with a coefficient of &lt;math&gt;0&lt;/math&gt; meaning that the element does not appear). The set of these sums forms, with the operations

*&lt;math&gt;\sum_{i=1}^n r_i m_i + \sum_{i=1}^n s_i m_i := \sum_{i=1}^n (r_i + s_i)m_i&lt;/math&gt; (any two such sums can be written so that they contain the same elements of &lt;math&gt;M&lt;/math&gt;; if necessary, terms of the form &lt;math&gt;0 \cdot m&lt;/math&gt; with &lt;math&gt;m \in M&lt;/math&gt; can be added), with &lt;math&gt;r_i, s_i \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,

*&lt;math&gt;r \cdot \sum_{i=1}^n r_i m_i = \sum_{i=1}^n (rr_i)m_i&lt;/math&gt; with &lt;math&gt;r_i, r \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,
a natural &lt;math&gt;R&lt;/math&gt;-module (in the case of a field, this is a vector space; in the case of integers, an abelian group).
This &lt;math&gt;R&lt;/math&gt;-module is also called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;''.


The definition is somewhat unsatisfactory in that the elements of the free &lt;math&gt;R&lt;/math&gt;-module are undefined sums &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;; note that we can only define the operations once we have the elements, but we need the operations to describe the elements. One could argue that the sums are merely &quot;formal,&quot; but this is mathematically unsatisfactory since it is unclear what a formal sum actually is. The main idea of the precise definition is to assign the formal sums an exact meaning.

==Preliminary Considerations for the Exact Definition==
How can we assign the sum &lt;math&gt;s = \sum_{i=1}^n r_i m_i&lt;/math&gt; a mathematically precise object with the same meaning? The idea is to think about what we actually need from the sums. These are the pairs &lt;math&gt;(m_i, r_i)&lt;/math&gt;. Remembering that the &lt;math&gt;m_i&lt;/math&gt; were pairwise distinct, we see that we are dealing with a map &lt;math&gt;m_i \mapsto r_i&lt;/math&gt;. Extending this map to all of &lt;math&gt;R&lt;/math&gt; by assigning &lt;math&gt;0&lt;/math&gt; to other elements gives us a map &lt;math&gt;\phi_s \colon M \to R&lt;/math&gt; with the property that its so-called ''support'' &lt;math&gt;\mathrm{supp}(\phi_s) := \phi_s^{-1}[R \setminus {0}]&lt;/math&gt;, i.e., the set of elements not mapped to zero, is finite. Conversely, if &lt;math&gt;\phi \colon M \to R&lt;/math&gt; is a map with finite support &lt;math&gt;\mathrm{supp}(\phi) = {m_1, \ldots, m_n}&lt;/math&gt;, we can assign it the formal sum &lt;math&gt;s_\phi := \sum_{i=1}^n \phi(m_i) m_i&lt;/math&gt;. One can see that these constructions are inverses of each other, so formal sums correspond exactly to maps with finite support. We can thus define:

==Definition==
A ''formal linear combination of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt;'' is a map &lt;math&gt;\phi \colon M \to R&lt;/math&gt; with

&lt;center&gt;&lt;math&gt;|\mathrm{supp}(\phi)| &lt; \infty &lt;/math&gt;&lt;/center&gt;  
The set of all such maps is denoted by &lt;math&gt;R^{(M)}&lt;/math&gt; and is an &lt;math&gt;R&lt;/math&gt;-submodule of &lt;math&gt;R^M&lt;/math&gt;. It is called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;.''
===Notation===
Let &lt;math&gt;\phi \in R^{(M)}&lt;/math&gt;. Since sums are more convenient to work with than maps (as the sum compactly encodes all the information), we want to write &lt;math&gt;\phi&lt;/math&gt; as a sum. For &lt;math&gt;m \in M&lt;/math&gt;, consider the map &lt;math&gt;\phi_m \colon M \to R&lt;/math&gt;, which maps &lt;math&gt;m&lt;/math&gt; to &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;M \setminus {m}&lt;/math&gt; to &lt;math&gt;0&lt;/math&gt;. If &lt;math&gt;{m_1, \ldots, m_n}&lt;/math&gt; is a listing of the support of &lt;math&gt;\phi&lt;/math&gt; with pairwise distinct &lt;math&gt;m_i&lt;/math&gt;, then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i) \phi_{m_i}&lt;/math&gt; (note that this is an actual sum in the &lt;math&gt;R&lt;/math&gt;-module &lt;math&gt;R^{(M)}&lt;/math&gt; with pointwise operations). The notation is often further simplified by identifying the elements &lt;math&gt;m \in M&lt;/math&gt; with the map &lt;math&gt;\phi_m&lt;/math&gt;, so &lt;math&gt;\phi_m&lt;/math&gt; is simply called &lt;math&gt;m&lt;/math&gt;. Then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i)m_i&lt;/math&gt;, allowing us to work with convenient sums while ensuring that everything rests on a solid mathematical foundation.
== Page Information ==
You can display this page as 
'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Formal%20linear%20combination/&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Formal%20linear%20combination&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''
=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Formal%20linear%20combination&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Formal%20linear%20combination&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''  were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' 
and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Formal%20linear%20combination
 https://en.wikiversity.org/wiki/Complex%20Analysis/Formal%20linear%20combination]
--&gt;
* [https://en.wikiversity.org/wiki/Complex%20Analysis/Formal%20linear%20combination This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Formale Linearkombination
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Formale Linearkombination|Kurs:Funktionentheorie/Formale Linearkombination]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Formale Linearkombination
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Formale Linearkombination&quot; srclang=&quot;de&quot; date=&quot;12/19/2024&quot; time=&quot;02:18&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Formale Linearkombination]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
      <sha1>gkzpgl952k61hanykcvq1gu73kg16s7</sha1>
    </revision>
    <revision>
      <id>2692603</id>
      <parentid>2692602</parentid>
      <timestamp>2024-12-19T13:33:56Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <origin>2692603</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6772" sha1="l2ch9zl66vks7is1nl56nsuosypkn2i" xml:space="preserve">In many areas of mathematics, especially in [[Algebra]] and its applications, such as [[Complex Analysis]] and [[Topology]], it is often helpful to consider &quot;linear combinations&quot; of elements of a set that does not have a module structure. In this article, we aim to briefly explain how these linear combinations can be mathematically represented as actual linear combinations.

Let &lt;math&gt;M&lt;/math&gt; be a set and &lt;math&gt;R&lt;/math&gt; a ring (in many classical cases, &lt;math&gt;R&lt;/math&gt; is a field or the ring &lt;math&gt;\mathbb Z&lt;/math&gt; of integers).

==Classical Definition==
A ''formal linear combination'' of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt; is a sum of the form &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;, where &lt;math&gt;r_i \in R&lt;/math&gt; and &lt;math&gt;m_i \in M&lt;/math&gt;, with the &lt;math&gt;m_i&lt;/math&gt; being pairwise distinct. Two sums of this form are considered equal if the same elements of &lt;math&gt;M&lt;/math&gt; appear with the same coefficients (with a coefficient of &lt;math&gt;0&lt;/math&gt; meaning that the element does not appear). The set of these sums forms, with the operations

*&lt;math&gt;\sum_{i=1}^n r_i m_i + \sum_{i=1}^n s_i m_i := \sum_{i=1}^n (r_i + s_i)m_i&lt;/math&gt; (any two such sums can be written so that they contain the same elements of &lt;math&gt;M&lt;/math&gt;; if necessary, terms of the form &lt;math&gt;0 \cdot m&lt;/math&gt; with &lt;math&gt;m \in M&lt;/math&gt; can be added), with &lt;math&gt;r_i, s_i \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,

*&lt;math&gt;r \cdot \sum_{i=1}^n r_i m_i = \sum_{i=1}^n (rr_i)m_i&lt;/math&gt; with &lt;math&gt;r_i, r \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,
a natural &lt;math&gt;R&lt;/math&gt;-module (in the case of a field, this is a vector space; in the case of integers, an abelian group).
This &lt;math&gt;R&lt;/math&gt;-module is also called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;''.


The definition is somewhat unsatisfactory in that the elements of the free &lt;math&gt;R&lt;/math&gt;-module are undefined sums &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;; note that we can only define the operations once we have the elements, but we need the operations to describe the elements. One could argue that the sums are merely &quot;formal,&quot; but this is mathematically unsatisfactory since it is unclear what a formal sum actually is. The main idea of the precise definition is to assign the formal sums an exact meaning.

==Preliminary Considerations for the Exact Definition==
How can we assign the sum &lt;math&gt;s = \sum_{i=1}^n r_i m_i&lt;/math&gt; a mathematically precise object with the same meaning? The idea is to think about what we actually need from the sums. These are the pairs &lt;math&gt;(m_i, r_i)&lt;/math&gt;. Remembering that the &lt;math&gt;m_i&lt;/math&gt; were pairwise distinct, we see that we are dealing with a map &lt;math&gt;m_i \mapsto r_i&lt;/math&gt;. Extending this map to all of &lt;math&gt;R&lt;/math&gt; by assigning &lt;math&gt;0&lt;/math&gt; to other elements gives us a map &lt;math&gt;\phi_s \colon M \to R&lt;/math&gt; with the property that its so-called ''support'' &lt;math&gt;\mathrm{supp}(\phi_s) := \phi_s^{-1}[R \setminus {0}]&lt;/math&gt;, i.e., the set of elements not mapped to zero, is finite. Conversely, if &lt;math&gt;\phi \colon M \to R&lt;/math&gt; is a map with finite support &lt;math&gt;\mathrm{supp}(\phi) = {m_1, \ldots, m_n}&lt;/math&gt;, we can assign it the formal sum &lt;math&gt;s_\phi := \sum_{i=1}^n \phi(m_i) m_i&lt;/math&gt;. One can see that these constructions are inverses of each other, so formal sums correspond exactly to maps with finite support. We can thus define:

==Definition==
A ''formal linear combination of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt;'' is a map &lt;math&gt;\phi \colon M \to R&lt;/math&gt; with

&lt;center&gt;&lt;math&gt;|\mathrm{supp}(\phi)| &lt; \infty &lt;/math&gt;&lt;/center&gt;  
The set of all such maps is denoted by &lt;math&gt;R^{(M)}&lt;/math&gt; and is an &lt;math&gt;R&lt;/math&gt;-submodule of &lt;math&gt;R^M&lt;/math&gt;. It is called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;.''
===Notation===
Let &lt;math&gt;\phi \in R^{(M)}&lt;/math&gt;. Since sums are more convenient to work with than maps (as the sum compactly encodes all the information), we want to write &lt;math&gt;\phi&lt;/math&gt; as a sum. For &lt;math&gt;m \in M&lt;/math&gt;, consider the map &lt;math&gt;\phi_m \colon M \to R&lt;/math&gt;, which maps &lt;math&gt;m&lt;/math&gt; to &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;M \setminus {m}&lt;/math&gt; to &lt;math&gt;0&lt;/math&gt;. If &lt;math&gt;{m_1, \ldots, m_n}&lt;/math&gt; is a listing of the support of &lt;math&gt;\phi&lt;/math&gt; with pairwise distinct &lt;math&gt;m_i&lt;/math&gt;, then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i) \phi_{m_i}&lt;/math&gt; (note that this is an actual sum in the &lt;math&gt;R&lt;/math&gt;-module &lt;math&gt;R^{(M)}&lt;/math&gt; with pointwise operations). The notation is often further simplified by identifying the elements &lt;math&gt;m \in M&lt;/math&gt; with the map &lt;math&gt;\phi_m&lt;/math&gt;, so &lt;math&gt;\phi_m&lt;/math&gt; is simply called &lt;math&gt;m&lt;/math&gt;. Then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i)m_i&lt;/math&gt;, allowing us to work with convenient sums while ensuring that everything rests on a solid mathematical foundation.
== Page Information ==
You can display this page as 
'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Formal%20linear%20combination/&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Formal%20linear%20combination&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''
=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Formal%20linear%20combination&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Formal%20linear%20combination&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''  were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' 
and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Formal%20linear%20combination
 https://en.wikiversity.org/wiki/Complex%20Analysis/Formal%20linear%20combination]
--&gt;
* [https://en.wikiversity.org/wiki/Complex%20Analysis/Formal%20linear%20combination This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Formale Linearkombination
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Formale Linearkombination|Formale Linearkombination]] - URL:
https://de.wikiversity.org/wiki/Formale Linearkombination
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Formale Linearkombination&quot; srclang=&quot;de&quot; date=&quot;12/19/2024&quot; time=&quot;02:18&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Formale Linearkombination]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
      <sha1>l2ch9zl66vks7is1nl56nsuosypkn2i</sha1>
    </revision>
    <revision>
      <id>2692609</id>
      <parentid>2692603</parentid>
      <timestamp>2024-12-19T13:59:55Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <origin>2692609</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5484" sha1="jixoyjleykb04syvy162ranx1szl9vl" xml:space="preserve">In many areas of mathematics, especially in [[Algebra]] and its applications, such as [[Complex Analysis]] and [[Topology]], it is often helpful to consider &quot;linear combinations&quot; of elements of a set that does not have a module structure. In this article, we aim to briefly explain how these linear combinations can be mathematically represented as actual linear combinations.

Let &lt;math&gt;M&lt;/math&gt; be a set and &lt;math&gt;R&lt;/math&gt; a ring (in many classical cases, &lt;math&gt;R&lt;/math&gt; is a field or the ring &lt;math&gt;\mathbb Z&lt;/math&gt; of integers).

==Classical Definition==
A ''formal linear combination'' of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt; is a sum of the form &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;, where &lt;math&gt;r_i \in R&lt;/math&gt; and &lt;math&gt;m_i \in M&lt;/math&gt;, with the &lt;math&gt;m_i&lt;/math&gt; being pairwise distinct. Two sums of this form are considered equal if the same elements of &lt;math&gt;M&lt;/math&gt; appear with the same coefficients (with a coefficient of &lt;math&gt;0&lt;/math&gt; meaning that the element does not appear). The set of these sums forms, with the operations

*&lt;math&gt;\sum_{i=1}^n r_i m_i + \sum_{i=1}^n s_i m_i := \sum_{i=1}^n (r_i + s_i)m_i&lt;/math&gt; (any two such sums can be written so that they contain the same elements of &lt;math&gt;M&lt;/math&gt;; if necessary, terms of the form &lt;math&gt;0 \cdot m&lt;/math&gt; with &lt;math&gt;m \in M&lt;/math&gt; can be added), with &lt;math&gt;r_i, s_i \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,

*&lt;math&gt;r \cdot \sum_{i=1}^n r_i m_i = \sum_{i=1}^n (rr_i)m_i&lt;/math&gt; with &lt;math&gt;r_i, r \in R&lt;/math&gt;, &lt;math&gt;m_i \in M&lt;/math&gt;, &lt;math&gt;n \in \mathbb N&lt;/math&gt;,
a natural &lt;math&gt;R&lt;/math&gt;-module (in the case of a field, this is a vector space; in the case of integers, an abelian group).
This &lt;math&gt;R&lt;/math&gt;-module is also called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;''.


The definition is somewhat unsatisfactory in that the elements of the free &lt;math&gt;R&lt;/math&gt;-module are undefined sums &lt;math&gt;\sum_{i=1}^n r_i m_i&lt;/math&gt;; note that we can only define the operations once we have the elements, but we need the operations to describe the elements. One could argue that the sums are merely &quot;formal,&quot; but this is mathematically unsatisfactory since it is unclear what a formal sum actually is. The main idea of the precise definition is to assign the formal sums an exact meaning.

==Preliminary Considerations for the Exact Definition==
How can we assign the sum &lt;math&gt;s = \sum_{i=1}^n r_i m_i&lt;/math&gt; a mathematically precise object with the same meaning? The idea is to think about what we actually need from the sums. These are the pairs &lt;math&gt;(m_i, r_i)&lt;/math&gt;. Remembering that the &lt;math&gt;m_i&lt;/math&gt; were pairwise distinct, we see that we are dealing with a map &lt;math&gt;m_i \mapsto r_i&lt;/math&gt;. Extending this map to all of &lt;math&gt;R&lt;/math&gt; by assigning &lt;math&gt;0&lt;/math&gt; to other elements gives us a map &lt;math&gt;\phi_s \colon M \to R&lt;/math&gt; with the property that its so-called ''support'' &lt;math&gt;\mathrm{supp}(\phi_s) := \phi_s^{-1}[R \setminus {0}]&lt;/math&gt;, i.e., the set of elements not mapped to zero, is finite. Conversely, if &lt;math&gt;\phi \colon M \to R&lt;/math&gt; is a map with finite support &lt;math&gt;\mathrm{supp}(\phi) = {m_1, \ldots, m_n}&lt;/math&gt;, we can assign it the formal sum &lt;math&gt;s_\phi := \sum_{i=1}^n \phi(m_i) m_i&lt;/math&gt;. One can see that these constructions are inverses of each other, so formal sums correspond exactly to maps with finite support. We can thus define:

==Definition==
A ''formal linear combination of elements from &lt;math&gt;M&lt;/math&gt; over &lt;math&gt;R&lt;/math&gt;'' is a map &lt;math&gt;\phi \colon M \to R&lt;/math&gt; with

&lt;center&gt;&lt;math&gt;|\mathrm{supp}(\phi)| &lt; \infty &lt;/math&gt;&lt;/center&gt;  
The set of all such maps is denoted by &lt;math&gt;R^{(M)}&lt;/math&gt; and is an &lt;math&gt;R&lt;/math&gt;-submodule of &lt;math&gt;R^M&lt;/math&gt;. It is called the ''free &lt;math&gt;R&lt;/math&gt;-module over &lt;math&gt;M&lt;/math&gt;.''
===Notation===
Let &lt;math&gt;\phi \in R^{(M)}&lt;/math&gt;. Since sums are more convenient to work with than maps (as the sum compactly encodes all the information), we want to write &lt;math&gt;\phi&lt;/math&gt; as a sum. For &lt;math&gt;m \in M&lt;/math&gt;, consider the map &lt;math&gt;\phi_m \colon M \to R&lt;/math&gt;, which maps &lt;math&gt;m&lt;/math&gt; to &lt;math&gt;1&lt;/math&gt; and &lt;math&gt;M \setminus {m}&lt;/math&gt; to &lt;math&gt;0&lt;/math&gt;. If &lt;math&gt;{m_1, \ldots, m_n}&lt;/math&gt; is a listing of the support of &lt;math&gt;\phi&lt;/math&gt; with pairwise distinct &lt;math&gt;m_i&lt;/math&gt;, then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i) \phi_{m_i}&lt;/math&gt; (note that this is an actual sum in the &lt;math&gt;R&lt;/math&gt;-module &lt;math&gt;R^{(M)}&lt;/math&gt; with pointwise operations). The notation is often further simplified by identifying the elements &lt;math&gt;m \in M&lt;/math&gt; with the map &lt;math&gt;\phi_m&lt;/math&gt;, so &lt;math&gt;\phi_m&lt;/math&gt; is simply called &lt;math&gt;m&lt;/math&gt;. Then &lt;math&gt;\phi = \sum_{i=1}^n \phi(m_i)m_i&lt;/math&gt;, allowing us to work with convenient sums while ensuring that everything rests on a solid mathematical foundation.
=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Formale Linearkombination
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Formale Linearkombination|Formale Linearkombination]] - URL:
https://de.wikiversity.org/wiki/Formale Linearkombination
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Formale Linearkombination&quot; srclang=&quot;de&quot; date=&quot;12/19/2024&quot; time=&quot;02:18&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Formale Linearkombination]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
      <sha1>jixoyjleykb04syvy162ranx1szl9vl</sha1>
    </revision>
  </page>
  <page>
    <title>Winding number</title>
    <ns>0</ns>
    <id>317347</id>
    <revision>
      <id>2692605</id>
      <timestamp>2024-12-19T13:49:35Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <comment>New resource with &quot;== Definition == Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb{C}&lt;/math&gt;, and let &lt;math&gt;z \in \mathbb{C}&lt;/math&gt; be a point that &lt;math&gt;\Gamma&lt;/math&gt; does not intersect. Then  &lt;center&gt;&lt;math&gt;n(\Gamma, z) := \frac{1}{2\pi i} \int_\Gamma \frac{1}{w-z} \, dw&lt;/math&gt;&lt;/center&gt;   is called the *winding number* of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.=== Motivation === First, consider the case where &lt;math&gt;\Gamma = \gamma&lt;/math&gt; consists of a single closed curve. Then &lt;ma...&quot;</comment>
      <origin>2692605</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2140" sha1="osavo1focgnprj42rw25y8awgecwuso" xml:space="preserve">== Definition ==
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb{C}&lt;/math&gt;, and let &lt;math&gt;z \in \mathbb{C}&lt;/math&gt; be a point that &lt;math&gt;\Gamma&lt;/math&gt; does not intersect. Then

&lt;center&gt;&lt;math&gt;n(\Gamma, z) := \frac{1}{2\pi i} \int_\Gamma \frac{1}{w-z} \, dw&lt;/math&gt;&lt;/center&gt;  
is called the *winding number* of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.=== Motivation ===
First, consider the case where &lt;math&gt;\Gamma = \gamma&lt;/math&gt; consists of a single closed curve. Then &lt;math&gt;\gamma&lt;/math&gt; is [[w:en:Homologous|homologous]] in &lt;math&gt;\mathbb{C} \setminus {z}&lt;/math&gt; to an &lt;math&gt;n&lt;/math&gt;-fold (for some &lt;math&gt;n \in \mathbb{Z}&lt;/math&gt;) traversed circle &lt;math&gt;\partial D_r(z)&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;r &gt; 0&lt;/math&gt;. Now,
:&lt;math&gt;\int_\gamma \frac{1}{w-z} , dw = \int_{n \cdot \partial B(z)} \frac{1}{w-z} , dw = n \int_{\partial B(z)} \frac{1}{w-z} , dw = 2\pi i \cdot n.&lt;/math&gt;
Thus, this integral counts how many times the curve &lt;math&gt;\gamma&lt;/math&gt; winds around the point &lt;math&gt;z&lt;/math&gt;.

=== Task ===
Let the closed integration path &lt;math&gt;\gamma : [-\pi, +\pi] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;\gamma(t) := \left(2 + \cos(t)\right) \cdot e^{2it}.&lt;/math&gt;

1. Plot the trace of the integration path.

2. Determine the winding number &lt;math&gt;n(\gamma, 1+i)&lt;/math&gt;.

3. Determine the winding number &lt;math&gt;n(\gamma, 0)&lt;/math&gt;.

4. Determine the winding number &lt;math&gt;n(\gamma, 1)&lt;/math&gt;.

== Additivity of the Integral ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \cdot \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, due to the additivity of the integral, we have
:&lt;math&gt;n(\Gamma, z) = \sum_{i=1}^k n_i \cdot n(\gamma_i, z).&lt;/math&gt;
Thus, the winding number also counts how many times the point &lt;math&gt;z&lt;/math&gt; is encircled.

== Length of the Cycle ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, the length of the cycle is defined additively over the lengths of the individual integration paths:
:&lt;math&gt;L(\Gamma) := \sum_{i=1}^k n_i \cdot L(\gamma_i).&lt;/math&gt;

== See Also ==

*[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]</text>
      <sha1>osavo1focgnprj42rw25y8awgecwuso</sha1>
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      <parentid>2692605</parentid>
      <timestamp>2024-12-19T13:56:12Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
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      <format>text/x-wiki</format>
      <text bytes="4030" sha1="e6i7ptjgth3sjmaj5wkit9mhfzocvim" xml:space="preserve">== Definition ==
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb{C}&lt;/math&gt;, and let &lt;math&gt;z \in \mathbb{C}&lt;/math&gt; be a point that &lt;math&gt;\Gamma&lt;/math&gt; does not intersect. Then

&lt;center&gt;&lt;math&gt;n(\Gamma, z) := \frac{1}{2\pi i} \int_\Gamma \frac{1}{w-z} \, dw&lt;/math&gt;&lt;/center&gt;  
is called the *winding number* of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.=== Motivation ===
First, consider the case where &lt;math&gt;\Gamma = \gamma&lt;/math&gt; consists of a single closed curve. Then &lt;math&gt;\gamma&lt;/math&gt; is [[w:en:Homologous|homologous]] in &lt;math&gt;\mathbb{C} \setminus {z}&lt;/math&gt; to an &lt;math&gt;n&lt;/math&gt;-fold (for some &lt;math&gt;n \in \mathbb{Z}&lt;/math&gt;) traversed circle &lt;math&gt;\partial D_r(z)&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;r &gt; 0&lt;/math&gt;. Now,
:&lt;math&gt;\int_\gamma \frac{1}{w-z} , dw = \int_{n \cdot \partial B(z)} \frac{1}{w-z} , dw = n \int_{\partial B(z)} \frac{1}{w-z} , dw = 2\pi i \cdot n.&lt;/math&gt;
Thus, this integral counts how many times the curve &lt;math&gt;\gamma&lt;/math&gt; winds around the point &lt;math&gt;z&lt;/math&gt;.

=== Task ===
Let the closed integration path &lt;math&gt;\gamma : [-\pi, +\pi] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;\gamma(t) := \left(2 + \cos(t)\right) \cdot e^{2it}.&lt;/math&gt;

1. Plot the trace of the integration path.

2. Determine the winding number &lt;math&gt;n(\gamma, 1+i)&lt;/math&gt;.

3. Determine the winding number &lt;math&gt;n(\gamma, 0)&lt;/math&gt;.

4. Determine the winding number &lt;math&gt;n(\gamma, 1)&lt;/math&gt;.

== Additivity of the Integral ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \cdot \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, due to the additivity of the integral, we have
:&lt;math&gt;n(\Gamma, z) = \sum_{i=1}^k n_i \cdot n(\gamma_i, z).&lt;/math&gt;
Thus, the winding number also counts how many times the point &lt;math&gt;z&lt;/math&gt; is encircled.

== Length of the Cycle ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, the length of the cycle is defined additively over the lengths of the individual integration paths:
:&lt;math&gt;L(\Gamma) := \sum_{i=1}^k n_i \cdot L(\gamma_i).&lt;/math&gt;

== See Also ==

*[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]
== Page Information ==
You can display this page as 
'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Winding%20number&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Winding%20number&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Winding%20 number&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Winding%20number&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''  were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' 
and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Winding number https://en.wikiversity.org/wiki/Complex%20Analysis/Winding number]
--&gt;
* [https://en.wikiversity.org/wiki/Complex%20Analysis/Winding number This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Umlaufzahl|Kurs:Funktionentheorie/Umlaufzahl]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Umlaufzahl&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;02:18&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Umlaufzahl]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
      <sha1>e6i7ptjgth3sjmaj5wkit9mhfzocvim</sha1>
    </revision>
    <revision>
      <id>2692607</id>
      <parentid>2692606</parentid>
      <timestamp>2024-12-19T13:57:05Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
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      <origin>2692607</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3141" sha1="dj25ntaci0v36t6y10f3fyneiw3rqdm" xml:space="preserve">== Definition ==
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb{C}&lt;/math&gt;, and let &lt;math&gt;z \in \mathbb{C}&lt;/math&gt; be a point that &lt;math&gt;\Gamma&lt;/math&gt; does not intersect. Then

&lt;center&gt;&lt;math&gt;n(\Gamma, z) := \frac{1}{2\pi i} \int_\Gamma \frac{1}{w-z} \, dw&lt;/math&gt;&lt;/center&gt;  
is called the *winding number* of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.=== Motivation ===
First, consider the case where &lt;math&gt;\Gamma = \gamma&lt;/math&gt; consists of a single closed curve. Then &lt;math&gt;\gamma&lt;/math&gt; is [[w:en:Homologous|homologous]] in &lt;math&gt;\mathbb{C} \setminus {z}&lt;/math&gt; to an &lt;math&gt;n&lt;/math&gt;-fold (for some &lt;math&gt;n \in \mathbb{Z}&lt;/math&gt;) traversed circle &lt;math&gt;\partial D_r(z)&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;r &gt; 0&lt;/math&gt;. Now,
:&lt;math&gt;\int_\gamma \frac{1}{w-z} , dw = \int_{n \cdot \partial B(z)} \frac{1}{w-z} , dw = n \int_{\partial B(z)} \frac{1}{w-z} , dw = 2\pi i \cdot n.&lt;/math&gt;
Thus, this integral counts how many times the curve &lt;math&gt;\gamma&lt;/math&gt; winds around the point &lt;math&gt;z&lt;/math&gt;.

=== Task ===
Let the closed integration path &lt;math&gt;\gamma : [-\pi, +\pi] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;\gamma(t) := \left(2 + \cos(t)\right) \cdot e^{2it}.&lt;/math&gt;

1. Plot the trace of the integration path.

2. Determine the winding number &lt;math&gt;n(\gamma, 1+i)&lt;/math&gt;.

3. Determine the winding number &lt;math&gt;n(\gamma, 0)&lt;/math&gt;.

4. Determine the winding number &lt;math&gt;n(\gamma, 1)&lt;/math&gt;.

== Additivity of the Integral ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \cdot \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, due to the additivity of the integral, we have
:&lt;math&gt;n(\Gamma, z) = \sum_{i=1}^k n_i \cdot n(\gamma_i, z).&lt;/math&gt;
Thus, the winding number also counts how many times the point &lt;math&gt;z&lt;/math&gt; is encircled.

== Length of the Cycle ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, the length of the cycle is defined additively over the lengths of the individual integration paths:
:&lt;math&gt;L(\Gamma) := \sum_{i=1}^k n_i \cdot L(\gamma_i).&lt;/math&gt;

== See Also ==

*[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]
== Page Information ==
You can display this page as 
'''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Winding%20number&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Winding%20number&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''
=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Umlaufzahl|Kurs:Funktionentheorie/Umlaufzahl]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Umlaufzahl&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;02:18&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Umlaufzahl]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
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      <text bytes="2840" sha1="km0tjv4pqy2afof1b8c6dfwmagrxjx4" xml:space="preserve">== Definition ==
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb{C}&lt;/math&gt;, and let &lt;math&gt;z \in \mathbb{C}&lt;/math&gt; be a point that &lt;math&gt;\Gamma&lt;/math&gt; does not intersect. Then

&lt;center&gt;&lt;math&gt;n(\Gamma, z) := \frac{1}{2\pi i} \int_\Gamma \frac{1}{w-z} \, dw&lt;/math&gt;&lt;/center&gt;  
is called the *winding number* of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.=== Motivation ===
First, consider the case where &lt;math&gt;\Gamma = \gamma&lt;/math&gt; consists of a single closed curve. Then &lt;math&gt;\gamma&lt;/math&gt; is [[w:en:Homologous|homologous]] in &lt;math&gt;\mathbb{C} \setminus {z}&lt;/math&gt; to an &lt;math&gt;n&lt;/math&gt;-fold (for some &lt;math&gt;n \in \mathbb{Z}&lt;/math&gt;) traversed circle &lt;math&gt;\partial D_r(z)&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;r &gt; 0&lt;/math&gt;. Now,
:&lt;math&gt;\int_\gamma \frac{1}{w-z} , dw = \int_{n \cdot \partial B(z)} \frac{1}{w-z} , dw = n \int_{\partial B(z)} \frac{1}{w-z} , dw = 2\pi i \cdot n.&lt;/math&gt;
Thus, this integral counts how many times the curve &lt;math&gt;\gamma&lt;/math&gt; winds around the point &lt;math&gt;z&lt;/math&gt;.

=== Task ===
Let the closed integration path &lt;math&gt;\gamma : [-\pi, +\pi] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;\gamma(t) := \left(2 + \cos(t)\right) \cdot e^{2it}.&lt;/math&gt;

1. Plot the trace of the integration path.

2. Determine the winding number &lt;math&gt;n(\gamma, 1+i)&lt;/math&gt;.

3. Determine the winding number &lt;math&gt;n(\gamma, 0)&lt;/math&gt;.

4. Determine the winding number &lt;math&gt;n(\gamma, 1)&lt;/math&gt;.

== Additivity of the Integral ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \cdot \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, due to the additivity of the integral, we have
:&lt;math&gt;n(\Gamma, z) = \sum_{i=1}^k n_i \cdot n(\gamma_i, z).&lt;/math&gt;
Thus, the winding number also counts how many times the point &lt;math&gt;z&lt;/math&gt; is encircled.

== Length of the Cycle ==
For a cycle &lt;math&gt;\Gamma = \sum_{i=1}^k n_i \gamma_i&lt;/math&gt; with closed &lt;math&gt;\gamma_i&lt;/math&gt;, the length of the cycle is defined additively over the lengths of the individual integration paths:
:&lt;math&gt;L(\Gamma) := \sum_{i=1}^k n_i \cdot L(\gamma_i).&lt;/math&gt;

== See Also ==

*[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]

=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl
Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Umlaufzahl|Kurs:Funktionentheorie/Umlaufzahl]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Umlaufzahl
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Umlaufzahl&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;02:18&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie/Umlaufzahl]]
&lt;/noinclude&gt;

[[Category:Wiki2Reveal]]</text>
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|Source={{own|Young1lim}}
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{{Information
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|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Source={{own|Young1lim}}
|Date=2024-12-19
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{{Information
|Description=C04.SA1: Applications of Pointers 1A (20241214 - 20241213)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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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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      <comment>{{Information
|Description=C04.SA1: Applications of Pointers 1A (20241216 - 20241214)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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{{Information
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|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}
== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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      <comment>{{Information
|Description=C04.SA1: Applications of Pointers 1A (20241217 - 20241216)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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      <text bytes="280" sha1="k39u8y6f4okwea1n9ftq120vtonyro5" xml:space="preserve">== Summary ==
{{Information
|Description=C04.SA1: Applications of Pointers 1A (20241217 - 20241216)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}
== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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    <title>File:VLSI.Arith.5A.CSkip.20241216.pdf</title>
    <ns>6</ns>
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    <revision>
      <id>2692623</id>
      <timestamp>2024-12-19T14:29:22Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=VLSI.Arith: Carry Skip Adders 1A (20241216- 20241214)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
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      <text bytes="276" sha1="c3eowe805izu4icsbkk8wdup5sjaw1h" xml:space="preserve">== Summary ==
{{Information
|Description=VLSI.Arith: Carry Skip Adders 1A (20241216- 20241214)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  <page>
    <title>File:VLSI.Arith.5A.CSkip.20241217.pdf</title>
    <ns>6</ns>
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    <revision>
      <id>2692627</id>
      <timestamp>2024-12-19T14:31:03Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=VLSI.Arith: Carry Skip Adders 1A (20241217- 20241216)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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      <text bytes="276" sha1="luvjthfy4fp82nikx5hz1mxzz3cl9vp" xml:space="preserve">== Summary ==
{{Information
|Description=VLSI.Arith: Carry Skip Adders 1A (20241217- 20241216)
|Source={{own|Young1lim}}
|Date=2024-12-19
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  </page>
  <page>
    <title>Boolf prop/3-ary/squad</title>
    <ns>0</ns>
    <id>317354</id>
    <revision>
      <id>2692648</id>
      <timestamp>2024-12-19T19:52:59Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;20&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !c...&quot;</comment>
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      <text bytes="5801" sha1="mwwsr42sagqifcmi0izicfa6h56xhg6" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;20&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| squad
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 1, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930889952488499230423029593188005934847229955.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 170
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 3, 4, 5, 16, 17, 238, 239, 250, 251, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_27140096050294567090646637084536785591997023268932163910966462456505609420860.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 153
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 7, 18, 19, 20, 21, 234, 235, 236, 237, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1357352650530606598947218565922298041682654699338805583030703314046163878080.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 136
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 9, 32, 33, 64, 65, 190, 191, 222, 223, 246, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_339234656657409796350799267682912243180441980833726968996884567057930126080.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[10, 11, 12, 13, 34, 35, 48, 49, 68, 69, 80, 81, 174, 175, 186, 187, 206, 207, 220, 221, 242, 243, 244, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_106010828942001592264551517252273245832597465068946641349143452266518297600.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 238
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 15, 50, 51, 84, 85, 170, 171, 204, 205, 240, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5300541194412286014878813768484650261679549215412055478847655584850821120.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 150
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[22, 23, 232, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_20705239040371691362304267586831076357353326916511159665487572683980800.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 135
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[24, 25, 36, 37, 66, 67, 188, 189, 218, 219, 230, 231]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5177573507594122483391826915683192345043009907725837058281872629432320.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 45
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[26, 27, 28, 29, 38, 39, 52, 53, 70, 71, 82, 83, 172, 173, 184, 185, 202, 203, 216, 217, 226, 227, 228, 229]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1617912756187374168092891647995110219353319034674222990352784325345280.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 30
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[30, 31, 54, 55, 86, 87, 168, 169, 200, 201, 224, 225]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_80879844822266053283405129357078610224829424474093477939445822914560.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[40, 41, 72, 73, 96, 97, 158, 159, 182, 183, 214, 215]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_78984218769807837609955415061230177455423158000941432336988241920.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 130
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[42, 43, 76, 77, 112, 113, 142, 143, 178, 179, 212, 213]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_19746054689003844161525772393371441188087545220498281399047946240.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 27
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[44, 45, 56, 57, 74, 75, 88, 89, 98, 99, 100, 101, 154, 155, 156, 157, 166, 167, 180, 181, 198, 199, 210, 211]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4937718880094579942623044377107019784445803548159211535356395520.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 46
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[46, 47, 58, 59, 78, 79, 92, 93, 114, 115, 116, 117, 138, 139, 140, 141, 162, 163, 176, 177, 196, 197, 208, 209]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1234429719161563544230119874054978799700774637947426872607375360.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 235
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[60, 61, 90, 91, 102, 103, 152, 153, 164, 165, 194, 195]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_75325220894809374730225284686057089418103311941988725555200.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 190
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[62, 63, 94, 95, 118, 119, 136, 137, 160, 161, 192, 193]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18831305210544547464837931740571053495131825066503537950720.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 232
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[104, 105, 150, 151]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4281743078117940490403668863359757250902097920.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 189
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[106, 107, 108, 109, 120, 121, 134, 135, 146, 147, 148, 149]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1338110050115187140608094783220386605773619200.svg|420px]]
|-
|class=&quot;size&quot;| 12
|class=&quot;prop&quot;| 142
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[110, 111, 122, 123, 124, 125, 130, 131, 132, 133, 144, 145]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_66922732295181095471268452244212479678218240.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 126
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[126, 127, 128, 129]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1276058875953519237987654777869130792960.svg|420px]]
|}

[[Category:Boolf prop/3-ary|squad]]</text>
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  <page>
    <title>Boolf prop/3-ary/company</title>
    <ns>0</ns>
    <id>317355</id>
    <revision>
      <id>2692649</id>
      <timestamp>2024-12-19T19:59:21Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !cla...&quot;</comment>
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&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| company
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 1, 126, 127, 128, 129, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86844066927987146567678238756515930891228547375183942267580842783803978022915.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 170
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 4, 5, 14, 15, 16, 17, 42, 43, 50, 51, 76, 77, 84, 85, 110, 111, 112, 113, 122, 123, 124, 125, 130, 131, 132, 133, 142, 143, 144, 145, 170, 171, 178, 179, 204, 205, 212, 213, 238, 239, 240, 241, 250, 251, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_27145396591508725431350519742466862937384369440431134780118052605969186406460.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 153
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 18, 19, 20, 21, 24, 25, 36, 37, 60, 61, 66, 67, 90, 91, 102, 103, 106, 107, 108, 109, 120, 121, 134, 135, 146, 147, 148, 149, 152, 153, 164, 165, 188, 189, 194, 195, 218, 219, 230, 231, 234, 235, 236, 237, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1357357828104114268394922852559926565150399615546410821754293924513292484800.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 136
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[8, 9, 32, 33, 62, 63, 64, 65, 94, 95, 118, 119, 136, 137, 160, 161, 190, 191, 192, 193, 222, 223, 246, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_339234656657409815182104478227459708018373721404780464128709633561468076800.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[10, 11, 12, 13, 34, 35, 46, 47, 48, 49, 58, 59, 68, 69, 78, 79, 80, 81, 92, 93, 114, 115, 116, 117, 138, 139, 140, 141, 162, 163, 174, 175, 176, 177, 186, 187, 196, 197, 206, 207, 208, 209, 220, 221, 242, 243, 244, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_106010828943236021983713080796503365706652443868647415987090879139125672960.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 150
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[22, 23, 104, 105, 150, 151, 232, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_20705239040371691362304271868574154475293817320180023025244823586078720.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 45
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[26, 27, 28, 29, 38, 39, 44, 45, 52, 53, 56, 57, 70, 71, 74, 75, 82, 83, 88, 89, 98, 99, 100, 101, 154, 155, 156, 157, 166, 167, 172, 173, 180, 181, 184, 185, 198, 199, 202, 203, 210, 211, 216, 217, 226, 227, 228, 229]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1617917693906254262672834271039487326373103480477771149564319681740800.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 30
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[30, 31, 40, 41, 54, 55, 72, 73, 86, 87, 96, 97, 158, 159, 168, 169, 182, 183, 200, 201, 214, 215, 224, 225]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_80958829041035861121015084772139840402284847632094419371782811156480.svg|420px]]
|}

[[Category:Boolf prop/3-ary|company]]</text>
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  <page>
    <title>Boolf prop/3-ary/octant</title>
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    <revision>
      <id>2692653</id>
      <timestamp>2024-12-19T20:06:17Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| octant !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 32 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list small...&quot;</comment>
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&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| octant
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 6, 10, 12, 18, 20, 24, 30, 34, 36, 40, 46, 48, 54, 58, 60, 66, 68, 72, 78, 80, 86, 90, 92, 96, 102, 106, 108, 114, 116, 120, 126]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86504076467481452450826130951662474305.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 7, 11, 13, 19, 21, 25, 31, 35, 37, 41, 47, 49, 55, 59, 61, 67, 69, 73, 79, 81, 87, 91, 93, 97, 103, 107, 109, 115, 117, 121, 127]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_173008152934962904901652261903324948610.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 4, 8, 14, 16, 22, 26, 28, 32, 38, 42, 44, 50, 52, 56, 62, 64, 70, 74, 76, 82, 84, 88, 94, 98, 100, 104, 110, 112, 118, 122, 124]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_26923379172831368703632071525593596180.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 9, 15, 17, 23, 27, 29, 33, 39, 43, 45, 51, 53, 57, 63, 65, 71, 75, 77, 83, 85, 89, 95, 99, 101, 105, 111, 113, 119, 123, 125]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_53846758345662737407264143051187192360.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[128, 134, 138, 140, 146, 148, 152, 158, 162, 164, 168, 174, 176, 182, 186, 188, 194, 196, 200, 206, 208, 214, 218, 220, 224, 230, 234, 236, 242, 244, 248, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_29435811888664441966364245270295815197988642535827134593354722855461406638080.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[129, 135, 139, 141, 147, 149, 153, 159, 163, 165, 169, 175, 177, 183, 187, 189, 195, 197, 201, 207, 209, 215, 219, 221, 225, 231, 235, 237, 243, 245, 249, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_58871623777328883932728490540591630395977285071654269186709445710922813276160.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[130, 132, 136, 142, 144, 150, 154, 156, 160, 166, 170, 172, 178, 180, 184, 190, 192, 198, 202, 204, 210, 212, 216, 222, 226, 228, 232, 238, 240, 246, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9161551190440956508159416399266820752987924897079407265310013611365713838080.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[131, 133, 137, 143, 145, 151, 155, 157, 161, 167, 171, 173, 179, 181, 185, 191, 193, 199, 203, 205, 211, 213, 217, 223, 227, 229, 233, 239, 241, 247, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18323102380881913016318832798533641505975849794158814530620027222731427676160.svg|420px]]
|}

[[Category:Boolf prop/3-ary|octant]]</text>
      <sha1>cqbc8vcls2jv95ltkcajkb32fh6tjxc</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/consul</title>
    <ns>0</ns>
    <id>317357</id>
    <revision>
      <id>2692657</id>
      <timestamp>2024-12-19T20:13:21Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| consul !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 32 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list small...&quot;</comment>
      <origin>2692657</origin>
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      <text bytes="3339" sha1="22ga50bzd5bs5oievbvo84ph9aktl48" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| consul
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 14, 15, 50, 51, 60, 61, 84, 85, 90, 91, 102, 103, 104, 105, 150, 151, 152, 153, 164, 165, 170, 171, 194, 195, 204, 205, 240, 241, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86849367469181558929018338465098072013517570674993328172038707360759325704195.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 12, 13, 48, 49, 62, 63, 86, 87, 88, 89, 100, 101, 106, 107, 148, 149, 154, 155, 166, 167, 168, 169, 192, 193, 206, 207, 242, 243, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_21732218896774435804810363594541994255593366971344855158125389901454660087820.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[4, 5, 10, 11, 54, 55, 56, 57, 80, 81, 94, 95, 98, 99, 108, 109, 146, 147, 156, 157, 160, 161, 174, 175, 198, 199, 200, 201, 244, 245, 250, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5512562842108565134317626013918721218214467978946355785360823781044728630320.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 8, 9, 52, 53, 58, 59, 82, 83, 92, 93, 96, 97, 110, 111, 144, 145, 158, 159, 162, 163, 172, 173, 196, 197, 202, 203, 246, 247, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1696173182187268540957397984723517243197055529240641257364553210254443152320.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[16, 17, 30, 31, 34, 35, 44, 45, 68, 69, 74, 75, 118, 119, 120, 121, 134, 135, 136, 137, 180, 181, 186, 187, 210, 211, 220, 221, 224, 225, 238, 239]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1325221238350303760652067221981340365549956035431476386074830363229880320.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[18, 19, 28, 29, 32, 33, 46, 47, 70, 71, 72, 73, 116, 117, 122, 123, 132, 133, 138, 139, 182, 183, 184, 185, 208, 209, 222, 223, 226, 227, 236, 237]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_331627565200081742394887269402276807612336138645854501961325007380152320.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[20, 21, 26, 27, 38, 39, 40, 41, 64, 65, 78, 79, 114, 115, 124, 125, 130, 131, 140, 141, 176, 177, 190, 191, 214, 215, 216, 217, 228, 229, 234, 235]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_84115428522608461362455658191217636228505462875315475196770315869880320.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[22, 23, 24, 25, 36, 37, 42, 43, 66, 67, 76, 77, 112, 113, 126, 127, 128, 129, 142, 143, 178, 179, 188, 189, 212, 213, 218, 219, 230, 231, 232, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_25882832294020502849540256028287938132713478431020204876828713492152320.svg|420px]]
|}

[[Category:Boolf prop/3-ary|consul]]</text>
      <sha1>22ga50bzd5bs5oievbvo84ph9aktl48</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/patron dominion</title>
    <ns>0</ns>
    <id>317358</id>
    <revision>
      <id>2692664</id>
      <timestamp>2024-12-19T20:23:09Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| patron dominion !class=&quot;prop&quot;| patron principality...&quot;</comment>
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      <text bytes="3666" sha1="hfpe298ibkt7q5j8adcyphdfaxa7jur" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| patron dominion
!class=&quot;prop&quot;| patron principality
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 0)
|class=&quot;prop&quot;| (0, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022336315325202904699959177005221122795925822929966558420718528929726465.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 1)
|class=&quot;prop&quot;| (42, 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 31, 41, 55, 73, 87, 97, 127, 129, 159, 169, 183, 201, 215, 225, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044672630650405809399918354010442245591851645859933116841437057859452930.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (44, 1)
|class=&quot;prop&quot;| (8, 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 4, 14, 16, 26, 28, 38, 42, 44, 50, 52, 56, 70, 74, 76, 82, 84, 88, 98, 100, 110, 112, 122, 124, 130, 132, 142, 144, 154, 156, 166, 170, 172, 178, 180, 184, 198, 202, 204, 210, 212, 216, 226, 228, 238, 240, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9048466069808806445868260805100377992290565271178205085963067390096289382420.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (44, 0)
|class=&quot;prop&quot;| (2, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 15, 17, 27, 29, 39, 43, 45, 51, 53, 57, 71, 75, 77, 83, 85, 89, 99, 101, 111, 113, 123, 125, 131, 133, 143, 145, 155, 157, 167, 171, 173, 179, 181, 185, 199, 203, 205, 211, 213, 217, 227, 229, 239, 241, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18096932139617612891736521610200755984581130542356410171926134780192578764840.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (6, 0)
|class=&quot;prop&quot;| (8, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 10, 12, 18, 20, 24, 34, 36, 46, 48, 58, 60, 66, 68, 78, 80, 90, 92, 102, 106, 108, 114, 116, 120, 134, 138, 140, 146, 148, 152, 162, 164, 174, 176, 186, 188, 194, 196, 206, 208, 218, 220, 230, 234, 236, 242, 244, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_487789552349116763459545311118809976952350686471686079247128267884139385920.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (6, 1)
|class=&quot;prop&quot;| (2, 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[7, 11, 13, 19, 21, 25, 35, 37, 47, 49, 59, 61, 67, 69, 79, 81, 91, 93, 103, 107, 109, 115, 117, 121, 135, 139, 141, 147, 149, 153, 163, 165, 175, 177, 187, 189, 195, 197, 207, 209, 219, 221, 231, 235, 237, 243, 245, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_975579104698233526919090622237619953904701372943372158494256535768278771840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (42, 1)
|class=&quot;prop&quot;| (0, 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 22, 32, 62, 64, 94, 104, 118, 136, 150, 160, 190, 192, 222, 232, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113085120632150062291155594166442760724283005074033548050578292795018051840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (42, 0)
|class=&quot;prop&quot;| (42, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 23, 33, 63, 65, 95, 105, 119, 137, 151, 161, 191, 193, 223, 233, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226170241264300124582311188332885521448566010148067096101156585590036103680.svg|420px]]
|}

[[Category:Boolf prop/3-ary|patron dominion]]</text>
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    </revision>
    <revision>
      <id>2692678</id>
      <parentid>2692664</parentid>
      <timestamp>2024-12-19T20:33:49Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692678</origin>
      <model>wikitext</model>
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      <text bytes="3888" sha1="cqio1yut6a08dqtrksr9082cehr1y0d" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| patron dominion
!class=&quot;prop&quot;| patron principality
!class=&quot;prop&quot;| patron king index and quadrant
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 0)
|class=&quot;prop&quot;| (0, 0)
|class=&quot;prop&quot;| (0, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022336315325202904699959177005221122795925822929966558420718528929726465.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 1)
|class=&quot;prop&quot;| (42, 2)
|class=&quot;prop&quot;| (0, 3)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 31, 41, 55, 73, 87, 97, 127, 129, 159, 169, 183, 201, 215, 225, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044672630650405809399918354010442245591851645859933116841437057859452930.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (44, 1)
|class=&quot;prop&quot;| (8, 2)
|class=&quot;prop&quot;| (2, 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 4, 14, 16, 26, 28, 38, 42, 44, 50, 52, 56, 70, 74, 76, 82, 84, 88, 98, 100, 110, 112, 122, 124, 130, 132, 142, 144, 154, 156, 166, 170, 172, 178, 180, 184, 198, 202, 204, 210, 212, 216, 226, 228, 238, 240, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9048466069808806445868260805100377992290565271178205085963067390096289382420.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (44, 0)
|class=&quot;prop&quot;| (2, 0)
|class=&quot;prop&quot;| (2, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 15, 17, 27, 29, 39, 43, 45, 51, 53, 57, 71, 75, 77, 83, 85, 89, 99, 101, 111, 113, 123, 125, 131, 133, 143, 145, 155, 157, 167, 171, 173, 179, 181, 185, 199, 203, 205, 211, 213, 217, 227, 229, 239, 241, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18096932139617612891736521610200755984581130542356410171926134780192578764840.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (6, 0)
|class=&quot;prop&quot;| (8, 0)
|class=&quot;prop&quot;| (2, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 10, 12, 18, 20, 24, 34, 36, 46, 48, 58, 60, 66, 68, 78, 80, 90, 92, 102, 106, 108, 114, 116, 120, 134, 138, 140, 146, 148, 152, 162, 164, 174, 176, 186, 188, 194, 196, 206, 208, 218, 220, 230, 234, 236, 242, 244, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_487789552349116763459545311118809976952350686471686079247128267884139385920.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| (6, 1)
|class=&quot;prop&quot;| (2, 2)
|class=&quot;prop&quot;| (2, 3)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[7, 11, 13, 19, 21, 25, 35, 37, 47, 49, 59, 61, 67, 69, 79, 81, 91, 93, 103, 107, 109, 115, 117, 121, 135, 139, 141, 147, 149, 153, 163, 165, 175, 177, 187, 189, 195, 197, 207, 209, 219, 221, 231, 235, 237, 243, 245, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_975579104698233526919090622237619953904701372943372158494256535768278771840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (42, 1)
|class=&quot;prop&quot;| (0, 2)
|class=&quot;prop&quot;| (0, 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 22, 32, 62, 64, 94, 104, 118, 136, 150, 160, 190, 192, 222, 232, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113085120632150062291155594166442760724283005074033548050578292795018051840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (42, 0)
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      <comment>New resource with &quot;== 5.1 - What are Healthy Behaviors? ==  * '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly).  * '''Health education''' - efforts to educate the public on maintaining healthy habits or red...&quot;</comment>
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      <text bytes="2914" sha1="4nsgngkea4nsqr3nqv7hg5zpwgz78ma" xml:space="preserve">== 5.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology.

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.</text>
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      <text bytes="3077" sha1="ojpxcg8ev92tzfxdn5t6sexn4yt2lkx" xml:space="preserve">== 5.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.</text>
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      <text bytes="7400" sha1="apvirhdrvlen1q2u4yaamdcykqolv33" xml:space="preserve">== 5.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in.
** '''Health Belief Model''' - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not. Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior -'''</text>
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      <text bytes="7873" sha1="owbntcdh0vgjlsx76gmdn8r1wms8xgi" xml:space="preserve">== 5.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in.
** '''Health Belief Model''' - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not. Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior -''' Behavior originates from intention, which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture. 
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' - Splits into two phases, when a decision to act is made and when the action is carried out.</text>
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      <text bytes="8153" sha1="2xi8cltylsfb899emfmv5m3uleajtud" xml:space="preserve">== 5.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' - Splits into two phases, when a decision to act is made and when the action is carried out.</text>
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      <text bytes="9412" sha1="c6nzhdxro6n8tumus1npnm8vnjsjzjn" xml:space="preserve">== 7.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out.

== 7.3 - Comparing the Models and Their Limitations ==
Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. 

* '''HBM''' lacks quantified data vs. the '''TPB''', but has recieved a lot of empirical support. Stuff like the beliefs about severity have low predictive value.
* The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change.
* The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness. 
* The '''TTM''' has cannot progress/digress without skipping steps. 

'''Changing Behaviors: Interventions'''

* '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. 
* What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''.</text>
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      <text bytes="9488" sha1="90adtes6dnmcvk9l8honeokrzsdskmu" xml:space="preserve">== 7.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out. Most predictive of behavioral intentions due to emphasis on self-efficacy. 

== 7.3 - Comparing the Models and Their Limitations ==
Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. 

* '''HBM''' lacks quantified data vs. the '''TPB''', but has recieved a lot of empirical support. Stuff like the beliefs about severity have low predictive value.
* The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change.
* The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness. 
* The '''TTM''' has cannot progress/digress without skipping steps. 

'''Changing Behaviors: Interventions'''

* '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. 
* What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''.</text>
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      <text bytes="9567" sha1="m3lpvim4jkdi5xc0rkjwu8mgcq00p9v" xml:space="preserve">== 7.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out. Most predictive of behavioral intentions due to emphasis on self-efficacy. 

== 7.3 - Comparing the Models and Their Limitations ==
Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. 

* '''HBM''' lacks rigorously quantified data vs. the '''TPB''', but has recieved a lot of empirical support. Stuff like the beliefs about severity have low predictive value.
* The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change.
* The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness, but TPB DOES include congitive elements (attitudes, perceived beliefs, etc.). 
* The '''TTM''' does not allow individuals to skip steps. 

'''Changing Behaviors: Interventions'''

* '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. 
* What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''.</text>
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      <text bytes="9576" sha1="sw3ewr2kyzhgb7pc8de5mc44i0zmkv1" xml:space="preserve">== 7.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out. Most predictive of behavioral intentions due to emphasis on self-efficacy. 

== 7.3 - Comparing the Models and Their Limitations ==
Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. 

* '''HBM''' lacks rigorously quantified data vs. the '''TPB''', but has recieved a lot of empirical support. For TPB, stuff like the beliefs about severity have low predictive value.
* The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change.
* The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness, but TPB DOES include congitive elements (attitudes, perceived beliefs, etc.). 
* The '''TTM''' does not allow individuals to skip steps. 

'''Changing Behaviors: Interventions'''

* '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. 
* What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''.</text>
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      <text bytes="9628" sha1="gu7q13ga4udf0jno9kouqx8i999xhog" xml:space="preserve">== 7.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out. Most predictive of behavioral intentions due to emphasis on self-efficacy. 

== 7.3 - Comparing the Models and Their Limitations ==
Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. 

* '''HBM''' lacks rigorously quantified data vs. the '''TPB''', but has recieved a lot of empirical support. For TPB, stuff like the beliefs about severity have low predictive value.
* The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change.
* The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness, but TPB DOES include congitive elements (attitudes, perceived beliefs, etc.). 
* The '''TTM''' does not allow individuals to skip steps, bringing criticisms that it is circular and flawed. 

'''Changing Behaviors: Interventions'''

* '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. 
* What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''.</text>
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      <text bytes="9628" sha1="80fln2h7n8w2c4auwju9krxean5f4p8" xml:space="preserve">== 7.1 - What are Healthy Behaviors? ==

* '''Healthy behaviors''' are behaviors that maintain and uphold health. These can be small, such as avoiding your phone once you've gotten up from your bed in the morning, to huge, such as avoiding harmful drugs. Others are ''episodic'' (getting a flu-shot; essentially short-term) or ''long-term'' (eating well and exercising regularly). 
* '''Health education''' - efforts to educate the public on maintaining healthy habits or reducing unhealthy habits, and accounts for the person's interpersonal relationships, institutions, and other aspects of their surrounding environment. More focus is put on the person's individual factors in health psychology, but the shift from the individual and their habits to social impacts came about in the 1970s (which is an overall benefit in the assessment of health psychology).

'''The Healthy People Programs''' 

* The '''[[w:Healthy_People_program|Healthy People Program]],''' a science-based, 10-year objective for promoting national health, lists the major health concerns in the US ('''leading health indicators'''). Consists of a statement (a national health objective) and goals to reduce health threats.

'''What Determines Health Behaviors?'''

* Healthy habits on a personal level + help of medical institutions and medical professionals (like regular check-ups). Figuring out the factors behind our health habits include a thorough examination of our social, cultural, and economic backgrounds, and it cannot be traced to one origin. 
** '''Biological factors''' can include overweight parents (inherit metabolic rates) or a gene (dopamine D&lt;sub&gt;2&lt;/sub&gt; receptor gene is associated with alcoholism).
** '''Social factors''' include what/who you are exposed to and how these things can influence you.
** '''Psychological factors''' can include The Big Five personality traits.

''In a demonstration of the long-term impact of personality, Hampson et al. (2007) studied 1,054 participants in the Hawaii Personality and Health study. This population-based longitudinal study of personality and health spanned 40 years from childhood to midlife. The study found that childhood agreeableness and conscientiousness influenced adult health status mediated by healthy eating habits and smoking. Similarly, Caspi et al. (1997) followed individuals from infancy until the age of 21. Results showed that a constellation of adolescent personality traits (with developmental origins in childhood) did link to different health-risk behaviors at 21. The study also determined that associations between personality and different health-risk behaviors were not seen simply because the same people engaged in different health-risk behaviors. Instead, the associations implicated the same personality type in different but related behaviors. Therefore, in planning campaigns, perhaps health professionals need to design programs that appeal to the unique psychological makeup of persons most at risk for particular behaviors (Caspi et al., 1997).''

NOTE: Social &amp; psychological factors can interact.

== 7.2 - Changing Health Behaviors ==

* When setting a goal, bear in mind that you should set a goal through a &quot;behavior contract&quot; (what is the method to achieving the goal?), monitoring and documenting your progress, and then reinforcing achievements through rewards (a candy bar is a classic example). You should include ''difficulty, time frame,'' and ''type of goal setting'' (either self-set or prescribed by a doctor). 
** Self-monitoring is crucial, and don't forget to weigh in the biopsychosocial factors in your progress (physical health, mental health, and social support). 
** Include barriers, both physical and mental - then attach a solution. Account for info in previous chapters, such as good coping methods when facing stress that may harm your progress. This is essentially diving into the context of the behavior you want to change. 

'''Importance of Theory'''

''Scientific theories guide our search to understand why behaviors are difficult to change and to predict successful change. At the core, a theory explains behavior and suggests ways to influence and change behavior. If you want to successfully adopt healthy behaviors, you can rely on explanatory or predictive theories to identify key factors, and theories and models (analogous to theories) of behavior change to help focus on the process. There are many available options, so there is no need to begin from scratch or use personal brainstorming. You can certainly start without reading the next section, but the reality is that theoretically informed health behavior change programs are more effective than those without a theoretical basis (Glanz et al., 2015). Mind you, not all published research will explicitly use a theory. In one review, only 68% of articles were informed by theory (Painter et al., 2008). However, without theory one cannot identify the factors most plausibly related to what we are interested in (Rothman et al., 2008). In the next section I discuss some of the most common theories and models used in health psychology. Make sure to note the variables that can help you change your own health behaviors.''

'''Key Theories of Health Behavior Change'''

* Many theories derived to explain the reasoning behind certain behaviors and why others may avoid such behaviors, and a lot of these theories originate from the '''Social Cognitive Theory''' (a comprehensive theory of behavior change that the traits of people, their environments, and their health behaviors all interact with each other and determien whether each person performs a health behavior). ''[[w:Self-efficacy|Self-efficacy]]'' (self-confidence) is the most &quot;central determinant&quot; of health behavior change. In addition to SCT, we also have...
** '''Transtheoretical Model''' (TTM) - People go through 6 stages when they want to change a behavior. It's like a roadmap for making changes in your life. Stages include precontemplation, contemplation, preparation, action, maintenance, and termination phase (&lt;20% of smokers, Snow et. al. 1992). Interventions are tailored directly to the stage the person is in. Note the prefix &quot;''Trans-''&quot; is included because it identifies common themes across different intervention theories through the six separate stages. 
** '''Health Belief Model''' (HBM) - People may/may not believe that it is easy to change their behavior, and this effects if they do the behavior or not (essentially, confidence is key to implementing positive health behavior changes). Mixes behaviorist (do we get a reward after doing this?), cognitive (expectations of an activity achieving a certain outcome), and social views. [''How does the HBM explain health behavior? The model, built on Hochbaum’s (1958) surveys, suggests that '''individuals will perform healthy behaviors if they believe they are susceptible to the health issue''', if they believe not performing the behavior will have severe consequences, if they believe that their behavior will be beneficial in reducing the severity or susceptibility, if they believe there are benefits to taking action, and if they believe that the anticipated benefits of the behavior outweigh its costs (or barriers). Individuals must also receive a trigger or cue in order to act (Aiken et al., 2012).''] - Issues with this theory surround inconsistency with measuring the various components with one another and the simultaneous measuring of health beliefs and health behaviors. '''Perceived barriers''' are the most critical component of the HBM &amp; culture can play a role as well.
** '''Theory of Planned Behavior''' (TPB) '''-''' Behavior originates from intention (which is a probability), which is dependent on their attitude toward the behavior, their preceptions of the social norms in accordance to that behavior [normative beliefs], and perceived control (self-efficacy). Also influenced by culture.
** '''Precaution Adoption Process Model''' - 7 stages from lack of awareness to action.
** '''Health Action Process Approach''' (HAPA) - Splits into two phases, when a decision to act is made and when the action is carried out. Most predictive of behavioral intentions due to emphasis on self-efficacy. 

== 7.3 - Comparing the Models and Their Limitations ==
Generally, the models don't include changes in mindset, the translation between beliefs/intentions and action, or the intention-behavior gap. 

* '''HBM''' lacks rigorously quantified data vs. the '''TPB''', but has recieved a lot of empirical support. For HBM, stuff like the beliefs about severity have low predictive value.
* The '''TPB''' and the '''TTM''' do not include information about the elements behind behavior change.
* The '''TPB''' has no emotional aspects, like HBM's perceived susceptibility to illness, but TPB DOES include congitive elements (attitudes, perceived beliefs, etc.). 
* The '''TTM''' does not allow individuals to skip steps, bringing criticisms that it is circular and flawed. 

'''Changing Behaviors: Interventions'''

* '''Interventions''' are medically backed programs that aim to assess levels of behaviors, introduce ways to change them, measure whether change has occurred, and assess the impact of the change. 
* What determines successful interventions? They should be '''based on theory''', be done at the '''appropriate level''', the '''size''' (duration and intensity) matters, should '''target people at risk''', appropriate for the '''risk group/risk factor''', be '''effective in objectives, prevent dropouts''', be '''ethical''', prevent '''relapse''', and be '''culturally sensitive'''.</text>
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    <revision>
      <id>2692695</id>
      <timestamp>2024-12-19T20:44:45Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| nameless 3 !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 16 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list...&quot;</comment>
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      <text bytes="4924" sha1="5ii1ba39gu9phqewgpxz4mrz94lngwf" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 3
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 6, 8, 14, 99, 101, 107, 109, 130, 132, 138, 140, 225, 231, 233, 239]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_900731818149502875539824974059074912173005084369928909466332538842661185.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 7, 9, 15, 98, 100, 106, 108, 131, 133, 139, 141, 224, 230, 232, 238]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_450365909074751437769912487032161033135462977738267072956465202331681410.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 4, 10, 12, 97, 103, 105, 111, 128, 134, 136, 142, 227, 229, 235, 237]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_277148251738308577089176915100246220267051134605474329441893104715043860.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 5, 11, 13, 96, 102, 104, 110, 129, 135, 137, 143, 226, 228, 234, 236]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_138574125869154288544588457558649735542646982850970673946668084316088360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[16, 22, 24, 30, 115, 117, 123, 125, 146, 148, 154, 156, 241, 247, 249, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_59030360434245820451377969499935533444170061209267661010785569265592643420160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[17, 23, 25, 31, 114, 116, 122, 124, 147, 149, 155, 157, 240, 246, 248, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_29515180217122910225688984750139705467565701709055070893274903500009072885760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[18, 20, 26, 28, 113, 119, 121, 127, 144, 150, 152, 158, 243, 245, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18163187825921790908116298308009736291421463157504365654303906510605114408960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[19, 21, 27, 29, 112, 118, 120, 126, 145, 151, 153, 159, 242, 244, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9081593912960895454058149154563669068522912668121214087768839573739166760960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[32, 38, 40, 46, 67, 69, 75, 77, 162, 164, 170, 172, 193, 199, 201, 207]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_209717968986781122735634565628935075002331022649724367966044160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[33, 39, 41, 47, 66, 68, 74, 76, 163, 165, 171, 173, 192, 198, 200, 206]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_104858984504658738991638544313817947561408053406517045032386560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[34, 36, 42, 44, 65, 71, 73, 79, 160, 166, 168, 174, 195, 197, 203, 205]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_64528605864189463103844648519167365888674027531967340174376960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[35, 37, 43, 45, 64, 70, 72, 78, 161, 167, 169, 175, 194, 196, 202, 204]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_32264302968716308829341424132472515640125275531045313980661760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[48, 54, 56, 62, 83, 85, 91, 93, 178, 180, 186, 188, 209, 215, 217, 223]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_13744076815517687659602546893057889075352765900372336179022670069760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[49, 55, 57, 63, 82, 84, 90, 92, 179, 181, 187, 189, 208, 214, 216, 222]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6872038408497315118556023640150373011384438188049501063242485596160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[50, 52, 58, 60, 81, 87, 89, 95, 176, 182, 184, 190, 211, 213, 219, 221]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4228946713915520653973562885352152490880141068335011605667968450560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[51, 53, 59, 61, 80, 86, 88, 94, 177, 183, 185, 191, 210, 212, 218, 220]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_2114473359357792015439719571945718784991250057202585697036649103360.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 3]]</text>
      <sha1>5ii1ba39gu9phqewgpxz4mrz94lngwf</sha1>
    </revision>
    <revision>
      <id>2692696</id>
      <parentid>2692695</parentid>
      <timestamp>2024-12-19T20:48:41Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692696</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5005" sha1="258upp3g97h42sy2og9vl7nlcbr4cfb" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;source lang=&quot;python&quot;&gt;
val = boolf.patron_index(3) ^ boolf.praetor(3)
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 3
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 6, 8, 14, 99, 101, 107, 109, 130, 132, 138, 140, 225, 231, 233, 239]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_900731818149502875539824974059074912173005084369928909466332538842661185.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 7, 9, 15, 98, 100, 106, 108, 131, 133, 139, 141, 224, 230, 232, 238]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_450365909074751437769912487032161033135462977738267072956465202331681410.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 4, 10, 12, 97, 103, 105, 111, 128, 134, 136, 142, 227, 229, 235, 237]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_277148251738308577089176915100246220267051134605474329441893104715043860.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 5, 11, 13, 96, 102, 104, 110, 129, 135, 137, 143, 226, 228, 234, 236]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_138574125869154288544588457558649735542646982850970673946668084316088360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[16, 22, 24, 30, 115, 117, 123, 125, 146, 148, 154, 156, 241, 247, 249, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_59030360434245820451377969499935533444170061209267661010785569265592643420160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[17, 23, 25, 31, 114, 116, 122, 124, 147, 149, 155, 157, 240, 246, 248, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_29515180217122910225688984750139705467565701709055070893274903500009072885760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[18, 20, 26, 28, 113, 119, 121, 127, 144, 150, 152, 158, 243, 245, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18163187825921790908116298308009736291421463157504365654303906510605114408960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[19, 21, 27, 29, 112, 118, 120, 126, 145, 151, 153, 159, 242, 244, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9081593912960895454058149154563669068522912668121214087768839573739166760960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[32, 38, 40, 46, 67, 69, 75, 77, 162, 164, 170, 172, 193, 199, 201, 207]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_209717968986781122735634565628935075002331022649724367966044160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[33, 39, 41, 47, 66, 68, 74, 76, 163, 165, 171, 173, 192, 198, 200, 206]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_104858984504658738991638544313817947561408053406517045032386560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[34, 36, 42, 44, 65, 71, 73, 79, 160, 166, 168, 174, 195, 197, 203, 205]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_64528605864189463103844648519167365888674027531967340174376960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[35, 37, 43, 45, 64, 70, 72, 78, 161, 167, 169, 175, 194, 196, 202, 204]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_32264302968716308829341424132472515640125275531045313980661760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[48, 54, 56, 62, 83, 85, 91, 93, 178, 180, 186, 188, 209, 215, 217, 223]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_13744076815517687659602546893057889075352765900372336179022670069760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[49, 55, 57, 63, 82, 84, 90, 92, 179, 181, 187, 189, 208, 214, 216, 222]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6872038408497315118556023640150373011384438188049501063242485596160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[50, 52, 58, 60, 81, 87, 89, 95, 176, 182, 184, 190, 211, 213, 219, 221]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4228946713915520653973562885352152490880141068335011605667968450560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[51, 53, 59, 61, 80, 86, 88, 94, 177, 183, 185, 191, 210, 212, 218, 220]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_2114473359357792015439719571945718784991250057202585697036649103360.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 3]]</text>
      <sha1>258upp3g97h42sy2og9vl7nlcbr4cfb</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/nameless 4</title>
    <ns>0</ns>
    <id>317363</id>
    <revision>
      <id>2692697</id>
      <timestamp>2024-12-19T20:52:02Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| nameless 4 !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 16 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list...&quot;</comment>
      <origin>2692697</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5010" sha1="6u7mtzvugutlzxqqclgw7oo4x8y4d6l" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 4
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 29, 33, 60, 71, 90, 102, 123, 132, 153, 165, 184, 195, 222, 226, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044733237871097417045490205538760474164395213398111671384344995827810305.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 28, 32, 61, 70, 91, 103, 122, 133, 152, 164, 185, 194, 223, 227, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022538488595551830631329085771900754604884405385804489715918477750960130.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 31, 35, 62, 69, 88, 100, 121, 134, 155, 167, 186, 193, 220, 224, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474011183309467774446211280426351952526512916562604391779217547965954523140.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 30, 34, 63, 68, 89, 101, 120, 135, 154, 166, 187, 192, 221, 225, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237005634622148888141557384886856634270301385308963421289001083797013266440.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 25, 37, 56, 67, 94, 98, 127, 128, 157, 161, 188, 199, 218, 230, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618504514523967775163109878064631098965107124286829118090211615581262577680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 24, 36, 57, 66, 95, 99, 126, 129, 156, 160, 189, 198, 219, 231, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809254846048737684642796634945773391589149634534855653076559633879136206880.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 27, 39, 58, 65, 92, 96, 125, 130, 159, 163, 190, 197, 216, 228, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904626128630991945261973200393752252521981432525136844492347768437081112640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 26, 38, 59, 64, 93, 97, 124, 131, 158, 162, 191, 196, 217, 229, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452313711512184424103090603022120544737747767387152777146909167455213453440.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 21, 41, 52, 79, 82, 110, 115, 140, 145, 173, 176, 203, 214, 234, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226184031303361284863412941307989648842752207751176135687412462083400270080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 20, 40, 53, 78, 83, 111, 114, 141, 144, 172, 177, 202, 215, 235, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113133426169253495190283695657846014730802627040050815513080582594618196480.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 23, 43, 54, 77, 80, 108, 113, 142, 147, 175, 178, 201, 212, 232, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56546007825840321216257311299859142575657892008703747172159499166302602240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 22, 42, 55, 76, 81, 109, 112, 143, 146, 174, 179, 200, 213, 233, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28283356542313373798311729864485873022686041594398287226769861131519395840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 17, 45, 48, 75, 86, 106, 119, 136, 149, 169, 180, 207, 210, 238, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14576488286272863347581805008971331325909835609498165357011196770267566080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 16, 44, 49, 74, 87, 107, 118, 137, 148, 168, 181, 206, 211, 239, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7950811794896582635665191441795677836192789899220656086234903284654612480.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 19, 47, 50, 73, 84, 104, 117, 138, 151, 171, 182, 205, 208, 236, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3644122071568215842645115616310007478820130206976227073562935641997066240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 18, 46, 51, 72, 85, 105, 116, 139, 150, 170, 183, 204, 209, 237, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1987702948724145670411417459853621617593601306213883306005486651130019840.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 4]]</text>
      <sha1>6u7mtzvugutlzxqqclgw7oo4x8y4d6l</sha1>
    </revision>
    <revision>
      <id>2692701</id>
      <parentid>2692697</parentid>
      <timestamp>2024-12-19T20:58:33Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692701</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5228" sha1="ewnfnssm94j5hojj0f0hwbjidf88agu" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

[[File:Set of 3-ary Boolean functions 1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|thumb|500px|The twins have the transposed pattern of [[Boolf prop/3-ary/nameless 1|nameless 1]]]]

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 4
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 29, 33, 60, 71, 90, 102, 123, 132, 153, 165, 184, 195, 222, 226, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044733237871097417045490205538760474164395213398111671384344995827810305.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 28, 32, 61, 70, 91, 103, 122, 133, 152, 164, 185, 194, 223, 227, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022538488595551830631329085771900754604884405385804489715918477750960130.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 31, 35, 62, 69, 88, 100, 121, 134, 155, 167, 186, 193, 220, 224, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474011183309467774446211280426351952526512916562604391779217547965954523140.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 30, 34, 63, 68, 89, 101, 120, 135, 154, 166, 187, 192, 221, 225, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237005634622148888141557384886856634270301385308963421289001083797013266440.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 25, 37, 56, 67, 94, 98, 127, 128, 157, 161, 188, 199, 218, 230, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618504514523967775163109878064631098965107124286829118090211615581262577680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 24, 36, 57, 66, 95, 99, 126, 129, 156, 160, 189, 198, 219, 231, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809254846048737684642796634945773391589149634534855653076559633879136206880.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 27, 39, 58, 65, 92, 96, 125, 130, 159, 163, 190, 197, 216, 228, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904626128630991945261973200393752252521981432525136844492347768437081112640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 26, 38, 59, 64, 93, 97, 124, 131, 158, 162, 191, 196, 217, 229, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452313711512184424103090603022120544737747767387152777146909167455213453440.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 21, 41, 52, 79, 82, 110, 115, 140, 145, 173, 176, 203, 214, 234, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226184031303361284863412941307989648842752207751176135687412462083400270080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 20, 40, 53, 78, 83, 111, 114, 141, 144, 172, 177, 202, 215, 235, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113133426169253495190283695657846014730802627040050815513080582594618196480.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 23, 43, 54, 77, 80, 108, 113, 142, 147, 175, 178, 201, 212, 232, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56546007825840321216257311299859142575657892008703747172159499166302602240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 22, 42, 55, 76, 81, 109, 112, 143, 146, 174, 179, 200, 213, 233, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28283356542313373798311729864485873022686041594398287226769861131519395840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 17, 45, 48, 75, 86, 106, 119, 136, 149, 169, 180, 207, 210, 238, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14576488286272863347581805008971331325909835609498165357011196770267566080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 16, 44, 49, 74, 87, 107, 118, 137, 148, 168, 181, 206, 211, 239, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7950811794896582635665191441795677836192789899220656086234903284654612480.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 19, 47, 50, 73, 84, 104, 117, 138, 151, 171, 182, 205, 208, 236, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3644122071568215842645115616310007478820130206976227073562935641997066240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 18, 46, 51, 72, 85, 105, 116, 139, 150, 170, 183, 204, 209, 237, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1987702948724145670411417459853621617593601306213883306005486651130019840.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 4]]</text>
      <sha1>ewnfnssm94j5hojj0f0hwbjidf88agu</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/reverse lictor</title>
    <ns>0</ns>
    <id>317364</id>
    <revision>
      <id>2692702</id>
      <timestamp>2024-12-19T21:00:22Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| reverse lictor !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-li...&quot;</comment>
      <origin>2692702</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2498" sha1="8mkf70fqmpk75ksxuxbfkdg1ktel7wb" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| reverse lictor
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 6, 9, 15, 16, 22, 25, 31, 32, 38, 41, 47, 48, 54, 57, 63, 64, 70, 73, 79, 80, 86, 89, 95, 96, 102, 105, 111, 112, 118, 121, 127, 128, 134, 137, 143, 144, 150, 153, 159, 160, 166, 169, 175, 176, 182, 185, 191, 192, 198, 201, 207, 208, 214, 217, 223, 224, 230, 233, 239, 240, 246, 249, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_58916414368174388287159143894326669525708211469837256548343833657493908718145.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 7, 8, 14, 17, 23, 24, 30, 33, 39, 40, 46, 49, 55, 56, 62, 65, 71, 72, 78, 81, 87, 88, 94, 97, 103, 104, 110, 113, 119, 120, 126, 129, 135, 136, 142, 145, 151, 152, 158, 161, 167, 168, 174, 177, 183, 184, 190, 193, 199, 200, 206, 209, 215, 216, 222, 225, 231, 232, 238, 241, 247, 248, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_29630477401537996448512785818199377656204129745064351246535495289733778653570.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 4, 11, 13, 18, 20, 27, 29, 34, 36, 43, 45, 50, 52, 59, 61, 66, 68, 75, 77, 82, 84, 91, 93, 98, 100, 107, 109, 114, 116, 123, 125, 130, 132, 139, 141, 146, 148, 155, 157, 162, 164, 171, 173, 178, 180, 187, 189, 194, 196, 203, 205, 210, 212, 219, 221, 226, 228, 235, 237, 242, 244, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18128127497899811780664351967485129084833295836873002014875025740767356528660.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 19, 21, 26, 28, 35, 37, 42, 44, 51, 53, 58, 60, 67, 69, 74, 76, 83, 85, 90, 92, 99, 101, 106, 108, 115, 117, 122, 124, 131, 133, 138, 140, 147, 149, 154, 156, 163, 165, 170, 172, 179, 181, 186, 188, 195, 197, 202, 204, 211, 213, 218, 220, 227, 229, 234, 236, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117069969703998907234703328676731586524347613865954229703229319918085739560.svg|420px]]
|}

[[Category:Boolf prop/3-ary|reverse lictor]]</text>
      <sha1>8mkf70fqmpk75ksxuxbfkdg1ktel7wb</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/nameless 5</title>
    <ns>0</ns>
    <id>317365</id>
    <revision>
      <id>2692704</id>
      <timestamp>2024-12-19T21:01:47Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| nameless 5 !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list s...&quot;</comment>
      <origin>2692704</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2465" sha1="chev4amtg2y0od0uub5nfcqshpafbei" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 5
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1766820104831717178943502833750131901118166179564640985811358930205474815.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_115790322390251417039241401712648644271680138743948311646133218849945997475840.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_411369862324345633660459182594392904092538379119254067153469440.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_26959535297288315447571852990506133362608595213959434544969773219840.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 5]]</text>
      <sha1>chev4amtg2y0od0uub5nfcqshpafbei</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/weight quadrant</title>
    <ns>0</ns>
    <id>317366</id>
    <revision>
      <id>2692708</id>
      <timestamp>2024-12-19T21:24:35Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| weight quadrant !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-l...&quot;</comment>
      <origin>2692708</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2424" sha1="rork52tda5mdk69pytaccje8zowx8fp" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| weight quadrant
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_140350834813144189858090274002849666665.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 64, 67, 69, 70, 73, 74, 76, 79, 81, 82, 84, 87, 88, 91, 93, 94, 97, 98, 100, 103, 104, 107, 109, 110, 112, 115, 117, 118, 121, 122, 124, 127]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_199931532107794273605284333428918544790.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[128, 131, 133, 134, 137, 138, 140, 143, 145, 146, 148, 151, 152, 155, 157, 158, 161, 162, 164, 167, 168, 171, 173, 174, 176, 179, 181, 182, 185, 186, 188, 191, 193, 194, 196, 199, 200, 203, 205, 206, 208, 211, 213, 214, 217, 218, 220, 223, 224, 227, 229, 230, 233, 234, 236, 239, 241, 242, 244, 247, 248, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47758914269546354982683078068829456703964492329985949123974750078192834314240.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[129, 130, 132, 135, 136, 139, 141, 142, 144, 147, 149, 150, 153, 154, 156, 159, 160, 163, 165, 166, 169, 170, 172, 175, 177, 178, 180, 183, 184, 187, 189, 190, 192, 195, 197, 198, 201, 202, 204, 207, 209, 210, 212, 215, 216, 219, 221, 222, 225, 226, 228, 231, 232, 235, 237, 238, 240, 243, 245, 246, 249, 250, 252, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_68033174967769840440887906939858451148965209968733676452019459322288527114240.svg|420px]]
|}

[[Category:Boolf prop/3-ary|weight quadrant]]</text>
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    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/consul weight</title>
    <ns>0</ns>
    <id>317367</id>
    <revision>
      <id>2692715</id>
      <timestamp>2024-12-19T21:54:06Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| consul weight !class=&quot;block&quot;| block |- |class=&quot;siz...&quot;</comment>
      <origin>2692715</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1599" sha1="6skh5ygmataaepxv8vppaw1gs5bupn0" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| consul weight
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 14, 15, 50, 51, 60, 61, 84, 85, 90, 91, 102, 103, 104, 105, 150, 151, 152, 153, 164, 165, 170, 171, 194, 195, 204, 205, 240, 241, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86849367469181558929018338465098072013517570674993328172038707360759325704195.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_27246106960121351242888641675682696814173384906326642419872288512862618598460.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_1696588925180991231161155327651110737640896370842162427341711305577693184960.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[22, 23, 24, 25, 36, 37, 42, 43, 66, 67, 76, 77, 112, 113, 126, 127, 128, 129, 142, 143, 178, 179, 188, 189, 212, 213, 218, 219, 230, 231, 232, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_25882832294020502849540256028287938132713478431020204876828713492152320.svg|420px]]
|}

[[Category:Boolf prop/3-ary|consul weight]]</text>
      <sha1>6skh5ygmataaepxv8vppaw1gs5bupn0</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/super family</title>
    <ns>0</ns>
    <id>317368</id>
    <revision>
      <id>2692719</id>
      <timestamp>2024-12-19T22:00:42Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;30&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !cl...&quot;</comment>
      <origin>2692719</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7863" sha1="lhgblmtth9h3svkbend40scz70u6d0g" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;30&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| super family
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044618658097711785492504343953926634992332820282019728792003956564819969.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 2, 4, 8, 16, 32, 64, 127, 128, 191, 223, 239, 247, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47267576113963700113891037153651775520612896549968809291683242518090289840406.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 64
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 12, 48, 63, 192, 207, 243, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7251140353850694982956619829190887723939603004798329462225855145563696140296.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 32
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 10, 80, 95, 160, 175, 245, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1865790500405973852040010052731675649071549611688552189837639037244453946400.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 96
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 9, 96, 111, 144, 159, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017703909312349373839979361158694184527283310552997901882537932975729803840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 144
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 11, 13, 14, 31, 47, 79, 112, 143, 176, 208, 224, 241, 242, 244, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_491183510968748886914222582355400445141562388892438978338438373955237603456.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 16
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1766847064778384329583297500742918515827483896875618958121606201292652544.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[17, 34, 68, 119, 136, 187, 221, 238]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_441715136187929672385227939396476489229670412240637915799880587110645760.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 72
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[18, 33, 72, 123, 132, 183, 222, 237]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_220862623083964841117825181273752891404574814892778593448892297302507520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 132
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[19, 35, 49, 50, 55, 59, 76, 115, 140, 179, 196, 200, 205, 206, 220, 236]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_110429626701289142201079274539005267349618993373418675459981188318887936.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[20, 40, 65, 125, 130, 190, 215, 235]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_55214023430471913853505242527422408066944382012676641618461839930163200.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 130
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[21, 42, 69, 81, 84, 87, 93, 117, 138, 162, 168, 171, 174, 186, 213, 234]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_27606998551198811826420364711086183859762040803214039045580343165845504.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 150
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[22, 41, 73, 97, 104, 107, 109, 121, 134, 146, 148, 151, 158, 182, 214, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_13803519021654050844148718343557808094930851019423434676786484062715904.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 104
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 43, 77, 113, 142, 178, 212, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6901752928808793455382809704201156581457135193295845546850465947320320.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 106
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[24, 36, 66, 126, 129, 189, 219, 231]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3451715671729414988927884610456227198687578716693276844744939656970240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 134
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[25, 38, 70, 98, 100, 103, 110, 118, 137, 145, 152, 155, 157, 185, 217, 230]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1725647211281027101089626623465410251044343107306914444838430639652864.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 146
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[26, 37, 74, 82, 88, 91, 94, 122, 133, 161, 164, 167, 173, 181, 218, 229]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_863139542515497779353218538749042464615518405464264876455950081327104.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 44
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[27, 39, 78, 114, 141, 177, 216, 228]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_431464458966078985463312770443698226589406926778204414258540036751360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 148
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[28, 44, 52, 56, 61, 62, 67, 124, 131, 188, 193, 194, 199, 203, 211, 227]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_215682878043348066241619524874697944951485786300229849877179956985856.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 74
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[29, 46, 71, 116, 139, 184, 209, 226]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_107840609420905739710342992751301905453067183737627442360307721175040.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 24
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[30, 45, 75, 120, 135, 180, 210, 225]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_53921538838860133290917051075637649238682330167094182737661971660800.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[51, 204]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_25711008708143844408671393477458601640355247902776485178507264.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 98
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[53, 58, 83, 92, 163, 172, 197, 202]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6628619438566337606216619207215223250303276857697676002590720.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 36
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[54, 57, 99, 108, 147, 156, 198, 201]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3615610599582819642227709477770141097221288688500395380572160.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 20
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[60, 195]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_50216813883093446110686315385661331328819996477216882950144.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[85, 170]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1496577676626844588240573307387100039795808514605056.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 66
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[86, 89, 101, 106, 149, 154, 166, 169]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_841848492689529729307025561319410174516754502385664.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 18
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[90, 165]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_46768052394588893382519152586960342009264740499456.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[102, 153]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_11417981541647684119068688668513567077874794496.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 22
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[105, 150]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1427247692706000445877493272790343030885318656.svg|420px]]
|}

[[Category:Boolf prop/3-ary|super family]]</text>
      <sha1>lhgblmtth9h3svkbend40scz70u6d0g</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/super clan</title>
    <ns>0</ns>
    <id>317369</id>
    <revision>
      <id>2692721</id>
      <timestamp>2024-12-19T22:05:05Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span clas...&quot;</comment>
      <origin>2692721</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4859" sha1="dkatecr3mhspv6rrmtwa84phhtleb54" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;14&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;2&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| super clan
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044618658097711785492504343953926634992332820282019728792003956564819969.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 2, 4, 8, 16, 32, 64, 127, 128, 191, 223, 239, 247, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47267576113963700113891037153651775520612896549968809291683242518090289840406.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 17, 34, 48, 63, 68, 80, 95, 119, 136, 160, 175, 187, 192, 207, 221, 238, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117372569392856764669015109861959849500382286899122289979294063395260732456.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 9, 18, 20, 33, 40, 65, 72, 96, 111, 123, 125, 130, 132, 144, 159, 183, 190, 215, 222, 235, 237, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017979985958863810594950691582495359826754829749903357117605287112962474560.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 130
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[7, 11, 13, 14, 19, 21, 31, 35, 42, 47, 49, 50, 55, 59, 69, 76, 79, 81, 84, 87, 93, 112, 115, 117, 138, 140, 143, 162, 168, 171, 174, 176, 179, 186, 196, 200, 205, 206, 208, 213, 220, 224, 234, 236, 241, 242, 244, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_491321547594001374868250081994650536592771769926615611052943935486722336896.svg|420px]]
|-
|class=&quot;size&quot;| 6
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 51, 85, 170, 204, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1766847064804095338292937922828216753893183071823361306064178494985764864.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 150
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[22, 41, 73, 97, 104, 107, 109, 121, 134, 146, 148, 151, 158, 182, 214, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_13803519021654050844148718343557808094930851019423434676786484062715904.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 104
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 43, 77, 113, 142, 178, 212, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6901752928808793455382809704201156581457135193295845546850465947320320.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 106
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[24, 36, 66, 126, 129, 189, 219, 231]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3451715671729414988927884610456227198687578716693276844744939656970240.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 134
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[25, 26, 28, 37, 38, 44, 52, 56, 61, 62, 67, 70, 74, 82, 88, 91, 94, 98, 100, 103, 110, 118, 122, 124, 131, 133, 137, 145, 152, 155, 157, 161, 164, 167, 173, 181, 185, 188, 193, 194, 199, 203, 211, 217, 218, 227, 229, 230]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_2804469631839872946684464687089150660611347299071409171171560677965824.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 44
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[27, 29, 39, 46, 53, 58, 71, 78, 83, 92, 114, 116, 139, 141, 163, 172, 177, 184, 197, 202, 209, 216, 226, 228]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_539305075015604163739993369411619339257697360819108714316523760517120.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| 24
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[30, 45, 54, 57, 75, 86, 89, 99, 101, 106, 108, 120, 135, 147, 149, 154, 156, 166, 169, 180, 198, 201, 210, 225]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_53921542454470733715585185992876856315848988707793045754811854618624.svg|420px]]
|-
|class=&quot;size&quot;| 6
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[60, 90, 102, 153, 165, 195]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_50216813929872916486816856452299552604448852053559498244096.svg|420px]]
|-
|class=&quot;size&quot;| 2
|class=&quot;prop&quot;| 22
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[105, 150]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1427247692706000445877493272790343030885318656.svg|420px]]
|}

[[Category:Boolf prop/3-ary|super clan]]</text>
      <sha1>dkatecr3mhspv6rrmtwa84phhtleb54</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/ultra family</title>
    <ns>0</ns>
    <id>317370</id>
    <revision>
      <id>2692722</id>
      <timestamp>2024-12-19T22:17:46Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;18&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf...&quot;</comment>
      <origin>2692722</origin>
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      <text bytes="5513" sha1="q3ga2sy7j3b4e0iyuecu1y5rzbn9asb" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;18&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;6&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| ultra family
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 15, 240, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57897811465722876096115075801844696845150819816717157638686913610157857472513.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 2, 4, 7, 8, 11, 13, 14, 16, 31, 32, 47, 64, 79, 112, 127, 128, 143, 176, 191, 208, 223, 224, 239, 241, 242, 244, 247, 248, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47758759624932449000805259736007175965754458938861248270021680892045527443862.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 64
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 12, 48, 63, 192, 207, 243, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7251140353850694982956619829190887723939603004798329462225855145563696140296.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 32
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 10, 80, 95, 160, 175, 245, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1865790500405973852040010052731675649071549611688552189837639037244453946400.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 96
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 9, 96, 111, 144, 159, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017703909312349373839979361158694184527283310552997901882537932975729803840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[17, 30, 34, 45, 68, 75, 119, 120, 135, 136, 180, 187, 210, 221, 225, 238]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_441769057726768532518518856447552126878909094570805009982618249082306560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 72
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[18, 29, 33, 46, 71, 72, 116, 123, 132, 139, 183, 184, 209, 222, 226, 237]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_220970463693385746857535524266504193310027882076516220891252605023682560.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 132
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[19, 28, 35, 44, 49, 50, 52, 55, 56, 59, 61, 62, 67, 76, 115, 124, 131, 140, 179, 188, 193, 194, 196, 199, 200, 203, 205, 206, 211, 220, 227, 236]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_110645309579332490267320894063879965294570479159718905309858368275873792.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[20, 27, 39, 40, 65, 78, 114, 125, 130, 141, 177, 190, 215, 216, 228, 235]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_55645487889437992838968555297866106293533788939454846032720379966914560.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 130
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[21, 26, 37, 42, 69, 74, 81, 82, 84, 87, 88, 91, 93, 94, 117, 122, 133, 138, 161, 162, 164, 167, 168, 171, 173, 174, 181, 186, 213, 218, 229, 234]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28470138093714309605773583249835226324377559208678303922036293247172608.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 134
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[22, 25, 38, 41, 70, 73, 97, 98, 100, 103, 104, 107, 109, 110, 118, 121, 134, 137, 145, 146, 148, 151, 152, 155, 157, 158, 182, 185, 214, 217, 230, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_15529166232935077945238344967023218345975194126730349121624914702368768.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 104
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 24, 36, 43, 66, 77, 113, 126, 129, 142, 178, 189, 212, 219, 231, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10353468600538208444310694314657383780144713909989122391595405604290560.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[51, 60, 195, 204]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_25761225522026937854782079792844262971684067899253702061457408.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 98
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[53, 58, 83, 92, 163, 172, 197, 202]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6628619438566337606216619207215223250303276857697676002590720.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 36
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[54, 57, 99, 108, 147, 156, 198, 201]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3615610599582819642227709477770141097221288688500395380572160.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[85, 90, 165, 170]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1543345729021433481623092459974060381805073255104512.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 66
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[86, 89, 101, 106, 149, 154, 166, 169]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_841848492689529729307025561319410174516754502385664.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[102, 105, 150, 153]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_12845229234353684564946181941303910108760113152.svg|420px]]
|}

[[Category:Boolf prop/3-ary|ultra family]]</text>
      <sha1>q3ga2sy7j3b4e0iyuecu1y5rzbn9asb</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/ultra clan</title>
    <ns>0</ns>
    <id>317371</id>
    <revision>
      <id>2692724</id>
      <timestamp>2024-12-19T22:21:41Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;11&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span cla...&quot;</comment>
      <origin>2692724</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4214" sha1="75lpeli59qsn403xrleadnnth86ny01" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;11&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;8&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| ultra clan
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 15, 240, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57897811465722876096115075801844696845150819816717157638686913610157857472513.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 2, 4, 7, 8, 11, 13, 14, 16, 31, 32, 47, 64, 79, 112, 127, 128, 143, 176, 191, 208, 223, 224, 239, 241, 242, 244, 247, 248, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47758759624932449000805259736007175965754458938861248270021680892045527443862.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 17, 30, 34, 45, 48, 63, 68, 75, 80, 95, 119, 120, 135, 136, 160, 175, 180, 187, 192, 207, 210, 221, 225, 238, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117372623314395603529148400779010925138031525581452457073476801057232393256.svg|420px]]
|-
|class=&quot;size&quot;| 40
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 9, 18, 20, 27, 29, 33, 39, 40, 46, 65, 71, 72, 78, 96, 111, 114, 116, 123, 125, 130, 132, 139, 141, 144, 159, 177, 183, 184, 190, 209, 215, 216, 222, 226, 228, 235, 237, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017980525263932197579675865238258554826886872224013872949461905960720400960.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 130
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[19, 21, 26, 28, 35, 37, 42, 44, 49, 50, 52, 55, 56, 59, 61, 62, 67, 69, 74, 76, 81, 82, 84, 87, 88, 91, 93, 94, 115, 117, 122, 124, 131, 133, 138, 140, 161, 162, 164, 167, 168, 171, 173, 174, 179, 181, 186, 188, 193, 194, 196, 199, 200, 203, 205, 206, 211, 213, 218, 220, 227, 229, 234, 236]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_139115447673046799873094477313715191618948038368397209231894661523046400.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 134
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[22, 25, 38, 41, 70, 73, 97, 98, 100, 103, 104, 107, 109, 110, 118, 121, 134, 137, 145, 146, 148, 151, 152, 155, 157, 158, 182, 185, 214, 217, 230, 233]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_15529166232935077945238344967023218345975194126730349121624914702368768.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 104
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 24, 36, 43, 66, 77, 113, 126, 129, 142, 178, 189, 212, 219, 231, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10353468600538208444310694314657383780144713909989122391595405604290560.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[51, 60, 85, 90, 165, 170, 195, 204]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_25761225523570283583803513274467355431658128281058775316561920.svg|420px]]
|-
|class=&quot;size&quot;| 8
|class=&quot;prop&quot;| 44
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[53, 58, 83, 92, 163, 172, 197, 202]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6628619438566337606216619207215223250303276857697676002590720.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 24
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[54, 57, 86, 89, 99, 101, 106, 108, 147, 149, 154, 156, 166, 169, 198, 201]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3615610600424668134917239207077166658540698863017149882957824.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[102, 105, 150, 153]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_12845229234353684564946181941303910108760113152.svg|420px]]
|}

[[Category:Boolf prop/3-ary|ultra clan]]</text>
      <sha1>75lpeli59qsn403xrleadnnth86ny01</sha1>
    </revision>
  </page>
  <page>
    <title>User:Indexcard88/Archive/December 19th 2024 1</title>
    <ns>2</ns>
    <id>317372</id>
    <revision>
      <id>2692730</id>
      <timestamp>2024-12-19T22:27:19Z</timestamp>
      <contributor>
        <username>Indexcard88</username>
        <id>118020</id>
      </contributor>
      <comment>New resource with &quot;I believe I am deluding myself in small moments to move on with life, in terms of a will or plan of God.  I have no one to talk to.&quot;</comment>
      <origin>2692730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="131" sha1="71jpk8ztbo4xiui5lwuatjoh3l8idm4" xml:space="preserve">I believe I am deluding myself in small moments to move on with life, in terms of a will or plan of God.

I have no one to talk to.</text>
      <sha1>71jpk8ztbo4xiui5lwuatjoh3l8idm4</sha1>
    </revision>
    <revision>
      <id>2692752</id>
      <parentid>2692730</parentid>
      <timestamp>2024-12-20T01:40:32Z</timestamp>
      <contributor>
        <username>Indexcard88</username>
        <id>118020</id>
      </contributor>
      <origin>2692752</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="176" sha1="5mr5uyyu95f6nf9esn6dgt2omaxymmi" xml:space="preserve">I believe I am deluding myself in small moments to move on with life, in terms of a will or plan of God.

I have no one to talk to.


Maybe someone else can explain everything.</text>
      <sha1>5mr5uyyu95f6nf9esn6dgt2omaxymmi</sha1>
    </revision>
    <revision>
      <id>2692753</id>
      <parentid>2692752</parentid>
      <timestamp>2024-12-20T01:41:13Z</timestamp>
      <contributor>
        <username>Indexcard88</username>
        <id>118020</id>
      </contributor>
      <origin>2692753</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="250" sha1="84ral8d2bnnt3zffazs4wr82d1xre8k" xml:space="preserve">I believe I am deluding myself in small moments to move on with life, in terms of a will or plan of God.

I have no one to talk to.


Maybe someone else can explain everything.

Should I respect my parents? (I write this with 27 years of experience.)</text>
      <sha1>84ral8d2bnnt3zffazs4wr82d1xre8k</sha1>
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  </page>
  <page>
    <title>Boolf prop/3-ary/quaestor weight</title>
    <ns>0</ns>
    <id>317373</id>
    <revision>
      <id>2692732</id>
      <timestamp>2024-12-19T22:38:01Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !cl...&quot;</comment>
      <origin>2692732</origin>
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      <text bytes="2350" sha1="8gfn0r2l76h8gngj3rqqzl7lrsa6hnc" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| quaestor weight
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 24, 36, 60, 66, 90, 102, 126, 129, 153, 165, 189, 195, 219, 231, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896048070373769491417295362101480869679047472698551317454488802455720034305.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 2, 4, 8, 16, 25, 26, 28, 32, 37, 38, 44, 52, 56, 61, 62, 64, 67, 70, 74, 82, 88, 91, 94, 98, 100, 103, 110, 118, 122, 124, 127, 128, 131, 133, 137, 145, 152, 155, 157, 161, 164, 167, 173, 181, 185, 188, 191, 193, 194, 199, 203, 211, 217, 218, 223, 227, 229, 230, 239, 247, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47267578918433331953763983838116462609763557161316108363092413689650967806230.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_10135353148578338045338863257023010613823332690195375173998659421843838342760.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[7, 11, 13, 14, 19, 21, 22, 31, 35, 41, 42, 47, 49, 50, 55, 59, 69, 73, 76, 79, 81, 84, 87, 93, 97, 104, 107, 109, 112, 115, 117, 121, 134, 138, 140, 143, 146, 148, 151, 158, 162, 168, 171, 174, 176, 179, 182, 186, 196, 200, 205, 206, 208, 213, 214, 220, 224, 233, 234, 236, 241, 242, 244, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_491335351113023028919094230712994094400866700777635034487620721970785052800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 23, 43, 51, 77, 85, 105, 113, 142, 150, 170, 178, 204, 212, 232, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1773748817732904131748320733959665603180640652894150424401371991818403840.svg|420px]]
|}

[[Category:Boolf prop/3-ary|quaestor weight]]</text>
      <sha1>8gfn0r2l76h8gngj3rqqzl7lrsa6hnc</sha1>
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  </page>
  <page>
    <title>Boolf prop/3-ary/praetor weight</title>
    <ns>0</ns>
    <id>317374</id>
    <revision>
      <id>2692733</id>
      <timestamp>2024-12-19T22:39:15Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !cl...&quot;</comment>
      <origin>2692733</origin>
      <model>wikitext</model>
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      <text bytes="2348" sha1="5fx50fmc7tm6krnmt7qwfuhs9sp4546" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| praetor weight
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896486333794311352466587372716853682903935386526539088661048023915243372545.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 2, 4, 8, 16, 19, 21, 25, 32, 35, 38, 42, 49, 50, 55, 59, 64, 69, 70, 76, 81, 84, 87, 93, 98, 100, 103, 110, 115, 117, 118, 127, 128, 137, 138, 140, 145, 152, 155, 157, 162, 168, 171, 174, 179, 185, 186, 191, 196, 200, 205, 206, 213, 217, 220, 223, 230, 234, 236, 239, 247, 251, 253, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_47267715876236163882872165742917649077474356975346093231312192918052414226710.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_10134921732989211815014789048860877719080233925814714358111068399000360326760.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[7, 11, 13, 14, 22, 26, 28, 31, 37, 41, 44, 47, 52, 56, 61, 62, 67, 73, 74, 79, 82, 88, 91, 94, 97, 104, 107, 109, 112, 121, 122, 124, 131, 133, 134, 143, 146, 148, 151, 158, 161, 164, 167, 173, 176, 181, 182, 188, 193, 194, 199, 203, 208, 211, 214, 218, 224, 227, 229, 233, 241, 242, 244, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_491198393310191099810912325911807626690066886747650166267841493569338632320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1766900986317273406530518280719747121391491205567195105433173375773081600.svg|420px]]
|}

[[Category:Boolf prop/3-ary|praetor weight]]</text>
      <sha1>5fx50fmc7tm6krnmt7qwfuhs9sp4546</sha1>
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  </page>
  <page>
    <title>Boolf prop/3-ary/patron index weight</title>
    <ns>0</ns>
    <id>317375</id>
    <revision>
      <id>2692734</id>
      <timestamp>2024-12-19T22:41:27Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !cl...&quot;</comment>
      <origin>2692734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2362" sha1="8fzqqqlpu53g0zws2vdd5e35c7hm9eu" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| patron index weight
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022336315325202904699959177005221122795925822929966558420718528929726465.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 31, 41, 55, 73, 87, 97, 127, 129, 159, 169, 183, 201, 215, 225, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044672630650405809399918354010442245591851645859933116841437057859452930.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_28495716254678664234651741094728422524270783174138228120130589208348100395260.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[8, 10, 12, 15, 17, 18, 20, 22, 32, 34, 36, 39, 57, 58, 60, 62, 64, 66, 68, 71, 89, 90, 92, 94, 104, 106, 108, 111, 113, 114, 116, 118, 136, 138, 140, 143, 145, 146, 148, 150, 160, 162, 164, 167, 185, 186, 188, 190, 192, 194, 196, 199, 217, 218, 220, 222, 232, 234, 236, 239, 241, 242, 244, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_152977216833470867241282074320892336705607414945699404685735027192181986560.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[9, 11, 13, 14, 16, 19, 21, 23, 33, 35, 37, 38, 56, 59, 61, 63, 65, 67, 69, 70, 88, 91, 93, 95, 105, 107, 109, 110, 112, 115, 117, 119, 137, 139, 141, 142, 144, 147, 149, 151, 161, 163, 165, 166, 184, 187, 189, 191, 193, 195, 197, 198, 216, 219, 221, 223, 233, 235, 237, 238, 240, 243, 245, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_299328756858084712963861962107577328925206299087846614965997616786058078720.svg|420px]]
|}

[[Category:Boolf prop/3-ary|patron index weight]]</text>
      <sha1>8fzqqqlpu53g0zws2vdd5e35c7hm9eu</sha1>
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  </page>
  <page>
    <title>Boolf prop/3-ary/prefect weight</title>
    <ns>0</ns>
    <id>317376</id>
    <revision>
      <id>2692739</id>
      <timestamp>2024-12-19T22:52:36Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| prefect weight !class=&quot;block&quot;| block |- |class=&quot;si...&quot;</comment>
      <origin>2692739</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1567" sha1="nxejgwpex88mmu3ydp4xyt57quuabsb" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;32&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| prefect weight
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 232, 233, 234, 235, 236, 237, 238, 239, 248, 249, 250, 251, 252, 253, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_115341536334051360628963456711242601354708523820828309034750019264059304509695.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_450552903159932278392934366827363174655893616276452539978402249012076936960.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_104902516214593930612256546768980043855800194642683360636263464960.svg|420px]]
|-
|class=&quot;size&quot;| 32
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 144, 145, 146, 147, 148, 149, 150, 151]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5686777136587184680002270086479134205484728320.svg|420px]]
|}

[[Category:Boolf prop/3-ary|prefect weight]]</text>
      <sha1>nxejgwpex88mmu3ydp4xyt57quuabsb</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/guardian</title>
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&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

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|-
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      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| patron index consul !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;blo...&quot;</comment>
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&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
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|-
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|-
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|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 8, 9, 22, 23, 24, 25, 32, 33, 46, 47, 48, 49, 62, 63, 64, 65, 78, 79, 80, 81, 94, 95, 102, 103, 104, 105, 118, 119, 120, 121, 134, 135, 136, 137, 150, 151, 152, 153, 160, 161, 174, 175, 176, 177, 190, 191, 192, 193, 206, 207, 208, 209, 222, 223, 230, 231, 232, 233, 246, 247, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1696199083957552105439127648460824174057944906629265700798594354042592297920.svg|420px]]
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[[Category:Boolf prop/3-ary|patron index consul]]</text>
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{{Information
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|Source={{own|Young1lim}}
|Date=2024-12-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
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      <comment>Purpose: To analyze the current situation and factors affecting human capital and propose solutions to improve human capital to support Muong people in escaping poverty sustainably in Thanh Hoa province, Vietnam. Participants and methods: A survey of 400 representatives of Muong households in 3 mountainous districts of Thanh Hoa province regarding the current situation and factors affecting human capital, and in-depth interviews with managers and people to supplement quantitative research inform</comment>
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==First Heading==

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===Subheading===


==Second Heading==


==Third Heading, etc==


==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
{{reflist|35em}}</text>
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===Subheading===


==Second Heading==


==Third Heading, etc==


==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
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== Licensing ==
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== Licensing ==
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== Licensing ==
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== Licensing ==
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      <comment>{{Information
|Description=API.2A: Functions (20241220 - 20241219)
|Source={{own|Young1lim}}
|Date=2024-12-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2692769</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="262" sha1="iqcs226pg87iszb9ahvcpvmh500tr2i" xml:space="preserve">== Summary ==
{{Information
|Description=API.2A: Functions (20241220 - 20241219)
|Source={{own|Young1lim}}
|Date=2024-12-20
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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  </page>
  <page>
    <title>Boolf prop/3-ary/tribe</title>
    <ns>0</ns>
    <id>317387</id>
    <revision>
      <id>2692788</id>
      <timestamp>2024-12-20T11:03:39Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !c...&quot;</comment>
      <origin>2692788</origin>
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      <text bytes="2194" sha1="9f4nyzk0p4awt09ablnte73069hmknz" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| province
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 15, 51, 60, 85, 90, 102, 105, 150, 153, 165, 170, 195, 204, 240, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57897811465722901857340599372141125877898447968637535478756498579041934147585.svg|420px]]
|-
|class=&quot;size&quot;| 128
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_47758914269546354982683078068829456704164423862093743397580034411621752859030.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 17, 30, 34, 45, 48, 54, 57, 63, 68, 75, 80, 86, 89, 95, 99, 101, 106, 108, 119, 120, 135, 136, 147, 149, 154, 156, 160, 166, 169, 175, 180, 187, 192, 198, 201, 207, 210, 221, 225, 238, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117372623314399219139748825447145842377238602748110997772339818207115351080.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 9, 18, 20, 27, 29, 33, 39, 40, 46, 53, 58, 65, 71, 72, 78, 83, 92, 96, 111, 114, 116, 123, 125, 130, 132, 139, 141, 144, 159, 163, 172, 177, 183, 184, 190, 197, 202, 209, 215, 216, 222, 226, 228, 235, 237, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017980525263938826199114431575864771446094087447264176226319603636722991680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 104
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 24, 36, 43, 66, 77, 113, 126, 129, 142, 178, 189, 212, 219, 231, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10353468600538208444310694314657383780144713909989122391595405604290560.svg|420px]]
|}

[[Category:Boolf prop/3-ary|province]]</text>
      <sha1>9f4nyzk0p4awt09ablnte73069hmknz</sha1>
    </revision>
    <revision>
      <id>2692789</id>
      <parentid>2692788</parentid>
      <timestamp>2024-12-20T11:35:02Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <minor />
      <comment>Watchduck moved page [[Boolf prop/3-ary/province]] to [[Boolf prop/3-ary/tribe]]</comment>
      <origin>2692788</origin>
      <model>wikitext</model>
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      <text bytes="2194" sha1="9f4nyzk0p4awt09ablnte73069hmknz" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| province
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 15, 51, 60, 85, 90, 102, 105, 150, 153, 165, 170, 195, 204, 240, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57897811465722901857340599372141125877898447968637535478756498579041934147585.svg|420px]]
|-
|class=&quot;size&quot;| 128
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_47758914269546354982683078068829456704164423862093743397580034411621752859030.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 17, 30, 34, 45, 48, 54, 57, 63, 68, 75, 80, 86, 89, 95, 99, 101, 106, 108, 119, 120, 135, 136, 147, 149, 154, 156, 160, 166, 169, 175, 180, 187, 192, 198, 201, 207, 210, 221, 225, 238, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117372623314399219139748825447145842377238602748110997772339818207115351080.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 9, 18, 20, 27, 29, 33, 39, 40, 46, 53, 58, 65, 71, 72, 78, 83, 92, 96, 111, 114, 116, 123, 125, 130, 132, 139, 141, 144, 159, 163, 172, 177, 183, 184, 190, 197, 202, 209, 215, 216, 222, 226, 228, 235, 237, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017980525263938826199114431575864771446094087447264176226319603636722991680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 104
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 24, 36, 43, 66, 77, 113, 126, 129, 142, 178, 189, 212, 219, 231, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10353468600538208444310694314657383780144713909989122391595405604290560.svg|420px]]
|}

[[Category:Boolf prop/3-ary|province]]</text>
      <sha1>9f4nyzk0p4awt09ablnte73069hmknz</sha1>
    </revision>
    <revision>
      <id>2692791</id>
      <parentid>2692789</parentid>
      <timestamp>2024-12-20T11:35:31Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692791</origin>
      <model>wikitext</model>
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      <text bytes="2184" sha1="6ebllc0qtk3vgb7t89xiljyp7zni5q6" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| tribe
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 15, 51, 60, 85, 90, 102, 105, 150, 153, 165, 170, 195, 204, 240, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57897811465722901857340599372141125877898447968637535478756498579041934147585.svg|420px]]
|-
|class=&quot;size&quot;| 128
|class=&quot;prop&quot;| -1
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_47758914269546354982683078068829456704164423862093743397580034411621752859030.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 17, 30, 34, 45, 48, 54, 57, 63, 68, 75, 80, 86, 89, 95, 99, 101, 106, 108, 119, 120, 135, 136, 147, 149, 154, 156, 160, 166, 169, 175, 180, 187, 192, 198, 201, 207, 210, 221, 225, 238, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117372623314399219139748825447145842377238602748110997772339818207115351080.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 9, 18, 20, 27, 29, 33, 39, 40, 46, 53, 58, 65, 71, 72, 78, 83, 92, 96, 111, 114, 116, 123, 125, 130, 132, 139, 141, 144, 159, 163, 172, 177, 183, 184, 190, 197, 202, 209, 215, 216, 222, 226, 228, 235, 237, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017980525263938826199114431575864771446094087447264176226319603636722991680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 24, 36, 43, 66, 77, 113, 126, 129, 142, 178, 189, 212, 219, 231, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10353468600538208444310694314657383780144713909989122391595405604290560.svg|420px]]
|}

[[Category:Boolf prop/3-ary|tribe]]</text>
      <sha1>6ebllc0qtk3vgb7t89xiljyp7zni5q6</sha1>
    </revision>
    <revision>
      <id>2692792</id>
      <parentid>2692791</parentid>
      <timestamp>2024-12-20T11:43:59Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692792</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2187" sha1="1sppqafpfmj4olm7rj8rac9mqkcx5px" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;128&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| tribe
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 15, 51, 60, 85, 90, 102, 105, 150, 153, 165, 170, 195, 204, 240, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57897811465722901857340599372141125877898447968637535478756498579041934147585.svg|420px]]
|-
|class=&quot;size&quot;| 128
|class=&quot;prop&quot;| sharp
|class=&quot;block&quot;| [[File:Set_of_3-ary_Boolean_functions_47758914269546354982683078068829456704164423862093743397580034411621752859030.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[3, 5, 10, 12, 17, 30, 34, 45, 48, 54, 57, 63, 68, 75, 80, 86, 89, 95, 99, 101, 106, 108, 119, 120, 135, 136, 147, 149, 154, 156, 160, 166, 169, 175, 180, 187, 192, 198, 201, 207, 210, 221, 225, 238, 243, 245, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9117372623314399219139748825447145842377238602748110997772339818207115351080.svg|420px]]
|-
|class=&quot;size&quot;| 48
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 9, 18, 20, 27, 29, 33, 39, 40, 46, 53, 58, 65, 71, 72, 78, 83, 92, 96, 111, 114, 116, 123, 125, 130, 132, 139, 141, 144, 159, 163, 172, 177, 183, 184, 190, 197, 202, 209, 215, 216, 222, 226, 228, 235, 237, 246, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1017980525263938826199114431575864771446094087447264176226319603636722991680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[23, 24, 36, 43, 66, 77, 113, 126, 129, 142, 178, 189, 212, 219, 231, 232]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10353468600538208444310694314657383780144713909989122391595405604290560.svg|420px]]
|}

[[Category:Boolf prop/3-ary|tribe]]</text>
      <sha1>1sppqafpfmj4olm7rj8rac9mqkcx5px</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/province</title>
    <ns>0</ns>
    <id>317388</id>
    <redirect title="Boolf prop/3-ary/tribe" />
    <revision>
      <id>2692790</id>
      <timestamp>2024-12-20T11:35:02Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>Watchduck moved page [[Boolf prop/3-ary/province]] to [[Boolf prop/3-ary/tribe]]</comment>
      <origin>2692790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36" sha1="mpxh6e560v5b0fy4xi1h3rhrcs47z2g" xml:space="preserve">#REDIRECT [[Boolf prop/3-ary/tribe]]</text>
      <sha1>mpxh6e560v5b0fy4xi1h3rhrcs47z2g</sha1>
    </revision>
  </page>
</mediawiki>