Wikiversity:Requests for Deletion

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    <title>Wikiversity:Requests for Deletion</title>
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      <comment>/* Classical guitar pedagogy */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]])</comment>
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== [[Classical guitar pedagogy]] ==

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

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

== [[Concomitant Strabismus]] ==

Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC)</text>
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  <page>
    <title>User talk:Mu301</title>
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      <contributor>
        <username>Mu301</username>
        <id>3705</id>
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      <minor />
      <comment>-&gt; archive</comment>
      <origin>2818039</origin>
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      <text bytes="974" sha1="783nv7wjel9gzjv3kxs3p34jld05bt4" xml:space="preserve">{{ messagebox
| image = Information icon4.svg
| text  = If you leave a message here I will automatically get a notification. I'm very responsive to requests for help or to answer questions. Please feel free to contact me and I'll respond in a timely manner. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 16:59, 21 May 2026 (UTC)
}}
&lt;br&gt;
{{Talk header|noarchive=yes|disclaimer=yes|Wikiversity:Support staff|Wikiversity:Request custodian action|Wikiversity:Colloquium}}&lt;br&gt;
{{#switch: {{CURRENTDOW}}
|0={{/0}} &lt;!-- Fourth Annual Report of the Secretary of the Board of Education --&gt;
|1={{/1}} &lt;!-- Philosophical Question #3: Is the individual greater than the community? --&gt;
|2={{/2}} &lt;!-- consensus and the zeroth law of robotics --&gt;
|3={{/3}} &lt;!-- Kipple drives out nonkipple --&gt;
|4={{/4}} &lt;!-- (Possible materials to add to this resource) --&gt;
|5={{/5}} &lt;!-- Afterthoughts --&gt;
|6={{/6}} &lt;!-- Letter to John Adams --&gt;
}}
{{User talk:Mu301/Archive Index}}
__TOC__</text>
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  <page>
    <title>User:Mu301</title>
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      <minor />
      <comment>update</comment>
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      <text bytes="2025" sha1="qfdv3nwb0lfgl1wpl8aonzcx4og9jvp" xml:space="preserve">&lt;code lang=&quot;csh&quot;&gt;% whois !MU301&lt;/code&gt;&lt;sup&gt;[https://tools.ietf.org/html/rfc954]&lt;/sup&gt;

&lt;!-- I am the Curator of the historic [[w:Ladd Observatory|Ladd Observatory]] at [[w:Brown University|Brown University]]. --&gt; 

My research focus is late 19&lt;sup&gt;th&lt;/sup&gt; and early 20&lt;sup&gt;th&lt;/sup&gt; century history of science and technology.

Visit my [[/Learning blog/]] and join in on the conversation. I also have a personal blog called [https://fornaxchimiae.blogspot.com/ ''Fornax Chimiæ''] and I contribute to the [https://web.archive.org/web/20230401051800/https://blogs.brown.edu/ladd/author/mumbrichbrown-edu/ ''Ladd Observatory blog''].

[[File:Ada-Lovelace-Wikithon-2013-08.jpg|thumb|right|Presenting GLAM and Wikipedian in Residence at the Ada Lovelace Day Wikipedia edit-a-thon at Brown.[https://en.wikipedia.org/wiki/Wikipedia:Meetup/Ada_Lovelace_Edit-a-thon_2013_-_Brown] ]]
I'm a [[Special:ListUsers/bureaucrat|bureaucrat]] at en-wikiversity; &lt;code&gt;&lt;nowiki&gt;{{ping|Mu301}}&lt;/nowiki&gt;&lt;/code&gt; if you need assistance or help with anything here.

Where I'm currently most active
* '''[[User:Mu301|Wikiversity:User:Mu301]]'''
* '''[[Wikipedia:User:Mu301]]'''
* [[Outreach:User:Mu301]]

Projects that I'm involved with
* '''[[Outreach:GLAM]]''' ('''G'''alleries • '''L'''ibraries • '''A'''rchives • '''M'''useums)
&lt;!-- * [[commons:Commons:Ladd Observatory|Commons:Ladd Observatory]] --&gt;
* [[commons:Commons:Retro-Computing Society of Rhode Island|Commons:Retro-Computing Society of Rhode Island]]

&lt;small&gt;My Freenode irc nick and cloak is &lt;code lang=&quot;irc&quot;&gt;mikeu@wikimedia/mu301&lt;/code&gt;[https://en.wikiversity.org/w/index.php?title=User:Mu301&amp;diff=prev&amp;oldid=1512387]&lt;/small&gt;&lt;br&gt;
&lt;small&gt;My other accounts are [[User:Mikeu|Mikeu]] and [[User:Mu301Bot|Mu301Bot]] (w/bot flag for [[User:Mu301Bot|en-wv]] and [[betawikiversity:User:Mu301Bot|beta-wv]])&lt;/small&gt;

{{Userpage}}

[[Category:Wikiversity bureaucrats]]
[[Category:Wikiversity custodians]]
[[Category:Wikiversity bot owners]]
[[Category:Wikimedians]]
[[Category:Wikipedians]]</text>
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  <page>
    <title>Understanding Arithmetic Circuits</title>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2817987</origin>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260709.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260709.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
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  <page>
    <title>Complex analysis in plain view</title>
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        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
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      <text bytes="5296" sha1="oedo8o4j31o15wprbzaf2ycen75tihf" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

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

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

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

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

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

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

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

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

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

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

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

====Geometric Series Examples====

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

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

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

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

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

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260709.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

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

* Double Pole Case

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

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

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

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

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

[[Category:Complex analysis]]</text>
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  <page>
    <title>Algebra 1/Unit 1: Introduction To Algebra</title>
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      <contributor>
        <username>Namjoel</username>
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      <comment>I added clarification to make it easier to find the percentage.</comment>
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      <text bytes="24863" sha1="hwme9zy02mlej8etvr8gruq54p1rpv9" xml:space="preserve">[[File:Quadratic formula.svg|thumb|right|An example of a Algebra formula (quadratic formula)]]
{{mathematics}}
{{secondary education}}
{{lesson}}
{{complete}}

'''Algebra''' (from the Arabic word &quot;al-jabr&quot; (الجبر), meaning &quot;reunion of broken parts&quot;) can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is &lt;small&gt;IMPORTANT&lt;/small&gt; that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on.

Without further do, let's dig right into this!
==Algebra==
===What is Algebra?===
[[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]In Algebra, we use letters called '''[[Variable|Variables]]''' or P'''ronumeral''' to represent quantities that are not known. Calling them Variables is the most excepted word to use in most contexts. The constant is a number that stays the same.   

Imagine you have a bag full of jellybeans on a table; with 10 green jelly beans and a unknown amount of blue jellybeans. Let's call the blue jellybeans x. Well done, this is a pronumeral they are that simple.  Now, your friend comes over, tells you there is 20 total jellybeans in the bag.    

How many blue jellybeans are there! Imagine you have a bag full of jellybeans in a jar; with 10 green jelly beans and a unknown amount of blue jellybeans. Imagine you have a 10 green jelly beans and a object covering the amount of blue jellybeans. blue jellybeans are represented with the letter x.   

# So far we know that we have 10 green jelly beans we know that the object being covered is a unknown number represented by '''Variable''' x which we are unsure of. 
# Now, your friend comes over, whispers to you there is 20 total jellybeans in the bag. 
# Here is how you do it, if 10+x=20 then we just need to keep count the numbers that are missing to get to 20  from 10. Once you get to 20 how much new numbers did you count? 


Make your own digram  to help you see the equations better it:
[[File:Digram for Variables.png|border|left|937x937px]]
here is a summery   

# 10 green jelly beans unknown amount of x cover 10+x
# Friend whispered that there are 20 jelly beans  in the bag   10+x=20
# Count the steps we take from 10 till we get to 20 

10+x

=10+10  

=10  


this is what you sould be thinking in your head or recalling without so many steps.    

10+1= 11, 10+2= 12, 10+3=13, 10+4=14, 10+5=15 10+6= 16, 10+7= 17, 10+8= 18, 10+9= 19, 10+ 10  = 20   

Well done, this is a pronumeral they are that simple you did algebra.   

The core concept of algebra is the equal sign (=). Think of an equation as a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced.   

example 10+10=20-10  don't worry about decoding the numbers it is explained below on how to decode the numbers.   

To find the unknown number, you need to find unknown number represented by x completely by itself and do it in a way that is proves you are right with big quantities of numbers. You do this by using inverse operations (doing the opposite).


the two main family's of maths are opposite in roles:
*Addition (+) and Subtraction (-) are opposites. 
* Multiplication (times*, ×) and Division÷, /) are opposites.

To work out the number of blue jellybeans using proper algebra follow these steps: 

↓-----------↓ 

x+10=20

# Identify the goal: We want x by itself the '''[[Variable|Unknown]]'''.
# See the obstacle: There is a +10 next to the x, the Known.
# Do the opposite: Now take the known and move it to the end of the opposite side, and add a  minus symbol in the middle of both knowns because we are also counting the 20 as a known.


x=20-10

Anther way of thinking about it:

Plus is married to minus so minus dose all the heavy work because plus is just bring him self to minus house from pluses house.  He is bring 10 apples with him and we show him in his girlfriends house. Minus is doing the heavy work of catering to Plus.


Do 20-10

x=10       Good job.

Checking Your Work

In maths, you should always check your work. You can do this by working if the original equation equals the same number  by adding the veriable with the constant  two together now that you know the pronumeral or variable to be. 

Does (10 + x (10) = 20)? Yes! Your answer is correct.  

'''Example two'''

In algebra, a fraction line means division. So, this equation means &quot;x (unknown number) divided by 4 equals 3.&quot;

&lt;math&gt;\tfrac{x}{4}&lt;/math&gt;  = 3
# Identify the goal: Get x by itself.
# See the obstacle: The x is being divided by 4.
# Do the opposite: The opposite of division is multiplication. Multiply both sides by 4, because whatever you do to one side you must do to the other to balance the equation. Remember about the family relationship we move the number to the other house.

x = 3*4

x=12

Check Your Work

Put 12 back into the original equation:

* Does 12 / 4 = 3?   Yes! The answer is correct.

===== '''Important notes:''' =====
* Can be called variable OR pronumeral.
* And can be any letter from a to z.
* Solving for (pronumeral here) e.g., &quot;solving for x&quot; means finding the pronumeral.

====== '''Fun Fact''' ======
The letter that is most commonly used for variables is x and the reason for this dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, often called one of the main &quot;founders&quot; of Algebra you could say, used to call the unknown pronumeral &quot;'''shay'''&quot;. &quot;'''Shay&quot;''' comes from the Arabic word '''شَيْء''', which essentially means &quot;thing&quot;. When Al-Khwarizmi's works were translated to Latin in medieval Spain, &quot;shay&quot; was translated as '''&quot;xay&quot;,''' since the letter x was pronounced as &quot;sh&quot; in Spain. Later on, this word &quot;'''xay&quot;''' got abbreviated to &quot;'''x&quot;''' to represent the symbol of the unknown, so we normally use x for standard questions. For more information, visit this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page.

= Algebra problems =
Solve for x.&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{''x'' − 9 = 20
|type=&quot;{}&quot;}
''x''={ 29_3 }

{''x'' − 3 = 6
|type=&quot;{}&quot;}
''x''={ 9_3 }

{''x'' + 5 = 15
|type=&quot;{}&quot;}
''x''={ 10_3 }

{''x'' + 17 = 23
|type=&quot;{}&quot;}
''x''={ 6_3 }

{4''x'' = 12
|type=&quot;{}&quot;}
''x''={ 3_3 }

{''x''/2 = 0.5
|type=&quot;{}&quot;}
''x''={ 1_3 }

{''x''/50 = 2
|type=&quot;{}&quot;}
''x''={ 100_3 }

{''x''/9 = 5
|type=&quot;{}&quot;}
''x''={ 45_3 }
&lt;/quiz&gt;

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...

==Arithmetic==
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]]
'''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.
===Fractions===
[[File:Cake quarters.svg|thumb|left|A Cake with fractions]]
'''Fractions''' (from Latin ''fractus'', &quot;broken&quot;) are parts of a whole. On the left side in the image of the cake, there is only &lt;math&gt;3/4&lt;/math&gt;'s of the cake showing, the other &lt;math&gt;1/4&lt;/math&gt; has been eaten/taken away.

The number, ''3'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''denominator''' (Denominator: Number showing the all time total).
;
;Simplest form/reduced form
A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So &lt;math&gt;2/4&lt;/math&gt; is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: &lt;math&gt;1/2&lt;/math&gt;. Though, not every fraction can be divided by 2, there are fractions, such as: &lt;math&gt;5/35&lt;/math&gt;, &lt;math&gt;7/21&lt;/math&gt;, and &lt;math&gt;30/5&lt;/math&gt;. The two first fractions are not divisible by 2, and &lt;math&gt;30/5&lt;/math&gt; can not be divided by 2 on both sides, but only on &lt;math&gt;30&lt;/math&gt;. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, &lt;math&gt;\tfrac{2}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{2}&lt;/math&gt; represents the same quantity, a half! 

Here, we will present a few fractions for you to simplify.
====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)====
&lt;quiz display=simple points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{6}{8}=&lt;/math&gt;{ 3/4_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{4}{60}=&lt;/math&gt;{ 1/15_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{30}{90}=&lt;/math&gt;{ 1/3_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{8}{18}=&lt;/math&gt;{ 4/9_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{9}{72}=&lt;/math&gt;{ 1/8_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{64}{46}=&lt;/math&gt;{ 32/23|1 9/23_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{206}{340}=&lt;/math&gt;{ 103/170_7 }
&lt;/quiz&gt;
===== Adding or Subtracting Fractions =====
[[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]]
To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; and &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, you first have to multiply the 2 in &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation)

Alright, we got that out of the way, so once we have &lt;math&gt;\tfrac{12}{6}&lt;/math&gt; + &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, we can simply add. So &lt;math&gt;12&lt;/math&gt; + &lt;math&gt;4&lt;/math&gt; = &lt;math&gt;16&lt;/math&gt;, but don't add the denominators, they stay the same. So the answer is &lt;math&gt;\tfrac{16}{6}&lt;/math&gt;, and then we simplify down to &lt;math&gt;\tfrac{8}{3}&lt;/math&gt; dividing by 2 on both the numerator and denominator. 

But... did you notice something? &lt;math&gt;\tfrac{8}{3}&lt;/math&gt;? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]).

===== Multiplying Fractions =====
To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; and &lt;math&gt;\tfrac{2}{6}&lt;/math&gt;. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{3}&lt;/math&gt;. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; × &lt;math&gt;\tfrac{1}{3}&lt;/math&gt; &lt;math&gt;=&lt;/math&gt; &lt;math&gt;\tfrac{3}{12}&lt;/math&gt;. 

&lt;math&gt;\tfrac{3}{12}&lt;/math&gt; is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in &lt;math&gt;\tfrac{1}{4}&lt;/math&gt;.

===== Dividing Fractions =====
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as &quot;Keep, Change, Flip&quot; where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; ÷ &lt;math&gt;\tfrac{7}{12}&lt;/math&gt;. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; × (or •) &lt;math&gt;\tfrac{12}{7}&lt;/math&gt;. Multiply the numerators and denominators. The answer is &lt;math&gt;\tfrac{72}{56}&lt;/math&gt;, simplified down to &lt;math&gt;\tfrac{9}{7}&lt;/math&gt;.

===== Improper Fraction --&gt; Mixed Fraction =====
Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

{{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}}

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

===Decimals===

Ever wondered how to write 8&lt;math&gt;\tfrac{47}{100}&lt;/math&gt; as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern:
# 6&lt;math&gt;\tfrac{98}{100}&lt;/math&gt; = 6.98
# 2&lt;math&gt;\tfrac{56}{100}&lt;/math&gt; = 2.56
# 9&lt;math&gt;\tfrac{27}{100}&lt;/math&gt; = 9.27
# 5&lt;math&gt;\tfrac{83}{100}&lt;/math&gt; = 5.83
You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. 

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have &lt;math&gt;6.72&lt;/math&gt;, the &lt;math&gt;6&lt;/math&gt; is in the Ones place. Now, let's throw &lt;math&gt;9&lt;/math&gt; in the tens place, which is 10 times bigger than the Ones place: &lt;math&gt;96.72&lt;/math&gt;. But... that's doesn't seem enough, does it? Let's throw in a &lt;math&gt;6, 2, 8&lt;/math&gt; and a &lt;math&gt;3&lt;/math&gt; in there! And now, we have: &lt;math&gt;628,396.72&lt;/math&gt;.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, &lt;math&gt;628,396.72&lt;/math&gt;, is the number we need to break down. Let's start from the decimal point, and move left:
* The number &lt;math&gt;6&lt;/math&gt; is in the Ones place. '''x10'''
* The number &lt;math&gt;9&lt;/math&gt; is in the Tens place. '''x10'''
* The number &lt;math&gt;3&lt;/math&gt; is in the Hundreds place. '''x10'''
* The number &lt;math&gt;8&lt;/math&gt; is in the Thousands place. '''x10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten thousands place. '''x10'''
* The number &lt;math&gt;6&lt;/math&gt; is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a &lt;math&gt;5, 2, 4&lt;/math&gt; and a &lt;math&gt;7&lt;/math&gt;. Now, we have &lt;math&gt;628,396.725,247&lt;/math&gt;. Let's break this number up like we did above.

So, our number, &lt;math&gt;628,396.725,247&lt;/math&gt;, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'':
* The number &lt;math&gt;7&lt;/math&gt; is in the Tenths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Hundredths place. '''x-10'''
* The number &lt;math&gt;5&lt;/math&gt; is in the Thousandths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten Thousandths place. '''x-10'''
* The number &lt;math&gt;4&lt;/math&gt; is in the Hundred Thousandths place. '''x-10'''
* The number &lt;math&gt;7&lt;/math&gt; is in the Millionths place. 

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

==== Adding/Subtracting Decimals ====
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for &lt;math&gt;1.5&lt;/math&gt; + &lt;math&gt;2.5&lt;/math&gt; we'd line up the decimal points. But, if we had a problem like &lt;math&gt;1.15&lt;/math&gt; + &lt;math&gt;2.0&lt;/math&gt;, we'd add a &lt;math&gt;0&lt;/math&gt; after the &lt;math&gt;0&lt;/math&gt; that is behind the decimal. Adding a zero to a place in a decimal means &quot;no value&quot;. So &lt;math&gt;10&lt;/math&gt; basically means no ones, and &lt;math&gt;100&lt;/math&gt;, means no ones or hundreds. Same things goes for subtracting as well folks.
=====Sample problems for ''adding/subtracting decimals''=====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
6.8 - 2.5 = { 4.3_6 }

{
|type=&quot;{}&quot;}
3.4 + 5.6 = { 9_6 }

{
|type=&quot;{}&quot;}
9 + 4.50 = { 13.5_6 }

{
|type=&quot;{}&quot;}
41.89 + 25.00 = { 66.89_6 }

{
|type=&quot;{}&quot;}
9.01 + 3.089 = { 12.099_6 }

{
|type=&quot;{}&quot;}
10.90 + 11.1 = { 22_6 }

{
|type=&quot;{}&quot;}
9.5 + 3.44 = { 12.94_6 }

{
|type=&quot;{}&quot; coef=&quot;2.5&quot;}
9.00 x 2.00 = { 18_6 } 
&lt;/quiz&gt;

==== Multiplying Decimals ====
[[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]]
Multiplying decimals isn't as hard as it really seems to be. So, we have &lt;math&gt;9.83&lt;/math&gt; × &lt;math&gt;5.73&lt;/math&gt;. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;× &lt;math&gt;5.73&lt;/math&gt;
-------

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt;   &lt;math&gt;68810&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt; &lt;math&gt;491500
&lt;/math&gt;
-------

With the simple usage of addition, we should get:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;   &lt;math&gt;+&lt;/math&gt;  &lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+
&lt;/math&gt; &lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;563259&lt;/math&gt;

Now, we need to bring back our handy dandy decimal point, but where? In &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;. Now, that totals up to four numbers overall behind the decimal point. So in &lt;math&gt;563259&lt;/math&gt;, we need to move the decimal point four times (beginning from the right). So watch as follows:

&lt;math&gt;563259.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56325.9&lt;/math&gt;&lt;br&gt;
&lt;math&gt;5632.59&lt;/math&gt;&lt;br&gt;
&lt;math&gt;563.259&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56.3259&lt;/math&gt;

That simple. Now, review your work, your whole work should look like this:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;       × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;    &lt;math&gt;+&lt;/math&gt;&lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+&lt;/math&gt;&lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;56.3259&lt;/math&gt;

==== Dividing Decimals ====
;Dividing a decimal by a whole number
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. 

Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal!
;Dividing a decimal by a decimal
The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: &lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt;. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:&lt;br&gt;
&lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;694.5&lt;/math&gt; ÷ &lt;math&gt;57.8&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;6945&lt;/math&gt;. ÷ &lt;math&gt;578&lt;/math&gt;.


Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, &lt;math&gt;69.45&lt;/math&gt; divided by &lt;math&gt;5.78&lt;/math&gt; should get you &lt;math&gt;12.0155709&lt;/math&gt;!

A pretty simple one we could go is &lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;, here, we simply move our dots like so:&lt;br&gt;
&lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;04.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;4&lt;/math&gt;&lt;br&gt;
Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is &lt;math&gt;16&lt;/math&gt;.

{{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}}

===Percentages===
A good definition of &quot;percent&quot; is a fraction in which the denominator is the number &lt;math&gt;100&lt;/math&gt;. For example, the numbers &lt;math&gt;59%&lt;/math&gt;, &lt;math&gt;63%&lt;/math&gt;, &lt;math&gt;91%&lt;/math&gt;, and &lt;math&gt;85%&lt;/math&gt;, are the same as just saying &lt;math&gt;\tfrac{59}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{63}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{91}{100}&lt;/math&gt;, and &lt;math&gt;\tfrac{85}{100}&lt;/math&gt;. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.
====Converting Percentages====
Now that we got the basis of percentages and how they operate, we should look into changing percentages.
===== Percentage → Decimal =====
Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have &lt;math&gt;\tfrac{9}{100}&lt;/math&gt;, which, in percentage form, is &lt;math&gt;9%&lt;/math&gt;. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: &lt;math&gt;9.&lt;/math&gt;. Now, we have &lt;math&gt;9.&lt;/math&gt;, so then we move the decimal number two places to the left, like so: &lt;math&gt;9.&lt;/math&gt; → &lt;math&gt;.9&lt;/math&gt; → &lt;math&gt;.09&lt;/math&gt;. So now, we have &lt;math&gt;0.09&lt;/math&gt;. We added the 2 zeros in because there is no value in the tenths place, and because &lt;math&gt;.09&lt;/math&gt; does not look quite right. Looks a bit off.
===== Samples problems for ''converting percentages to decimals'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
59% = { 0.59_5 }

{
|type=&quot;{}&quot;}
63% = { 0.63_5 }

{
|type=&quot;{}&quot;}
91% = { 0.91_5 }

{
|type=&quot;{}&quot;}
85% = { 0.85_5 }

{
|type=&quot;{}&quot;}
9% = { 0.09_5 }

{
|type=&quot;{}&quot;}
9834% = { 98.34_5 }

{
|type=&quot;{}&quot;}
20% = { 0.2_5 }

{
|type=&quot;{}&quot;}
4% = { 0.04_5 }

{
|type=&quot;{}&quot;}
7.6% = { 0.076_5 }

{
|type=&quot;{}&quot;}
6% = { 0.06_5 }
&lt;/quiz&gt;
===== Decimal → Percentage =====
Now to convert a decimal into percentage we essentially do the complete opposite. We have &lt;math&gt;98.34&lt;/math&gt;. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: &lt;math&gt;98.34&lt;/math&gt; → &lt;math&gt;983.4&lt;/math&gt; → &lt;math&gt;9834.&lt;/math&gt; --Now, we have &lt;math&gt;9834.&lt;/math&gt;, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of &lt;math&gt;9834%&lt;/math&gt;.

==== Finding percent of a number ====
[[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]]
So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt;. We take the percentage, &lt;math&gt;20%&lt;/math&gt;, and divide it by &lt;math&gt;100&lt;/math&gt;. So we get &lt;math&gt;20/100&lt;/math&gt; = &lt;math&gt;.2&lt;/math&gt;. Then, we multiply &lt;math&gt;.2&lt;/math&gt; by &lt;math&gt;95&lt;/math&gt;, in which we get &lt;math&gt;19&lt;/math&gt;. So &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt; is &lt;math&gt;19&lt;/math&gt;. If that didn't make sense .2 x 95 is 2 x 95 = 190 but because we multipled 2 by ten in the tenths place to the ones place we -10 off of the number to reverse what we just did.  Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University.    

{{subpage navbar}}
[[Category:Speak Math Now!]]</text>
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  <page>
    <title>Social Victorians/People/Abercorn</title>
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    <id>263978</id>
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        <username>Scogdill</username>
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      <text bytes="43461" sha1="hxy66do47slu2hb3j0qv5ypgg3pywyi" xml:space="preserve">== Overview ==
The Dukedom of Abercorn is the last non-royal dukedom created. Queen Victoria created it in 1869. 

This page includes the Earl of Wicklow, the family of which married into the Abercorn family in 1816 when William Howard, 4th Earl of Wicklow married Lady Cecil Frances Hamilton — the daughter and only child of John Hamilton, 1st Marquess of Abercorn.&lt;ref&gt;{{Cite journal|date=2026-06-24|title=William Howard, 4th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=William_Howard,_4th_Earl_of_Wicklow&amp;oldid=1360966619|journal=Wikipedia|language=en}}&lt;/ref&gt; William Howard, 4th Earl of Wicklow was succeeded by his nephew, Charles Howard, 5th Earl of Wicklow (5 November 1839 – 20 June 1881).&lt;ref&gt;{{Cite journal|date=2024-08-26|title=Charles Howard, 5th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Charles_Howard,_5th_Earl_of_Wicklow&amp;oldid=1242455245|journal=Wikipedia|language=en}}&lt;/ref&gt; Also Ralph Howard, 7th Earl of Wicklow married Lady Gladys Mary Hamilton (daughter of the 2nd Duke of Abercorn) in 1902.&lt;ref name=&quot;:18&quot;&gt;{{Cite journal|date=2025-08-05|title=Cecil Howard, 6th Earl of Wicklow|url=https://en.wikipedia.org/w/index.php?title=Cecil_Howard,_6th_Earl_of_Wicklow&amp;oldid=1304372795|journal=Wikipedia|language=en}}&lt;/ref&gt;

The National Library of Ireland has papers from Sarah Howard and her children, including Lady Caroline Howard. 

== Also Known As ==
*Family name: Hamilton
*the Duke of Abercorn
**James Hamilton, 1st Duke of Abercorn (10 August 1868 – 31 October 1885)&lt;ref name=&quot;:0&quot;&gt;&quot;James Hamilton, 1st Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10144.htm#i101433|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Hamilton, 2nd Duke of Abercorn (31 October 1885 – 3 January 1913)&lt;ref name=&quot;:12&quot;&gt;&quot;James Hamilton, 2nd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101033|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
**James Albert Edward Hamilton, 3rd Duke of Abercorn (3 January 1913 – 12 September 1953)&lt;ref name=&quot;:13&quot;&gt;&quot;James Albert Edward Hamilton, 3rd Duke of Abercorn.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101031|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;
*the Duchess of Abercorn
**Louisa Russell Hamilton, Duchess of Abercorn (10 August 1868 – 31 October 1885)
**Maria Anna Curzon-Howe Hamilton (31 October 1885 – 3 January 1913)
*Dowager Duchess of Hamilton
**Louisa Russell Hamilton, Duchess of Abercorn (31 October 1885 – March 1905)
**Maria Anna Curzon-Howe Hamilton (3 January 1913 – )
*Subsidiary titles:
**Marquess of Hamilton (courtesy title for the heir apparent)
***James Albert Edward Hamilton, 3rd Duke of Abercorn (31 October 1885 – 12 September 1953)
**Viscount Strabane (courtesy title for the heir apparent of the Marquess of Hamilton)

== Acquaintances, Friends and Enemies ==
=== Friends ===
*The Royal Family, especially [[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales | Alexandra, Princess]] of Wales, in the generation of the 2nd duke.

== Timeline ==
A lot of people are treated on this page, so this timeline will be somewhat chaotic to read. These events probably didn't directly affect every single person treated on this page, but discussions about them probably circulated through the families. The detail about Lady Caroline Howard and her mother, the Hon. Susan Howard, is to make these people, whose papers are in the National Library of Ireland, more concrete and known.

'''1832 October 25''', James Hamilton and Louisa Russell married at Gordon Castle, Fochabers, Morayshire, in Scotland.&lt;ref name=&quot;:0&quot; /&gt;

'''1854 May 23''', Beatrix Frances Hamilton and George Frederick D'Arcy Lambton married.&lt;ref&gt;&quot;Lady Beatrix Frances Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1147.htm#i11470|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1855 April 10''', Harriet Georgiana Louisa Hamilton and Thomas George Anson married.&lt;ref name=&quot;:2&quot;&gt;&quot;Lady Harriett Georgiana Louisa Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1034.htm#i10332|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1858 October 26''', Katherine Elizabeth Hamilton and William Henry Edgcumbe married.&lt;ref&gt;&quot;Lady Katherine Elizabeth Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1135.htm#i11344|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1859 November 22''', Louisa Jane Hamilton and William Montagu Douglass Scott married.&lt;ref&gt;&quot;Lady Louisa Jane Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10359.htm#i103583|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1868''', the title the Duke of Abercorn was created.&lt;ref&gt;{{Cite journal|date=2020-07-06|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&amp;oldid=966293304|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1869 January 7''', James Hamilton (2nd Duke) and Maria Anna Curzon-Howe married at St. George's Church, St. George Street, Hanover Square, in London.&lt;ref name=&quot;:3&quot;&gt;&quot;Lady Mary Anna Curzon.&quot; {{Cite web|url=http://www.thepeerage.com/p10104.htm#i101034|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1869 November 8''', there may have been a double wedding: Albertha Frances Anne Hamilton and George Charles Spencer-Churchill married&lt;ref name=&quot;:8&quot;&gt;&quot;Lady Albertha Frances Anne Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p10595.htm#i105942|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;, and Maud Evelyn Hamilton and Henry Petty-Fitzmaurice married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Maud Evelyn Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1163.htm#i11629|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1871 January''' '''4, Wednesday''', Lady Caroline Howard was invited to a [[Social Victorians/Timeline/1870s#4 January 1871, Wednesday|ball hosted by Major Goodman and the Officers of the 5th Dragoon Guards]] (probably in Coventry?).

'''1871 February 17, Friday''', Lady Caroline Howard attended a [[Social Victorians/Timeline/1870s#Birmingham Tennis Court Club Ball|ball hosted by the &quot;bachelors of the Tennis Court Club&quot; in Birmingham]].

'''1871 May 9, Tuesday''', Lady Caroline Howard, Lady Alice Howard and Lady Louisa Howard were [[Social Victorians/Timeline/1870s#9 May 1871, Tuesday, Queen's Drawing-Room|presented to Queen Victoria at a Drawing-room]] by their mother, the Hon. Mrs. Sarah Howard.

'''1871 August 31, Thursday''', The Freeman's Journal reported that &quot;The Hon. Mrs. Howard, Lady Caroline Howard and suite have arrived at the Morrisson Hotel.&quot;&lt;blockquote&gt;The following are amongst the latest arrivals at the Morrisson Hotel: — Mrs. Percival Maxwell and the Misses Maxwell and suite, Mr and Mrs Herbert Read and suite, Rev H R Heywood, and Master H A Heywood, Mr F H Downing, Mr M Neil, Mr and Mrs Herbert and suite, Mr Abbott, Mr D'Arcy, Mr and Mrs G Woods and suite.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Freeman's Journal'' 31 August 1871, Thursday: 4 [of 4], Col. 1a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18710831/012/0004. Same print title, n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1871 November 28''', George Francis Hamilton and Maud Caroline Lascelles married.&lt;ref name=&quot;:6&quot;&gt;&quot;Rt. Hon. Lord Sir George Francis Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1133.htm#i11323|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1872 January 4, Thursday''', the Hon. Mrs. Howard and Lady Caroline Howard and their suites were reported to &quot;have arrived at Morrisson's Hotel in Dublin.&lt;ref&gt;&quot;Fashionable Miscellany.&quot; ''Dublin Evening Post'' 4 January 1872, Thursday: 3 [of 4], Col. 2c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000435/18720104/021/0003. Same print title, n.p.&lt;/ref&gt;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Morning Mail'' (Dublin) 5 January 1872, Friday: 3 [of 4, digital], Col. 2c [of 10 on digital image]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0006103/18720105/067/0003. The digital image has the last 2 columns of the prior page on this page, so the citation should be to p. 2 [of 4], Col. 8c [of 8].&lt;/ref&gt; Also at the Morrisson's Hotel at this time was Sir Roland Blennerhassett, Bart., M.P.&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening'' Mail 5 January 1872, Friday:  3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18720105/028/0003. Same print and digital title, print n.p.&lt;/ref&gt;

'''1872 February 28, Wednesday''', the Howards are back at Morrisson's Hotel:&lt;blockquote&gt;Lady Caroline Howard, Lady Louisa Howard, and the Hon Mrs Howard and suite, Shelton Abbey, have arrived at Morrrisson's Hotel.&lt;ref&gt;&quot;Fashionable.&quot; ''Dublin Evening Telegraph'' 28 February 1872, Wednesday: 4 [of 4], Col. 7b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002093/18720228/051/0004. Print title: ''The Evening Telegraph'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;'''1872 March 2, Saturday''', the ''Weekly Freeman and Irish Agriculturalist'' reported that &quot;Lady Caroline Howard, Lady Louisa Howard, and the Hon Mrs Howard and suite, Shelton Abbey, have arrived at Morrisson's Hotel.&quot; Two 1-sentence paragraphs later, the paper reported that the same group had &quot;left Morrisson's Hotel for Shelton Abbey.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Weekly Freeman's Journal'' 2 March 1872, Saturday: 7 [of 8], Col. 1a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001446/18720302/062/0007. Print title: ''Weekly Freeman and Irish Agriculturalist'', same p.&lt;/ref&gt; Shelton Abbey was the [[Social Victorians/People/Abercorn#Residences|ancestral seat and at this time the country residence]] of the Earls of Wicklow, Arklow, Co. Wicklow.

'''1872 December 13, Friday''', &quot;The Hon. Mrs. Howard, Lady Caroline Howard, Lady Louisa Howard, and Lady Alice Howard and suite have left Morrisson's Hotel.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Freeman's Journal'' 13 December 1872, Friday: 2 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000056/18721213/006/0002. Same print title and p.&lt;/ref&gt;

'''1873 January''' '''13, Monday''', the Hon. Mrs. Sarah Howard, the Hon. Lady Alice Howard and the Hon. Lady Louisa Howard attended the [[Social Victorians/Timeline/1870s#Ball at the Chief Secretary's Lodge|Marquis of Hartington's ball at the Chief Secretary's Lodge]]. It is not clear why Lady Caroline Howard's name is not mentioned.

'''1873 January 14, Tuesday''', &quot;Lord Dunally and suite, Hon. Mrs. Howard, Lady Alice Howard and suite, Lady Louise Howard and suite, and Lady Caroline Howard, have arrived at Morrisson's Hotel.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Dublin Evening Post'' 14 January 1873, Tuesday: 3 [of 4], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000435/18730114/049/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt; Since they attended a ball the night before, probably they had already arrived. Lady Catherine was with them.

'''1874 December 15, Tuesday''', the Right Hon. Sir Michael and Lady Lucy Hicks-Beach hosted a dinner in the Chief Secretary's Lodge, suggesting that this social event might have had a political purpose. Mr. LeFanu cannot be the Irish writer Sheridan Le Fanu, who died 7 February 1873.&lt;ref&gt;{{Cite journal|date=2026-06-28|title=Sheridan Le Fanu|url=https://en.wikipedia.org/w/index.php?title=Sheridan_Le_Fanu&amp;oldid=1361491348|journal=Wikipedia|language=en}}&lt;/ref&gt; (Perhaps this LeFanu is a relation, a son or brother? Another LeFanu with a first name gets mentioned at a social event about this time.)&lt;blockquote&gt;THE CHIEF SECRETARY’S LODGE.&lt;p&gt;The Right Hon. Sir Michael and Lady Lucy Hicks-Beach entertained the following at dinner on Tuesday evening at the Chief Secretary’s Lodge: — Sir Dominic Corrigan, Sir Arthur and Lady Olive Guinness, Lady Mary Fortescue, the Hon. Mrs. Howard and Lady Caroline Howard, Mr. and Mrs. Percy Bernard, Colonel Henry, R.A., and Mrs. Henry; Mr. Donnelly. C.B., and Mrs. Donnelly; Mr., Mrs., and Miss lsaac; Mr. LeFanu, Colonel Forster, Colonel Hillier, and Mr. Caulfield.&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Cork Constitution'' 17 December 1874, Thursday: 4 [of 4; n.p. in print], Col. 1a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001648/18741217/099/0004. Print title: ''The Cork Constitution''.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;'''1876 March 23''', Cecil Howard, 6th Earl of Wicklow and Francesca Maria Chamberlayne married.&lt;ref name=&quot;:18&quot; /&gt;

'''1877 July 25, Wednesday''', Miss Tottenham, Lady Caroline Howard, Miss Colley are reported to have arrived at Merton Lodge in Torquay.&lt;ref&gt;&quot;The Torquay Directory.&quot; ''Torquay Directory and South Devon Journal'' 25 July 1877, Wednesday: 4 [of 8], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001246/18770725/085/0004. Same print and digital title and p.&lt;/ref&gt;

'''1877 July 28, Saturday''', Lady Caroline Howard is listed as one of the guests at Merton Lodge in Lincombe Hill Road Middle, Torquay. Other guests listed are Miss Kelly, Mrs. Frank Webber, Miss Tottenham and Miss Colley.&lt;ref&gt;&quot;49. Lincombe Hill Road. Middle.&quot; &quot;Torquay Directory.&quot; ''Torquay Times and South Devon Advertiser'' 28 July 1877, Saturday: 2 [of 8, both print and digital], Col. 3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001420/18770728/039/0002.&lt;/ref&gt;

'''1877 December 6, Thursday''', donations from the Hon. Mrs. Sarah Howard (£2 2s.), Lady Alice Howard (£1), Lady Caroline Howard (£1) and Lady Louise Howard (£1) to the Church of Ireland Clergy Widows' and Orphans' Society.&lt;ref&gt;&quot;The Church.&quot; ''Cork Constitution'' 11 December 1877, Tuesday: 3 [of 4], Col. 2a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001648/18771211/076/0003. Same print title, n.p.&lt;/ref&gt;

'''1877 December 22, Saturday''', Sarah Howard, Lady Caroline Howard and Captain the Hon. Cecil Ralph Howard were visitors in Dagmar Terrace in Portsmouth. The following are all the people listed as visitors at Dagmar Terrace, with the odd numbering:&lt;blockquote&gt;D&lt;small&gt;AGMAR&lt;/small&gt; T&lt;small&gt;ER&lt;/small&gt;&lt;small&gt;RACE&lt;/small&gt;.

# Captain the Hon. Cecil Ralph Howard, late 60th Rifles, &amp; the Hon Mrs Howard Lady Caroline Howard
# Captain &amp; Mrs. Henderson
## [a] The Hon. Richard and Mrs. Bineham 
# [a] Captain and Mrs. Fearson and family
# Mr.and Mrs. Hall Mrs. and the Misses Buchannans
# The Rev Palms &amp; fam
# [a] Colonel Johnston [a] Mrs. Oldfield [a] Miss Flowers
# Captain Parkinson and family&lt;ref&gt;&quot;Visitors' List.&quot; ''Portsmouth Times and Naval Gazette'' 22 December 1877, Saturday: 3 [of 10, digital and print], Col. 5 [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001365/18771222/027/0003. Print title: ''Portsmouth Times and Naval Gazette, County Journal''.&lt;/ref&gt;
&lt;/blockquote&gt;

'''1878 January 18, Friday''', The ''Dublin Daily Express'' says,&lt;blockquote&gt;Lady Caroline Howard arrived yesterday at Kingstown from England.&lt;p&gt;

Captain the Hon. C. Howard and Mrs. Howard have arrived at Kingstown from England.&lt;ref&gt;&quot;The Court.&quot; ''Dublin Daily Express'' 19 January 1878, Saturday: 5 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001384/18780119/113/0005. Print title: ''The Daily Express'', same p.&lt;/ref&gt;&lt;/blockquote&gt;'''1878 January 26, Saturday''', visitors at Dagmar Terrace (in Portsmouth?) were Lady Caroline Howard, listed with Capt. the Hon. Cecil Ralph Howard, &quot;late 60th Rifles.&quot;&lt;ref&gt;&quot;Visitors' List.&quot; ''Portsmouth Times and Naval Gazette'' 26 January 1878, Saturday: 6 [of 10], Col. 6c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001365/18780126/051/0006. Print title: ''Portsmouth Times and Naval Gazette, County Journal'', same p.&lt;/ref&gt;

'''1878 July 20''', Claud John Hamilton and Carolina Chandos-Pole married.&lt;ref name=&quot;:5&quot;&gt;&quot;Lord Claud John Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110662|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1879 October 23, Thursday''', Lady Caroline Howard had &quot;arrived from England.&quot;&lt;ref&gt;&quot;The Court.&quot; ''Dublin Daily Express'' 23 October 1879, Thursday: 5 [of 8], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001384/18791023/061/0005. Print title: ''The Daily Express'', same p.&lt;/ref&gt;

'''1880 June 2''', Cecil Howard, 6th Earl of Wicklow and Fanny Catherine Wingfield married.&lt;ref name=&quot;:18&quot; /&gt;

'''1880 December 13, Monday''', Lady Caroline Howard &quot;arrived at Kingstown from London.&quot;&lt;ref&gt;&quot;Court.&quot; ''Dublin Daily Express'' 13 December 1880, Monday: 5 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001384/18801213/089/0005. Print title: ''Daily Express'', same p.&lt;/ref&gt;

'''1881 July 25, Monday''', the ''Irish Times'' says that Lady Caroline Howard and &quot;the Hon. Mrs. Howard and the Ladies Howard (2) have arrived at Kingstown from England.&quot;&lt;ref&gt;&quot;Fashionable Intelligence.&quot; ''Irish Times'' 25 July 1881, Monday: 6 [of 8, digital and print], Col. 3a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001683/18810725/124/0006. Same print title and p.&lt;/ref&gt;

'''1881 August 10, Wednesday''', the ''Dublin Evening Mail'' says that Lady Caroline Howard &quot;has left Kingstown for England.&quot;&lt;ref&gt;&quot;Fashion and Varieties.&quot; ''Dublin Evening Mail'' 10 August 1881, Wednesday: 3 [of 4], Col. 9c [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000433/18810810/046/0003. Same print and digital title, print p. is n.p.&lt;/ref&gt;

'''1881 October 22, Saturday''', Lady Caroline Howard is listed as one of the visitors staying at the Crown Hotel &quot;during the past week.&quot; The visitors listed are the following:&lt;blockquote&gt;Mr. Thomas Barber, Doctor and Mrs. Ayerst, Miss Noyce, Dr. Wilks, Mr. Nightingale, Mr. and Mrs. J. Hill, Lady Caroline Howard, the Hon. Mrs. Ross, Mr. Masters, Mr. Richardson and friend, Mr. Simpson, Mr. Wilson, &amp;c.&lt;ref&gt;&quot;Lyndhurst, Oct. 22.&quot; ''Hampshire Advertiser'' 22 October 1881, Saturday: 7 [of  8, both print and digital], Col. 2c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18811022/049/0007. Print title: ''Hampshire Advertiser County Newspaper''.&lt;/ref&gt;&lt;/blockquote&gt;

=== Fixing Things ===
'''1882 January 3, Tuesday''', the Howard women donated to feed poor people at Christmas:
&lt;blockquote&gt;ACKNOWLEDGMENTS.&lt;p&gt;

Mr J R Fowler acknowledges with thanks the following for free breakfasts to the poor in the Christian Union Buildings:— Mrs Barker, £5; Mrs Lovell, by Mrs Aimers, 10s; Mrs Jno Figgis, [illegible, shillings]; collected by Miss Carroll, 10s: Capt Thompson, 5s; Mrs O Stoney, 2s 6d; Mrs E H Smyth, £1; A Friend, per Dr Darley, £1; Mrs Lewers, £1; Mr Holmes, 10s; Mr Duffus, 10s; Mr W O'B Smyth, 10s; Hon Mrs Howard, £1; Lady Caroline Howard, £1; Lady Alice Howard, 10s; Lady Louisa Howard, 10s; T C Ratcliffe, per Mrs Smyly, £5; Mrs Hemphill, per Mr G Atkinson, 2s 6d; collected in box, 9d — Total, [illegible12] 10 s 9d. Number present last Sunday, 1,200.&lt;ref&gt;&quot;Acknowledgments.&quot; ''Dublin Daily Express'' 3 January 1882, Tuesday: 5 [of 8], Col. 4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001384/18820103/061/0005. Print title: ''The Daily Express'', same p.&lt;/ref&gt;&lt;/blockquote&gt;'''1882 March 16''', Georgiana Susan Hamilton and Edward Turnour married.&lt;ref&gt;&quot;Lady Georgiana Susan Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p1180.htm#i11791|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1882 June 1, Thursday''', the Hon. Sarah Howard and Lady Caroline Howard arrived in Kingstown from England.&lt;ref&gt;&quot;Court and Fashion.&quot; ''Evening Irish Times'' 1 June 1882, Thursday: 7 [of 8], Col. 5b [of 8]. ''British Newspaper Archives'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003464/18820601/108/0007. Print title ''Irish Times'', same p.&lt;/ref&gt;

'''1883 May 28, Monday''', the Hon. Mrs. Sarah Howard and Lady Caroline Howard &quot;left Kingstown for England,&quot; as did the Hon. Bourke.&lt;ref&gt;&quot;Court and Fashion.&quot; ''Evening Irish Times'' 28 May 1883, Monday: 6 [of 8], Col. 8b [of 8]. ''British Newspaper Archives'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003464/18830528/092/0006. Print title: ''Irish Times'', same p.&lt;/ref&gt;

'''1883 November 20''', the marriage between Albertha Frances Anne Hamilton Spencer-Churchill and George Charles Spencer-Churchill was annulled by petition from Albertha Frances Anne Hamilton Spencer-Churchill (married in 1869).&lt;ref name=&quot;:8&quot; /&gt;

'''1883 December 27, Thursday''', the Hon. Mrs. Sarah Howard and Lady Caroline Howard were invited to the ''déjeuner'' after the [[Social Victorians/Timeline/1883#Wedding of William Noble and Grace Elizabeth Lefroy|wedding of Colonel William Noble and Grace Elizabeth Lefroy]].

'''1887 November 14, Monday''', the standing committee of the Meath Hospital and County Dublin Infirmary met and accepted a number of donations, including £L1 1s each from the Hon. Mrs. Sarah Howard, Lady Alice M, Howard, Lady Caroline L. Howard and Lady Louisa F. Howard.&lt;ref&gt;&quot;Meath Hospital and County Dublin Infirmary.&quot; ''Dublin Daily Express'' 15 November 1887, Tuesday: 3 [of 8], Col. 2c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001384/18871115/034/0003. Print title: ''The Daily Express'', same p.&lt;/ref&gt;

'''1891 June 2''', Ernest William Hamilton and Pamela Campbell married.&lt;ref name=&quot;:7&quot;&gt;&quot;Pamela Campbell.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21063|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

'''1894 April 10''', Fanny Catherine Wingfield Howard, Dowager 6th Countess of Wicklow married her 2nd husband, Marcus Francis Beresford.&lt;ref name=&quot;:18&quot; /&gt;

'''1894 November 1''', James Albert Edward Hamilton and Rosaline Cecilia Caroline Bingham married at St. Paul's Church, Knightsbridge, in London.&lt;ref name=&quot;:14&quot;&gt;&quot;Lady Rosalind Cecilia Caroline Bingham.&quot; {{Cite web|url=https://www.thepeerage.com/p10104.htm#i101032|title=Person Page|website=www.thepeerage.com|access-date=2021-05-15}}&lt;/ref&gt;

'''1895 July 13 to August 7''', the general election of 1895. Following the election, the brother-in-law of Cecil Howard, 6th Earl of Wicklow's (brother of his first wife Francesca Chamberlayne) was unseated because of allegations of misconduct.&lt;ref&gt;{{Cite journal|date=2026-02-27|title=Thomas Chamberlayne (cricketer)|url=https://en.wikipedia.org/w/index.php?title=Thomas_Chamberlayne_(cricketer)&amp;oldid=1340809770|journal=Wikipedia|language=en}}&lt;/ref&gt;

'''1897 June 28, Monday''', according to the ''Morning Post'', James Hamilton, 2nd Duke and Maria, Duchess of Abercorn were invited to the [[Social Victorians/Diamond Jubilee Garden Party|Queen's Garden Party]], the official end of the Diamond Jubilee celebrations in London, as were James Albert Edward Hamilton, Marquis and Rosaline, Marchioness of Hamilton.&lt;ref&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''  ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004&lt;/nowiki&gt;'' and ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005&lt;/nowiki&gt;''.&lt;/ref&gt;

'''1897 July 2, Friday''', Alexandra Phyllis Hamilton attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton, the Marquess of Hamilton, and a Mr. Ronald Hamilton. Besides these, probably, a Mr. and Mrs. Hamilton also attended.

'''1902''', Ralph Howard, 7th Earl of Wicklow and Lady Gladys Mary Hamilton married. (She was the daughter of James Hamilton, 2nd Duke of Abercorn.)&lt;ref name=&quot;:18&quot; /&gt;

'''1902 January 14''', Gladys Mary Hamilton and Ralph Francis Forward-Howard married.&lt;ref&gt;&quot;Lady Gladys Mary Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21066|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;

'''1933 July 11''', Claud Nigel Hamilton and Violet Ruby Ashton married.&lt;ref name=&quot;:4&quot;&gt;&quot;Captain Lord Sir Claud Nigel Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2109.htm#i21081|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

[[File:Helen-Mary-Theresa-ne-Vane-Tempest-Stewart-Countess-of-Ilchester-when-Lady-Helen-Stewart-as-the-Archduchess-Marie-Christine-of-Austria.jpg|thumb|alt=Black-and-white photograph of a seated woman richly dressed in an historical costume with a white feather plume in her hair and a fan|Lady Helen Stewart as Arch-duchess Marie Christine of Austria. ©National Portrait Gallery, London.]]
=== Lady Alexandra Hamilton ===
Lady Alexandra Hamilton was one of the archduchesses — along with with 3 or 4 other young women — in [[Social Victorians/People/Londonderry#The Entourage of Maria Thérèse|the entourage of the Marchioness of Londonderry]], who led the Austrian procession as Marie Thérèse, Empress of the Holy Roman Empire.&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3a}} These young women were present at the ball as the daughters of Marie Thérèse, and the young men dressed as archdukes were present as her sons. Lady Alexandra Hamilton went as &quot;Archduchess Marie-Josepha in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&quot;&lt;ref name=&quot;:9&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 6b}} &lt;ref name=&quot;:10&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;

The newspapers report that the archduchesses were all dressed alike, but only one photograph exists of any of these young women in costume — that of [[Social Victorians/People/Londonderry#Helen Mary Theresa Vane-Tempest-Stewart|Helen Mary Theresa Vane-Tempest-Stewart]] (which is shown, right). The newspaper descriptions are on her page, with her portrait in costume, but they apply to all the archduchesses.

=== Lord Frederick Hamilton ===
[[File:Lord Frederick Spencer Hamilton Vanity Fair 1895-02-07.jpg|thumb|left|alt=Colored drawing of a man in a suit, his hands in his pockets, facing to the right|Lord Frederick Hamilton, ''Vanity Fair'', by &quot;Spy,&quot; 7 February 1895]]
Lord Frederick Spencer Hamilton was 6th son and 13th child of the 1st Duke of Abercorn. No photograph of him in costume exists.

He is shown (at left) as he looked in 7 February 1895 in a Spy caricature in ''Vanity Fair''. This caricature portrait, by Leslie Ward (&quot;Spy&quot;) is called ''The Pall Mall Magazine'' and is Number 647 in Vanity Fair's &quot;Statesmen&quot; series.&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}}&lt;/ref&gt; He was editor of the ''Pall Mall Gazette'' 1896–1900.&lt;ref&gt;{{Cite journal|date=2023-09-23|title=Lord Frederick Spencer Hamilton|url=https://en.wikipedia.org/w/index.php?title=Lord_Frederick_Spencer_Hamilton&amp;oldid=1176655264|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Frederick_Spencer_Hamilton.&lt;/ref&gt;

For the ball, Lord Frederick Hamilton was dressed
*as a &quot;gentleman of the Court of Queen Elizabeth,&quot; wearing &quot;crimson cloth of gold with jewelled belt.&quot;&lt;ref name=&quot;:15&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 36, Col. 3b}}
*as a &quot;Gentleman of the Court of Queen Elizabeth. Costume of crimson and cloth of g [sic] with jewelled belt.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1b}}
*&quot;in crimson cloth of gold and jeweled belt.&quot;&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7a}}
*&quot;as a gentleman of the court of Queen Elizabeth, was dressed in a costume of crimson cloth-of-gold, with a jewelled belt.&quot;&lt;ref name=&quot;:11&quot;&gt;“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.&lt;/ref&gt;

==== Memoirs ====

* Hamilton, Frederic [sic] Spencer. ''My Yesterdays'' (3 vols.). Hodder and Stoughton, 1920.
*# ''The Days Before Yesterday''. The Internet Archive has this: https://archive.org/details/daysbeforeyester00hamiuoft/page/n5/mode/2up.
*# ''Vanished Pomps of Yesterday''. The Internet Archive has this: https://archive.org/details/vanishedpompsofy028823mbp.
*# ''Here, There and Everywhere''. The Internet Archive has this: https://archive.org/details/herethereeverywh0000hami.

[[File:James Hamilton 3rd Duke of Abercorn.png|thumb|alt=Old colored drawing of a man in a 19th-century officer's uniform of the 1st Life Guards with white gloves, a red stripe down the side of his pants and unbuttoned jacket and a hat, holding a white or silver sword under his left arm, facing 1/4 to his right|&quot;He will be the 3rd Duke&quot; (James Hamilton, Marquis of Hamilton), ''Vanity Fair'' 16 February 1899]]
=== James Hamilton, Marquess of Hamilton ===
James Hamilton, Marquis of Hamilton was dressed in a &quot;black velvet tunic; breeches and cloak trimmed jet; large hat, feathers, wig, sword, &amp;c., of the period&quot; of Charles II.&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 3a}} No photograph of him in costume exists.

A caricature portrait (right) called ''He will be the 3rd Duke'' (James Hamilton, Marquess of Hamilton) by &quot;Hadge&quot; appeared in the 16 February 1899 issue of ''Vanity Fair'', as Number 739 in its &quot;Men of the Day&quot; series,&lt;ref name=&quot;:16&quot; /&gt; giving a sense of what he looked like at about the time of the ball.

In 1892 Hamilton joined the 1st Life Guards, so the uniform he is wearing in this portrait is likely that of an officer of the 1st Life Guards.&lt;ref&gt;{{Cite journal|date=2024-01-12|title=James Hamilton, 3rd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_3rd_Duke_of_Abercorn&amp;oldid=1195216640|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/James_Hamilton,_3rd_Duke_of_Abercorn.&lt;/ref&gt;

James Hamilton's wife Lady Rosalind Hamilton is not reported as having been present at the ball, perhaps because she was pregnant with her second child and gave birth in August, five weeks later, so she was around 8 months pregnant.

=== Ronald Hamilton ===
Mr. Ronald Hamilton, possibly Ronald James Hamilton, was dressed as a &quot;Gentleman of the Court of Queen Elizabeth, in black velvet trimmed with jet.&quot;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1c}}

== Demographics ==

=== Nationality ===
*The title Duke of Abercorn is in the peerage of Ireland; the Marquess of Hamilton is in the peerage of the U.K.

=== Residences ===

==== The Hon. Mrs. Sarah Howard and the Earls of Wicklow ====

* Shelton Abbey, Arklow, Co. Wicklow (east coast of Ireland) (until 1951)&lt;ref&gt;{{Cite journal|date=2026-06-30|title=Shelton Abbey Prison|url=https://en.wikipedia.org/w/index.php?title=Shelton_Abbey_Prison&amp;oldid=1361924427|journal=Wikipedia|language=en}}&lt;/ref&gt;

== Family ==
*James Hamilton, 1st Duke of Abercorn (21 January 1811 – 31 October 1885)&lt;ref name=&quot;:0&quot; /&gt;
*Louisa Russell Hamilton (– March 1905)
#Lady '''Harriet Georgiana Louisa Hamilton''' Anson (6 July 1834 – 23 April 1913)
#Lady Beatrix Frances Hamilton Lambton (21 July 1835 – 21 January 1871)
#Lady Louisa Jane Hamilton Scott (26 August 1836 – 16 March 1912)
#Lord '''James Hamilton, 2nd Duke of Abercorn''' (24 August 1838 – 3 January 1913)
#Lady Katherine Elizabeth Hamilton Edgcumbe (9 January 1840 – 3 September 1874)
#Lady Georgiana Susan Hamilton Turnour (7 July 1841 – 23 March 1913)
#Lord '''Claud John Hamilton''' (20 February 1843 – 26 January 1925)
#Rt. Hon. Lord Sir '''George Francis Hamilton''' (17 December 1845 – 22 September 1927)
#Lady Albertha Frances Anne Hamilton Spencer-Churchill (29 July 1847 – 7 January 1932)
#Lord Ronald Douglas Hamilton (17 March 1849 – DVP&lt;ref&gt;{{Cite journal|date=2020-07-27|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&amp;oldid=969822724|journal=Wikipedia|language=en}}&lt;/ref&gt; 6 November 1867)
#Lady Maud Evelyn Hamilton Petty-Fitzmaurice, the [[Social Victorians/People/Lansdowne | Marchioness of Lansdowne]] (17 December 1850 – 21 October 1932)&lt;ref name=&quot;:1&quot; /&gt;
#Lord Cosmo Hamilton (16 April 1853 – 16 April 1853)
#Lord '''Frederick Spencer Hamilton''' (13 October 1856 – 11 August 1928)
#Lord '''Ernest William Hamilton''' (5 September 1858 – 14 December 1939)


*Harriet Georgiana Louisa Hamilton Anson (6 July 1834 – 23 April 1913)&lt;ref name=&quot;:2&quot; /&gt;
*Thomas George Anson, 2nd Earl of Lichfield (15 August 1825 – 7 January 1892)
#Lady Evelyn Anson ( – 2 July 1895)
#Thomas Francis Anson, 3rd Earl of Lichfield (31 January 1856 – 29 July 1918)
#Hon. Sir George Augustus Anson (22 December 1857 – 25 May 1947)
#Major Hon. Henry James Anson (29 December 1858 – 26 February 1904)
#Lady Florence Beatrice Anson (1860 – 25 September 1946)
#Hon. Frederic William Anson (4 February 1862 – 2 April 1917)
#Hon. Claud Anson (11 January 1864 – 25 December 1947)
#Lady Beatrice Anson (1865 – 15 December 1919)
#Hon. Francis Anson (7 March 1867 – 13 April 1928)
#Lady Mary Maud Anson (1869 – 22 September 1961)
#Lady Edith Anson (1870 – 8 October 1932)
#Hon. William Anson (19 April 1872 – 22 June 1926)
#Hon. Alfred Anson (15 April 1876 – 25 March 1944)


*James Hamilton, 2nd Duke of Abercorn (24 August 1838 – 3 January 1913)&lt;ref name=&quot;:12&quot; /&gt;
*Maria Anna Curzon-Howe Hamilton (23 July 1848 – 10 May 1929)&lt;ref name=&quot;:3&quot; /&gt;
#James Albert Edward Hamilton, 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)
#Claud Penn Alexander Hamilton (18 October 1871 – 18 October 1871)
#Charlie Hamilton (10 April 1874 – 10 April 1874)
#'''Alexandra Phyllis Hamilton''' (23 January 1876 – 10 October 1918)
#Claud Francis Hamilton (25 October 1878 – 25 December 1878)
#Gladys Mary Hamilton Forward-Howard (10 December 1880 – 12 March 1917)
#Arthur John Hamilton (20 August 1883 – 6 November 1914)
#(unnamed son) Hamilton (31 October 1886 – 31 October 1886)
#Claud Nigel Hamilton (10 November 1889 – 22 August 1975)&lt;ref name=&quot;:4&quot; /&gt;


* '''James Albert Edward Hamilton''', Marquess of Hamilton and 3rd Duke of Abercorn (30 November 1869 – 12 September 1953)&lt;ref name=&quot;:13&quot; /&gt;
* Lady Rosalind Cecilia Caroline Bingham (26 February 1869 – 18 January 1958)&lt;ref name=&quot;:14&quot; /&gt;
*# Lady Mary Cecilia Rhodesia Hamilton (21 January 1896 – 5 September 1984)
*# Lady Cynthia Elinor Beatrix Hamilton (16 August 1897 – 4 December 1972)
*# Lady Katharine Hamilton (25 February 1900 – 28 April 1985)
*# James Edward Hamilton, 4th Duke of Abercorn (29 February 1904 – 4 June 1979)
*# Captain Lord Claud David Hamilton (13 February 1907 – 15 February 1968)


*Claud John Hamilton (20 February 1843 – 26 January 1925)&lt;ref name=&quot;:5&quot; /&gt;
*Carolina Chandos-Pole Hamilton (19 July 1857 – 21 September 1911)&lt;ref&gt;&quot;Carolina Chandos-Pole.&quot; {{Cite web|url=http://www.thepeerage.com/p11067.htm#i110663|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
#Colonel Gilbert Claud Hamilton (21 April 1879 – 30 March 1943)
#Ida Hamilton (23 July 1883 – November 1970)


*George Francis Hamilton (17 December 1845 – 22 September 1927)&lt;ref name=&quot;:6&quot; /&gt;
*Lady Maud Caroline Lascelles Hamilton (1846 – 14 April 1938)
#'''Ronald James Hamilton''' (26 September 1872 – 22 January 1958)
#Anthony George Hamilton (17 December 1874 – 11 July 1936)
#Robert Cecil Hamilton (31 January 1882 – 31 July 1947)


*Ernest William Hamilton (5 September 1858 – 14 December 1939)&lt;ref&gt;&quot;Lord Ernest William Hamilton.&quot; {{Cite web|url=http://www.thepeerage.com/p2107.htm#i21062|title=Person Page|website=www.thepeerage.com|access-date=2020-10-08}}&lt;/ref&gt;
*Pamela Campbell Hamilton ( – 11 May 1931)&lt;ref name=&quot;:7&quot; /&gt;
#Guy Ernest Frederick Hamilton (11 November 1894 – 23 November 1914)
#Mary Brenda Hamilton (28 March 1897 – 14 March 1985)
#Jean Barbara Hamilton (6 September 1898 – 2 November 1989)
#John George Peter Hamilton (15 October 1900 – 17 June 1967)


=== Earls of Wicklow ===

* Charles Hamilton (1772 – 29 September 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21387|title=Charles Hamilton. Person Page #2139|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Marianne '''Caroline Tighe''' ( – 29 July 1861)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p62375.htm#i623745|title=Marianne Caroline Tighe. Person Page #62375|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot; /&gt; – 13 March 1892)
*# Caroline Elizabeth Hamilton ( – 31 May 1909)
*# Mary Hamilton
*# Charles William Hamilton (1 April 1802 – 16 February 1880)
*# William Tighe Hamilton (31 March 1807 – )
*# Frederick John Henry Fownes Hamilton (27 July 1816 – 1893)


* Rev. Hon. Francis Howard (12 January 1797 – 16 February 1857)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21391|title=Rev. Hon. Francis Howard. Person Page #2140|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
* Frances Beresford ( – 17 November 1833)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3227.htm#i32266|title=Frances Beresford. Person Page #3227|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# William George Howard (25 April 1825 – 12 October 1864)
* '''Sarah Hamilton''' (1805&lt;ref name=&quot;:17&quot;&gt;{{Cite web|url=https://catalogue.nli.ie/Collection/vtls000572704|title=Tighe, Hamilton and Howard Papers,|date=1737|website=catalogue.nli.ie|language=English|access-date=2026-06-19}}&lt;/ref&gt; – 13 March 1892)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2141.htm#i21405|title=Sarah Hamilton. Person Page #2141|website=www.thepeerage.com|access-date=2026-06-19}}&lt;/ref&gt;
*# 4 unnamed daughters [per The Peerage; The NLI has 3 daughters]
*# Lady Alice Howard
*# Lady Louisa 'Loulie' Howard
*# Lady Caroline Howard (1836–1923)&lt;ref name=&quot;:17&quot; /&gt;
*# Charles Francis Arnold Howard, '''5th Earl of Wicklow''' (5 November 1839 – 20 June 1881)
*# Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)


* Cecil Ralph Howard, '''6th Earl of Wicklow''' (26 April 1842 – 24 July 1891)&lt;ref name=&quot;:18&quot; /&gt;
* Francesca Maria Chamberlayne ( – 1877)
*# Ralph Howard, 7th Earl of Wicklow (24 December 1877 – 11 October 1946)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2140.htm#i21394|title=Cecil Ralph Howard, 6th Earl of Wicklow. Person Page 2140.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
* Fanny Catherine Wingfield (c. 1860 – 3 February 1914)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p2139.htm#i21388|title=Fanny Catherine Wingfield. Person Page 2139.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;
*# Hon. Cecil Mervyn Malcolm Howard (18 November 1881 – 16 April 1882)
*# Hon. Hugh Melville Howard (28 March 1883 – 17 February 1919)
* Marcus Francis Beresford (26 December 1862 – 14 December 1896)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p3186.htm#i31858|title=Marcus Francis Beresford. Person Page #3186.|website=www.thepeerage.com|access-date=2026-06-28}}&lt;/ref&gt;

== Memoirs and Archives ==

# The Abercorn Papers: GB 0255 PRONI/D623 (found via https://iar.ie/archive/abercorn-papers). A descriptive list is available to search online at: http://www.proni.gov.uk/. The collection is arranged as follows: D623/A Correspondence D623/B Title deeds and leases D623/C Rentals, accounts and vouchers D623/D Maps, plans, surveys, inventories and valuations D623/E Photographs, illuminations, addresses and albums D623/F Material still at Baronscourt D623/G Miscellaneous
#Alexandra Phyllis Hamilton (#64 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]) attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her uncle Lord Frederick Spencer Hamilton (#84), the Marquess of Hamilton (#657), and a Mr. Ronald Hamilton (#105). Besides these, probably, a Mr. and Mrs. Hamilton also attended.

== Questions and Notes ==
#DVP = decessit vita patris, died while the father was still living
#Mr. Ronald Hamilton cannot be Frederick Hamilton's brother, who should be Lord Ronald Hamilton rather than Mr. Ronald Hamilton, and he died in 1867. He could be this Ronald Hamilton, who would be a Mr. Hamilton: http://www.thepeerage.com/p2163.htm#i21622. He was Lady Alexandra's cousin and nephew of the 1st Duke of Abercorn.
#A Mr. Hamilton is mentioned in the ''Gentlewoman'' article: &quot;Mr. Hamilton (Elizabethan costume), black velvet, trimmed gold.&quot;&lt;ref name=&quot;:15&quot; /&gt;{{rp|34, Col. 1c}} But a later reference in this same article to Mr. Ronald Hamilton matches the description in the ''Morning Post'' article, saying he wore black velvet with jet, rather than gold trim: &quot;'''Mr. Ronald Hamilton''' (gentleman of the Court of Queen Elizabeth), black velvet with jet.&quot;&lt;ref name=&quot;:15&quot; /&gt; (36, Col. 3b) I believe the other Mr. Hamilton is Mr. [[Social Victorians/People/Cole-Hamilton|Claud Cole-Hamilton]], particularly since Mrs. Hamilton was dressed as Amy Robsart and thus must be Lucy Charlewood Cole-Hamilton because of the description of her costume in the Album of photographs given to the Duchess of Devonshire later.
#Claud John Hamilton is probably who attended the social events, because the other Claud, of whatever generation either died too young or was born too late.
== Footnotes ==
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  </page>
  <page>
    <title>Social Victorians/Timeline/1870s</title>
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    <id>264241</id>
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      <parentid>2817951</parentid>
      <timestamp>2026-07-09T19:08:27Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <minor />
      <comment>/* Birmingham Tennis Court Club Ball */</comment>
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      <format>text/x-wiki</format>
      <text bytes="31258" sha1="7ay8vc5a38dcq714torrq7pchnzyxno" xml:space="preserve">==Time Line==
[[Social Victorians/Timeline/1840s|1840s]]  [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] 1870s [[Social Victorians/Timeline/1880s | 1880s]] [[Social Victorians/Timeline/1890s | 1890s]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==1870==
&quot;Until 1870 all of the money women earned belonged to their husbands, and until 1882 their property did too, even after a divorce or separation.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698 of 1203)

In 1870 Parliament debated and defeated the first bill for women's suffrage, but allowed &quot;women who owned property ... to stand for election to school boards.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698–699 of 1203)

&quot;The bulk of Irish farmers did not own their land, and instead leased it from landlords, the majority of whom lived in England. In 1870, only 3 percent of agricultural holdings were occupied by owners.&quot;&lt;ref name=&quot;:4&quot; /&gt; (742 of 1203)

Dante Gabriel Rossetti and Arthur Sullivan were at the same dinner party in 1870?

Another dinner party had as guests Charles Dickens, Dante Gabriel Rossetti, John Tenniel and George Du Maurier.

January
February
March
April
May
June
July
August
September
October
November
December

==1871==
Although Queen Victoria had opened Parliament for the first time in February 1866, when people saw her for the first time in years as her open carriage made its way, she was unpopular because it seemed she was not working. Gladstone was Prime Minister.&lt;blockquote&gt;Between 1871 and 1874, eighty-five Republican Clubs were founded in Britain, protesting, among other things, the &quot;expensiveness and uselessness of the monarchy&quot; and Bertie's &quot;immoral example.&quot;&lt;ref name=&quot;:4&quot;&gt;Baird, Julia. ''Victoria the Queen, an Intimate Biography of the Woman Who Ruled an Empire''. Random House, 2016. Apple Books: https://books.apple.com/us/book/victoria-the-queen/id953835024.&lt;/ref&gt; (617 of 1203)&lt;/blockquote&gt;&quot;The 1871 Royal Commission on the Contagious Diseases Acts ... declared there was no comparison to be made between prostitutes and their clients: 'With the one sex the offence is committed as a matter of gain, with the other it is an irregular indulgence of a natural impulse.'&quot;&lt;ref name=&quot;:4&quot; /&gt; (704 of 1203)

=== January ===
Germany is united under King William I of Prussia. Julia Baird says, &quot;At the same time, Italy captured and annexed the Papal States, which had been under the direct rule of the Pope since the 700s and had lost their protector in Napoleon III.&quot;&lt;ref name=&quot;:4&quot; /&gt; (646 of 1203)

==== 4 January 1871, Wednesday ====
&lt;blockquote&gt;INVITATION BALL.
&lt;p&gt;On Wednesday evening last Major Goodman and the Officers of the 5th Dragoon Guards gave an invitation ball, which was held in the Drapers’ Hall (kindly placed at their disposal by the Drapers’ Company). The following ladies and gentlemen were amongst those who received invitations The Marquis and Marchioness of Hertford; the Earl and Countess of Aylesford; Lady A. N. Finch, Lord Guernsey, and the Hon. Mr. Finch; Lord and Lady Leigh and Miss Leigh; Lord and Lady Henley and Miss Henley, Miss Elwes, Lord and Lady Wrottealey, Lord and Lady Manners; C. N. Newdegate, Esq., M.P.; Captain, Mrs., and Miss Adams; E. Petre, Esq., and Lady Gwendoline Petre; J. Beech, Esq., Mrs. and Miss Beech, and Mr. Beech, jun.; Mr. and Mrs. Turner; Mr. and Mrs. Fetherstone Dilke, Mrs. and the Misses Fetherstone, Mr. Fetherstone, and Mr. Beaumont Fetherstone; Mr. and Mrs. P. A. Muntz; Captain and Mrs. Boultbee, of Knowle; Mr. C. M. Caldecott, Mrs. Caldecott, and the Misses Caldecott; the Rev. A. Fanshawe and Mrs. Fanshawe; Captain and Mrs. Battine; the Rev. S. C. Spencer Smith; the Rev. R. H. Baynes, M.A., vicar of St. Michael’s; the Rev. H. T. Harris, (Christ Church); General and Mr. Richmond Jones; Colonel F. Chaplin, and the Officers of the 4th Dragoon Guards, stationed at Northampton; Captain Thornelow, and the Officers of the Royal Artillery, at Weedon; the officers of the 4th Royal Regiment at Weedon; Mr. and Mrs. E. Wood; Mr. and Mrs. Herbert Wood; the Colonel and officers of the First Warwickshire Militia; Mrs. and Miss Alston, and Mr. Alston, jun., of Elmdon; Mr. and Mrs. F. Paget; Mr. and Mrs. Gulson; Captain Thomson; Captain and Mrs. Raleigh King; Mrs. Phillipson; Lord and Lady Mountgarret; the Honourable Miss Butler; Mr. and Mrs. Courtenay Lord; the Hon. Mrs. Twistleton; Mr. and the Misses Conant; Captain and Mrs. J. Marsland; Major and Mrs. Edlman; Mr. and Mrs. Astley; Mr. T. Lant, Mr. R. Lant and Mr. J. Lant, Mrs. and Miss Lant; Mr. W. T. Cavendish; Mr. and Mrs. A. Rotherham; the Marquis of Ormonde, of the first Life Guards; the Earl of Calludon, of the First Life Guards; Mrs. and the Misses Hobson; Mr P. Hobson, and Mrs. Hobson; Mr. and Mrs. Soames; Mr. and Mrs. Adderley, Sir John Rae Reid; Capt. and Mrs. Townshend, of Caldecote Hall; Lieut.-Colonel Swinfen and the Officers of the 5th Dragoon Guards stationed at Leeds; Capt. Marsden and the Officers of the 5th Dragoon Guards stationed at Birmingham; Colonel, Mrs., and Miss Bourne; Mr. and Mrs. Wyley Lord; Captain and Mrs. Thursby; Mr. and Mrs Morrice; Lieut.-Colonel Wirgman; Mr. and Mrs. J. Rotherham; [[Social Victorians/People/Abercorn|Lady Caroline Howard]]; Mr. and Mrs. Rotherham; Mr and Mrs John Sankey and the Misses Sankey; Mrs. and the Misses Murphy; Mr. Bibby (4th Hussars), Captain Gist (7th Hussars), Mr. Gregg (8th Hussars), Mr. Hamilton (7th Dragoon Guards), Colonel Rattray, Mr and Mrs. R. Boyd, &amp;c, &amp;c.&lt;/p&gt;
&lt;p&gt;The string band of the 5th Dragoon Guards, under the direction of Mr. Sidney Jones, performed the following selection of music:— Quadrille, Barbe Bleue; Valse, Marian; Galop, Bonderbryllup; Lancers, Knight of St. Patrick; Valse, Hydropaten; Galop, Flick and Flock; Quadrille, Princess of Trebizonde; Valse, the Belle of the Ball; Galop, the Fox Hunters; Valse, the Dragoon Guards; Lancers, the Gaiety; Valse, the Beautiful Danube; Valse, Wiener Kinder; Quadrille, the Fest; Galop, the Village Rose; Valse, the Geraldine; Lancers, Merry Tunes; Galop, Barbe Bleue; Valse, Various; Galop, Glorioso.&lt;ref&gt;&quot;Invitation Ball.&quot; ''Coventry Standard'' 6 January 1871, Friday: 4 [of 4], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000683/18710106/100/0004. Same print title, n.p.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;

=== February ===

==== Birmingham Tennis Court Club Ball ====
1871 February 17, Friday, the &quot;bachelors of the Tennis Court Club&quot; hosted a ball in Birmingham:&lt;blockquote&gt;LEAMINGTON.&lt;p&gt;

B&lt;small&gt;ACHELORS'&lt;/small&gt; B&lt;small&gt;ALL&lt;/small&gt;.&lt;p&gt;— Last night the bachelors of the Tennis Court Club gave a grand ball at the Royal Assembly Rooms, Regent Street. The ball was one of the most brilliant of the season, nearly four hundred of the ''élite'' of the town and neighbourhood having accepted the invitation of the bachelors. The ballroom was specially fitted up for the occasion, and a splendid supper was served in the adjoining rooms, where refreshments were also provided. Coote and Tiney's band was specially engaged for the occasion, and played a selection of the newest and most popular dance music. Amongst the distinguished guests present were — The High Sheriff and Mrs. J. T. Arkwright, Lady Arbuthnott, Lord and Lady Conyers, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mountgarret and the Hon. Miss Butler, Sir John and Lady Blois, Sir Thomas Biddulph, the Hon. Miss Somerville, Sir William and Lady Fairfax, the Hon. Charles L. Butler, Rev. Sir John Rae, General and Mrs. Richmond Jones, Major Eldman, Major and Mrs. James Ashton, Major and Mrs. Boothby, Colonel Ruttie, Colonel Duberly, Colonel and Mrs. Machen, Colonel Rattray, Capt. and Mrs. Kennedy, Capt. W. J. Hall, Capt. Hodge, Capt. and Mrs. Morgan, Capt. and Mrs. Pearse, Capt. Roberts, Capt. Story, Mr. and Mrs. Featherstone Dilke (Maxstoke Castle) and Miss Dixie, Mr. C. M., Miss, and Miss M. A. Caldecott (Holbrooke Grange), Mr. and Mrs. J. Dugdale (Wroxhall Abbey), Mr. E. Greaves, M.P., Mr. and Mrs. C. L. Adderley (Hams Hall), and Capt. and Mrs. Hatherall. Several of the officers from the dragoons and artillery at Coventry and Birmingham were also present. The bachelors who gave the ball were twenty-eight in number.&lt;ref&gt;&quot;Leamington.&quot; &quot;District News.&quot; ''Birmingham Morning News'' 18 February 1871, Saturday: 7 [of 8, print and digital], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005826/18710218/114/0007. Print and digital title are the same.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;

=== March ===

=== April ===

==== 18 April 1871 ====
&lt;blockquote&gt;Karl Marx “was commissioned by the General Council of the International to write a pamphlet about the Paris [377–378] Commune.&quot;&lt;ref name=&quot;:3&quot;&gt;Smee, Sebastian. ''Paris in Ruins: Love, War, and the Birth of Impressionism''. W. W. Norton, 2024.&lt;/ref&gt;{{rp|377–378 of 667}}&lt;/blockquote&gt;

===May===

==== 9 May 1871, Tuesday, Queen's Drawing-Room ====
&lt;blockquote&gt;THE QUEEN'S DRAWING-ROOM.

The Queen held a Drawing-room at Buckingham Palace on Tuesday afternoon. The Priuce of Wales, Prince Arthur, Prince Leopold, and Princess Beatrice were present. Her Majesty, accompanied by the Prince of Wales and the other members of the royal family, entered the Throne Room shortly after three o'clock. The Queen wore a black moire antique dress with a train, long white tulle veil with a coronet of diamonds. Her Majesty also wore a necklace of diamonds and amethysts, the Riband and Star of the Order of the Garter, the Orders of Victoria and Albert and Louise of Prussia, and the Saxe Coburg and Gotha Family Order. Princess Beatrice wore a dress of white tulle over a rich white silk petticoat looped up with lilies of the valley and apple blossom; ornaments — pearls and diamonds.

The presentations to Her Majesty were about 280 in number, and included the following:— Mrs Atlay, by the Countess Grey; Miss Backhouse, by her mother, Mrs Backhouse; Miss Charlesworth, by her aunt, Frances Lady Hawke; Miss Backhouse Fox, by her aunt, Mrs Backhouse; [[Social Victorians/People/Abercorn|Lady Caroline Howard]], by her mother, [[Social Victorians/People/Abercorn|the Hon. Mrs Howard]]; the Hon. Gwendoline Fitz-Alan Howard, by the Duchess of Sutherland; [[Social Victorians/People/Abercorn|Lady Alice Howard]], by her mother, Hon. Mrs Howard; [[Social Victorians/People/Abercorn|Lady Louisa Howard]], by her mother, Hon. Mrs Howard; Miss Howard (of Corby), by the Hon. Mrs Philip Stourton; Miss Agnes Howard (of Corby), by the Hon. Mrs Philip Stourton; Sir Henry Ingilby, Bart., by Earl Russell; Mrs Frank Lascelles, by Lady Edward Cavendish; Mrs Gerald Liddell, marriage, by the Countess of Normanby.&lt;ref&gt;&quot;Court and Official News.&quot; ''Yorkshire Post and Leeds Intelligencer'' 11 May 1871, Thursday: 3 [of 4], Col. 4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000686/18710511/074/0003. Same print title and p.n.&lt;/ref&gt;&lt;/blockquote&gt;'''24 May 1871, Wednesday''': Derby Day. Baron Rothschild's Favonius won. The Prince of Wales attended.

June
July
August
September
===October===
'''October 1871'''&lt;blockquote&gt;At Londesborough Lodge near Scarborough, where Lady Londesborough gave a royal house party in October 1871, not only [ 41/42 ] were the bathrooms few but the drains seeped into the drinking water. Several guests, including the Prince [of Wales] and his groom and Lord Chesterfield, contracted typhoid fever. When Chesterfield and the groom died, the doctors abandoned hope for the Prince.&lt;ref name=&quot;:1&quot;&gt;Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973. Print.&lt;/ref&gt;{{rp|41–42}}&lt;/blockquote&gt;

The Prince of Wales recovered on 14 December 1871.

November
December

==1872==
January
February
March
April
===May===
'''29 May 1872, Wednesday''': Derby Day

June
July
===August===
'''August 1872''': The &quot;dance on the cruiser Ariadne&quot; probably occurred in August 1872:&lt;blockquote&gt;When his [the Prince of Wales'] brother, the Duke of Edinburgh, married the attractive Grand Duchess Marie, daughter of Tsar Alexander II of Russia, her family made a fuss because she was not granted precedence above the Princess of Wales. Albert Edward soothed ruffled feelings by inviting the Tsarevitch and his wife Marie Feodorovna (who was Alexandra's sister) to stay for two months and be entertained at Cowes. ...&lt;p&gt;&lt;/p&gt;

... At the dance on the cruiser Ariadne which the Prince gave in honour of the Tsarevitch and his Grand Duchess,&quot; Lord Randolph Churchill met the 19-year-old &quot;Miss Jennie Jerome of New York.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|42–43}}&lt;/blockquote&gt;

September
October
November
December

==1873==

=== January ===

==== 13 January 1873, Monday ====

==== Ball at the Chief Secretary's Lodge ====
On Tuesday, 14 January 1873, the Dublin Evening Telegraph reported that the Marquis of Hartington's ball had taken place the evening before.&lt;blockquote&gt;The Marquis of Hartington gave a ball last evening at the Chief Secretary's Lodge, to their Excellencies the Lord Lieutenant and the Countess Spencer, who were accompanied by the Dowager Countess Spencer, the Ladies Sarah and Victoria Spencer and the Hon Robert Spencer, Lord and Lady Charles Bruce, and Major Stirling, A D C.&lt;p&gt;

The following had the honour of receiving invitations to meet their Excellencies — The Duke of Leinster, the Marquis and Marchioness of Kildare, the Ladies Fitzgerald, the Marquis and Marchioness of Drogheda, the Earl and Countess of Listowel, Lord and Lady Edward Cavendish, the Earl of Charleville, the Lord Chancellor and Lady O'Hagan, Viscount, Viscountess, the Hon Misses, and Hon Henry Monck; the Archbishop of Dublin, the Hon Mrs and the Misses Trench; Lord Talbot de Malahide and the Hon Francis Talbot, Lord and Lady Sandhurst and Captain Bang, A D C; Lady Cloncurry, Hon Emily and Hon Mary Lawless, Viscount, Viscountess, Hon Georgiana, and Hon Beatrice [de?] Vesci; Lord and Lady Kilmaize [?], Hon Gertrude [?] Browze, Lord and Lady Ventry, Hon Norah Westenra, Lord and Lady Athlumney, Lord, Lady, and Hon D Plunket, M P; Viscountess and the Hon. Miss Netterivlle, Capt the Hon Mrs Vesey, Captain and Lady Julia Follett, Sir Arthur and Lady Olive Guiness and the Ladies White, the Hon H W L Corry, Lord and Lady and the Hon Miss O'Neill, Viscount Hawarden, the Hon Florence Maude, the Hon. Clementina Maude, the Hon Jenico and Mrs Preston, the Hon Henry Leeson, Colonel and the Hon Mrs Caulfield, Mr and the Hon Mrs Robert Hobart, Captain, Lady Mary and Miss Lindsay; Mr Ion [?] Trent Hamilton, M P; Mr Bagwell; the Hon Mrs and the Misses Bagwell, and Mr Bagwell; Colonel the Hon L and Mrs Curzon Smyth, Mr, Lady Margaret, and the Misses Stronge [?]; Mr and the Hon Mrs O'Hagan, Hon Charles Bourke, Hon Mrs Alfred and Lady Kathleen Bury, [[Social Victorians/People/Abercorn|Hon Mrs, Lady Alice, and Lady Louisa Howard]]; Captain, the Hon Mrs, and Miss Donaldson; Dr and Miss Bans, Mrs Grattan Bellew, Sir Edward and Miss Borough, Mr Arthur Cane, Sir Dominic, Lady, and Miss Corrigan; Mr Corrigan, Mr and Mrs Gustavus Cornwall and Miss Cornwall, Mr D'Arcy, M P, and Mrs D'Arcy; Mr Baron Dowse [?], and Mrs and Miss Dowse, Mr Baron Deasy and Mrs Deasy, Dr, Mrs, and Miss de Ricci; Dr and Miss Hatchell, Sir George and Lady Hudson, Mr, Mrs, and the Misses Huband; Mr Arthur Huband, Miss Caroline Huband, Mr and Mrs Arthur Hume, Dr Hughes, Mr Henry Jephsen and Miss Jephsen, Mr Kearney and the Misses Kearney, Captain Kearney, A D C; Captain Lascelles, A D C; Mr, Mrs, and Miss Kirwan; Mr Justice Lawson and Mrs Lawson, Mr and Mrs W Le Fanu, Mr, Mrs, and Miss Lentaigne; Sir George L'Estrange and the Misses L'Estrange, the Lord and Lady Mayoress, and the Misses Mackey; the Lord Chief Justice Monahan, Mrs and Miss Monahan; Sir J, Lady, and Miss Power; Mr John Talbot Power, M P; Col, Mrs, and Miss Radcliffe; the Master of the Rolls, Mrs and Miss Sullivan; Capt and Mrs Moorsom, A D C; General Sir Thomas and Lady Steel, Captain and Mrs Brownrigg, A D C, Mr Granville Milner, Capt, Mrs and Miss Talbot, Colonel, Mrs, and the Misses White; Sir John Stewart Wood, Lady and the Misses Wood; Mrs and the Misses Williams, Mr Justice Fitzgerald and the Hon Mrs Fitzgerald, Mr Fitzgerald, Mr Justice Barry and Mrs Barry, Mr Sergeant Sherlock, M P, Mrs and Miss Sherlock; Mr Sheriock, the Right Hon W H Conan, M P, and Mrs Cogan; Mr Justice Keogh and Mrs Keogh, Mr Keogh, Capt Keogh, R N; Lord Chief Baron and Miss Pigott, Dr, Mrs, and Miss Nugent; General Wardlaw, Colonel M'Kerlie, Mr Sergeant and Mrs and Miss Armstrong; Col, Mrs, and the Misses Maude; Col, Mrs, and Miss Hillier; Mr Heron, M P; Mr and Mrs Watters, Col and Mrs Wynyard, Dr and the Misses Kennedy, the Attorney General and Mrs Palles, the Solicitor General and Mrs Law, Col, Mrs, and Miss Lake; Lady and the Misses Butler, Mr Butler, Col and Mrs Colthurst Vesey, and Miss Walton; Mr, Lady Fanny and Miss Lambert; Mr E C Guinness, Mr and Mrs MMorer O'Ferrall, Mr and Mrs Leonard Morrogh, Sir Bernard and Lady Burke, Mr G and Mrs G Brooke and Miss Brooke, Mr and Mrs Roe, Mr Vance, M P, Mrs and Miss Vance; Col and Mrs Primrose, Lieut Col Ferdall [?], Col and Mrs Goodlake and Miss Alexander, Mr Alison, Mr, Mrs, and Miss Barton, Mr Justice Flanagan, Mrs and Miss Flanagan, Mer J. N. Lentaigne, Mr Johnson, Captain Harrison, Mr, Mrs, and the Misses Maturin; Mr Justice Morris and Mrs Morris, Mr and Mrs Mazlere [?] Brady, Major, Mrs, and Miss Wilkinson; Mr, Mrs, and Miss Donnelly; Mr and Mrs Cruise, Mrs Power, Mr Braon Fitzgerald and Mrs Fitzgerald, Mr Henry Yates Thompson, Mr Courtenay Boyle, Colonel Forster, Mr, Mrs, and Miss Taylor, Mr Bland and Mrs Godfrey Bland, Mr and Miss Dillon, Mr and Mrs Wallace, Mr M'Kenna, Mr Cullinane, Mr Armstrong, Mr C E [?] Dobbin, Mr J A Blake, Major and Mrs Papillon, Capt and Mrs Keane, Mr E Pretty, Mr, Mrs John L O Ferrall and Miss O'Ferrall, Mrs and Miss Walsh, Mr and Mrs R Howard Brook, Mrs and Miss Brook, Mrs and the Misses Blake, Mr and Mrs J Warren, Sir John Gray, M P, Lady, and Miss Gray; Colonel and Mrs Frank Chaplin, Mr, Mrs, and Miss Hemphill; Sir R, Lady and Miss Kane, Mrs and Miss Courtenay, Mr Arthur Courtenay, Mr G Courtenay, Mr E Hardtop, A D C; Mr Bellew, Dr and Mrs Nedley, Dr and Mrs Newell, Mr and Mrs Freeman, Mr and Mrs Geale, Captain Hutten, A D C; Mr and Mrs Adair and Miss Wadsworth, Captain and Mrs J M Benthall, Sir R, Lady, and the Misses M'Causlend [?]; Mr, Mrs, and the Misses Newell Barron; Mr Hawkins, Colonel Goodlake and the Officers of the Coldstream Guards; Captain Spain, R N, and the Officers (4) of her Majesty's ship Vanguard; Colonel Radcliffe and Officers (4), Royal Artillery; Colonel Spade and Officers (4) 1st King's Dragoon Guards; Colonel Ainslie and Officers (4), 1st Royal Dragoons; Colonel Thompson and Officers (4), 14th Hussars; Colonel Ross and Officers (4), 4th Battalion Rifle Brigade; Colonel Hawkins and Officers (4), Royal Engineers; Colonel Gloster and Officers (4), 97th Regiment; Lieutenant-Colonel Maunsell and Officers (4), 13th Regiment.&lt;ref&gt;&quot;Fashionable.&quot; ''Dublin Evening Telegraph'' 14 January 1873, Tuesday: 4 [of 4], Col. 7a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002093/18730114/044/0004. Print title ''The Evening Telegraph'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;February
March
April

===May===
'''28 May 1873, Wednesday''': Derby Day

June
July
August
September
October
November
December

==1874==
January
February
March
April
===May===

==== 1874 May, Early ====
&lt;blockquote&gt;As monarchists’ hopes flared, the Catholic Church, too, enjoyed a conspicuous revival. The National Assembly approved a design for a new basilica for Paris. Intended as an act of collective atonement, Sacré-Coeur was to perch atop Montmartre, immediately above where Nadar’s balloons had been launched and where the radicals’ insurrection had broken out. Excavations began in early May 1874 ....

But the focus of the penance the basilica was intended to embody gradually shifted from the moral decline of French society in general to the despicable excesses of the Commune. In 1872 Archbishop Darboy’s successor claimed to have had a vision as he climbed the Butte Montmartre. The clouds dispersed, and he realized that it was there, “where the martyrs” were (he meant the murdered generals Lecomte and Clément-Thomas), that a new church should be built. And when the Assembly voted to proceed with the construction, legislators specified that its purpose was to “expiate the crimes of the Commune.”&lt;ref name=&quot;:3&quot; /&gt; (464 of 667)&lt;/blockquote&gt;

===June===
'''3 June 1874, Wednesday''': Derby Day

June
July
August
September
October
November
===December===
'''8 December 1874, Tuesday''': &quot;CHATSWORTH, Tuesday, December 8th, 1874. — We are come to the last slide of the Chatsworth magic lantern: the Duke of Cambridge and his equerry, a funny little man called Tyrwhitt, of no particular age, in a grey wig; Lord Carlingford and Ly. Waldegrave, the Spencers, Mr. Leveson, Cavendish.&quot;&lt;ref&gt;{{Cite web|url=http://ladylucycavendish.blogspot.com/2010/12/08dec1874-chatsworth-magic-lantern.html|title=Lady Lucy Cavendish: 08Dec1874, The Chatsworth Magic Lantern|last=H|first=Denise|date=2010-12-04|website=Lady Lucy Cavendish|access-date=2025-06-18}}&lt;/ref&gt;

==1875==
Disraeli's progressive legislation for labor rights:&lt;blockquote&gt;In 1875, he passed a series of enlightened acts protecting labor rights, arguing they were as important as property rights. Two of the laws ensured that workers would have the same recourse as employers when contracts were breached, and made peaceful picketing legal, protecting unions from charges of conspiracy.&lt;ref name=&quot;:4&quot; /&gt; (578 of 1203)&lt;/blockquote&gt;After women who owned property were allowed by Parliament to stand for local school-board elections in 1870, &quot;Elizabeth Garrett Anderson, the first woman to qualify as a doctor in Britain — in 1865 — stood and was elected to her local board five years later.&quot;&lt;ref name=&quot;:4&quot; /&gt; (199 of 1203)

The relationship between Swinburne and Lord Houghton:&lt;blockquote&gt;...not all Lord Houghton's children appreciated the catholicity of &quot;Papa's&quot; taste in friends: &quot;Swinburne (in a very excited state) came in in the evening,&quot; wrote Florence Milnes to her brother in 1875: &quot;He is madder than ever, to my astonishment he flopped down on one knee in front of me, &amp; announced that my hair had grown darker. This was rather embarrassing, and he is also so deaf now, which does not make it easier to talk to him.&quot;&lt;ref name=&quot;:2&quot;&gt;Pope-Hennessy Lord Crewe.&lt;/ref&gt;{{rp|5}}&lt;/blockquote&gt;

January
February
March
April
===May===
'''26 May 1875, Wednesday''': Derby Day. The Prince and Princess of Wales attended, as did a number of others of the royal family, including Princess Louise and Lorne.

June
July
===August===
'''August through October 1875''' Richard Monckton Milnes (Lord Houghton) and son Robert Milnes toured the U.S. and Canada:&lt;blockquote&gt;They set off in the steamer s.s Sarmatian from Liverpool in August 1875, stopping at Ireland to pick up the usual load of emigrants bound for the U.S.A. The most interesting among the passengers was 'Mr. Butler, author of Erewhon, who is very amusing and clever though infidel,' but, although he played whist with Samuel Butler, the young man was far more interested in the Eustace Smiths (parents of his friend W. H. Smith), and in a Canadian family named Macpherson, the youngest of whose two daughters, the dark-eyed Isobel, caught his fancy: he saw them afterwards in Toronto, and when they parted she gave him two larger than carte-de-visite photographs of herself, he gave her a smaller one of himself together with the inevitable volume of his father's verse.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|10}}&lt;/blockquote&gt;September
October
November
December

==1876==
Disraeli pushed through the Cruelty to Animals Act in order to please Queen Victoria. This act &quot;forced researchers to demonstrate that any experiments with animals involving pain were absolutely necessary, and ensured they would be anesthetized if so.&quot;&lt;ref name=&quot;:4&quot; /&gt; (679 of 1203)

January
February
March
April
===May===
'''11 May 1876''': In the midst of the Aylesford scandal, the Prince of Wales returned from a journey to Egypt and India, etc.:&lt;blockquote&gt;However harassed and exhausted, the Prince and Princess of Wales would put up a good show. Within an hour of their arrival home they set forth to attend a gala performance at Covent Garden Opera House. It was a brave decision to face the public and allow an immediate opportunity for demonstration. The Prince and Princess were rewarded when the audience rose to its feet to give them a standing ovation before the start of every act, as well as at the end, of Verdi's Ballo in Maschera.&lt;ref name=&quot;:1&quot; /&gt;{{rp|63}}&lt;/blockquote&gt;

'''27 May 1877''': Lily Langtry:&lt;blockquote&gt;Her big moment on May 27, 1877, when Sir Allen Young, the arctic explorer, invited her to late supper in his house, where it had been arranged that the Prince of Wales should meet her after the opera. The result was all that could have been expected. Mrs. Langtry became the Prince's first openly recognised mistress.&lt;ref name=&quot;:1&quot; /&gt;{{rp|69}}&lt;/blockquote&gt;'''31 May 1877, Wednesday''': Derby Day. The Prince and Princess of Wales did not attend, as he was ill.

June
July
August
September
October
November
December

==1877==
&quot;In 1877, unemployment was 4.7 percent; by 1879, it had risen to 11.4 percent.&quot;&lt;ref name=&quot;:4&quot; /&gt; (690 of 1203)

January
February
March
April
===May===
'''30 May 1877, Wednesday''': Derby Day.

June
July
August
September
October
November
===December===
'''15 December 1877'''&lt;blockquote&gt;On Dec. 15, 1877, the Queen honoured Lord Beaconsfield, the Premier, with a visit at Hughenden Manor. Her Majesty, accompanied by Princess Beatrice and attended by General Ponsonby and the Marchioness of Ely, left Windsor at 12.40 and proceeded by special train to High Wycombe, which was reached at 1.15. The Premier received the Queen at the station. A lofty triumphal arch spanned the entrance to the station-yard, and beneath this the royal party drove into the gaily decorated little town. The reception along the route was of the heartiest, and the drive of two miles to Hughenden was one long triumph. Lord Beaconsfield, who had preceded the party, welcomed the Queen at his own door. Lunch was served, and her Majesty remained about two hours. Before leaving she planted a memorial tree.&lt;ref&gt;&quot;The Queen's Glorious Reign.&quot; ''Illustrated London News'' (London, England), Saturday, May 27, 1899; pp. 757–765?; Issue 3136. Queen's Glorious Reign [Supplement]: 762?&lt;/ref&gt;&lt;/blockquote&gt;

==1878==
January
February
March
April
May
===June===
'''5 June 1878, Wednesday''': Derby Day.

July
August
September
October
===November===
'''8 November 1878''': from the journal of George, Duke of Cambridge:&lt;blockquote&gt;''November'' 8. — Gave farewell diner to the Lornes; Louise and Lorne, Augusta, Mary and Francis, Arthur, Leopold, Gleichens, J. Macdonald and self, and played at Nap afterwards. It was a good and nice little dinner.&quot;&lt;ref&gt;Sheppard, Edgar, Ed. ''George, Duke of Cambridge: A Memoir of His Private Life, Based on the Journals and Correspondence of His Royal Highness''. Vol. 2, 1871–1904. New York: Longmans, Green, 1906. http://books.google.com/books?id=dFoMAAAAYAAJ.&lt;/ref&gt;&lt;/blockquote&gt;December

==1879==
===January===
'''12 January 1879'''&lt;blockquote&gt;On 12 January 1879 Robert Milnes came of age, an event celebrated at Fryston by a tenants' ball.&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}}&lt;/blockquote&gt;

'''28 January 1879''': Brett &quot;Harte kicked off his tour at the Crystal Palace in Sydenham on January 28, 1879.&quot;&lt;ref&gt;Nissen, Alex. ''Brett Harte: Prince and Pauper''. Jackson, MS: University Press of Mississippi, 2000.&lt;/ref&gt;{{rp|174}}

February
March

===April===
'''Early April 1879''' or so, probably, Bret Harte got &quot;an invitation to dine the same evening with Arthur Sullivan and the Prince of Wales&quot; as a dinner in Birmingham where Harte met T. Edgar Pemberton.&lt;ref&gt;Scharnhorst, Gary. ''Bret Harte: Opening the American Literary West''. Norman, OK: Univ. of Oklahoma Press, 2000.&lt;/ref&gt;{{rp|152}}
===May===
'''28 May 1879, Wednesday''': Derby Day; the Prince and Princess of Wales attended.

===June===
'''June 1879''', Robert Milnes became engaged to &quot;Sibyl Marcia, a daughter of a North-country baronet, Sir Frederick Graham of Netherby.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}} Parties must have followed.

July
August
September
October
November
===December===
'''28 December 1879''': The Tay Bridge Disaster: The Tay Bridge collapsed with a train on it. The weather was very bad, with gale-force winds and rain.

The ''Times'' reported that the average high temperature for the week ending December 31, 1879, was 53° F. and the low was 20° F.

In his column &quot;What the World Says&quot; in the 21 January 1880 World, Edmund Yates writes the following:&lt;blockquote&gt;How am I to describe better the magnificence of the Earl and Countess of Rosslyn’s ball at Euston Lodge last month, than by calling attention to the fact that M. Carlo, the eminent Knightsbridge coiffeur, arrived early in the day to crimp and powder the lacqueys? My informant adds, however, that the curled darlings were rather the worse for the festivities towards night. Was it not enough to turn their heads in every sense of the word?&lt;ref name=&quot;:0&quot;&gt;Edmund Yates, &quot;What the World Says,&quot; ''The World: A Journal for Men and Women''.&lt;/ref&gt;{{rp|21 Jan. 1880, p. 8, col. b.}}&lt;/blockquote&gt;

'''31 December 1879''': Edmund Yates, editor of The World: A Journal for Men and Women, in his column &quot;What the World Says,&quot; describes a private viewing at the Grosvenor Gallery:&lt;blockquote&gt;The private view at the Grosvenor on the last day of the year gave people something to do on a desperately wet afternoon. The artistic dresses were perhaps in greater force than ever; indeed the faces and the hair and the attitudes pursued me to my bed, and gave me many a nightmare. I suppose the plain woman of all time has had the ambition to be looked at: centuries of failure have at last been crowned with a real success. Besides the Cimabue Browns there was an interesting menagerie of real lions, artistic, literary, and clerical. The artists were numerous, and their host and hostess seemed to enjoy themselves very thoroughly.

Frequenters of the picture private views have a new sensation this winter. Last season they mobbed beauty: now hideously-attired unkempt dowdiness provokes the stare. The prize for the new style seems generally awarded to a rhubarb coloured flannel Ulster and a cart-wheel beaver hat, which pervaded both the private views last week. [2 private views last week, one at the Grosvenor]&lt;ref name=&quot;:0&quot; /&gt;{{rp|7 Jan. 1880, p. 9}}&lt;/blockquote&gt;

The official premiere of ''The Pirates of Penzance'' occurred in New York City on 31 December 1879 at the Fifth Avenue Theatre, to establish international copyright. Gilbert and Sullivan were there with the cast. The performance was a social event: attending were Mrs. Vanderbilt and Mrs. Astor.

==Works Cited==
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    </revision>
    <revision>
      <id>2818020</id>
      <parentid>2818010</parentid>
      <timestamp>2026-07-09T21:45:43Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <comment>/* February */</comment>
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      <format>text/x-wiki</format>
      <text bytes="34314" sha1="p9ktu0xopkci9sd2es8avy6c6s88otp" xml:space="preserve">==Time Line==
[[Social Victorians/Timeline/1840s|1840s]]  [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] 1870s [[Social Victorians/Timeline/1880s | 1880s]] [[Social Victorians/Timeline/1890s | 1890s]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==1870==
&quot;Until 1870 all of the money women earned belonged to their husbands, and until 1882 their property did too, even after a divorce or separation.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698 of 1203)

In 1870 Parliament debated and defeated the first bill for women's suffrage, but allowed &quot;women who owned property ... to stand for election to school boards.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698–699 of 1203)

&quot;The bulk of Irish farmers did not own their land, and instead leased it from landlords, the majority of whom lived in England. In 1870, only 3 percent of agricultural holdings were occupied by owners.&quot;&lt;ref name=&quot;:4&quot; /&gt; (742 of 1203)

Dante Gabriel Rossetti and Arthur Sullivan were at the same dinner party in 1870?

Another dinner party had as guests Charles Dickens, Dante Gabriel Rossetti, John Tenniel and George Du Maurier.

January
February
March
April
May
June
July
August
September
October
November
December

==1871==
Although Queen Victoria had opened Parliament for the first time in February 1866, when people saw her for the first time in years as her open carriage made its way, she was unpopular because it seemed she was not working. Gladstone was Prime Minister.&lt;blockquote&gt;Between 1871 and 1874, eighty-five Republican Clubs were founded in Britain, protesting, among other things, the &quot;expensiveness and uselessness of the monarchy&quot; and Bertie's &quot;immoral example.&quot;&lt;ref name=&quot;:4&quot;&gt;Baird, Julia. ''Victoria the Queen, an Intimate Biography of the Woman Who Ruled an Empire''. Random House, 2016. Apple Books: https://books.apple.com/us/book/victoria-the-queen/id953835024.&lt;/ref&gt; (617 of 1203)&lt;/blockquote&gt;&quot;The 1871 Royal Commission on the Contagious Diseases Acts ... declared there was no comparison to be made between prostitutes and their clients: 'With the one sex the offence is committed as a matter of gain, with the other it is an irregular indulgence of a natural impulse.'&quot;&lt;ref name=&quot;:4&quot; /&gt; (704 of 1203)

=== January ===
Germany is united under King William I of Prussia. Julia Baird says, &quot;At the same time, Italy captured and annexed the Papal States, which had been under the direct rule of the Pope since the 700s and had lost their protector in Napoleon III.&quot;&lt;ref name=&quot;:4&quot; /&gt; (646 of 1203)

==== 4 January 1871, Wednesday ====
&lt;blockquote&gt;INVITATION BALL.
&lt;p&gt;On Wednesday evening last Major Goodman and the Officers of the 5th Dragoon Guards gave an invitation ball, which was held in the Drapers’ Hall (kindly placed at their disposal by the Drapers’ Company). The following ladies and gentlemen were amongst those who received invitations The Marquis and Marchioness of Hertford; the Earl and Countess of Aylesford; Lady A. N. Finch, Lord Guernsey, and the Hon. Mr. Finch; Lord and Lady Leigh and Miss Leigh; Lord and Lady Henley and Miss Henley, Miss Elwes, Lord and Lady Wrottealey, Lord and Lady Manners; C. N. Newdegate, Esq., M.P.; Captain, Mrs., and Miss Adams; E. Petre, Esq., and Lady Gwendoline Petre; J. Beech, Esq., Mrs. and Miss Beech, and Mr. Beech, jun.; Mr. and Mrs. Turner; Mr. and Mrs. Fetherstone Dilke, Mrs. and the Misses Fetherstone, Mr. Fetherstone, and Mr. Beaumont Fetherstone; Mr. and Mrs. P. A. Muntz; Captain and Mrs. Boultbee, of Knowle; Mr. C. M. Caldecott, Mrs. Caldecott, and the Misses Caldecott; the Rev. A. Fanshawe and Mrs. Fanshawe; Captain and Mrs. Battine; the Rev. S. C. Spencer Smith; the Rev. R. H. Baynes, M.A., vicar of St. Michael’s; the Rev. H. T. Harris, (Christ Church); General and Mr. Richmond Jones; Colonel F. Chaplin, and the Officers of the 4th Dragoon Guards, stationed at Northampton; Captain Thornelow, and the Officers of the Royal Artillery, at Weedon; the officers of the 4th Royal Regiment at Weedon; Mr. and Mrs. E. Wood; Mr. and Mrs. Herbert Wood; the Colonel and officers of the First Warwickshire Militia; Mrs. and Miss Alston, and Mr. Alston, jun., of Elmdon; Mr. and Mrs. F. Paget; Mr. and Mrs. Gulson; Captain Thomson; Captain and Mrs. Raleigh King; Mrs. Phillipson; Lord and Lady Mountgarret; the Honourable Miss Butler; Mr. and Mrs. Courtenay Lord; the Hon. Mrs. Twistleton; Mr. and the Misses Conant; Captain and Mrs. J. Marsland; Major and Mrs. Edlman; Mr. and Mrs. Astley; Mr. T. Lant, Mr. R. Lant and Mr. J. Lant, Mrs. and Miss Lant; Mr. W. T. Cavendish; Mr. and Mrs. A. Rotherham; the Marquis of Ormonde, of the first Life Guards; the Earl of Calludon, of the First Life Guards; Mrs. and the Misses Hobson; Mr P. Hobson, and Mrs. Hobson; Mr. and Mrs. Soames; Mr. and Mrs. Adderley, Sir John Rae Reid; Capt. and Mrs. Townshend, of Caldecote Hall; Lieut.-Colonel Swinfen and the Officers of the 5th Dragoon Guards stationed at Leeds; Capt. Marsden and the Officers of the 5th Dragoon Guards stationed at Birmingham; Colonel, Mrs., and Miss Bourne; Mr. and Mrs. Wyley Lord; Captain and Mrs. Thursby; Mr. and Mrs Morrice; Lieut.-Colonel Wirgman; Mr. and Mrs. J. Rotherham; [[Social Victorians/People/Abercorn|Lady Caroline Howard]]; Mr. and Mrs. Rotherham; Mr and Mrs John Sankey and the Misses Sankey; Mrs. and the Misses Murphy; Mr. Bibby (4th Hussars), Captain Gist (7th Hussars), Mr. Gregg (8th Hussars), Mr. Hamilton (7th Dragoon Guards), Colonel Rattray, Mr and Mrs. R. Boyd, &amp;c, &amp;c.&lt;/p&gt;
&lt;p&gt;The string band of the 5th Dragoon Guards, under the direction of Mr. Sidney Jones, performed the following selection of music:— Quadrille, Barbe Bleue; Valse, Marian; Galop, Bonderbryllup; Lancers, Knight of St. Patrick; Valse, Hydropaten; Galop, Flick and Flock; Quadrille, Princess of Trebizonde; Valse, the Belle of the Ball; Galop, the Fox Hunters; Valse, the Dragoon Guards; Lancers, the Gaiety; Valse, the Beautiful Danube; Valse, Wiener Kinder; Quadrille, the Fest; Galop, the Village Rose; Valse, the Geraldine; Lancers, Merry Tunes; Galop, Barbe Bleue; Valse, Various; Galop, Glorioso.&lt;ref&gt;&quot;Invitation Ball.&quot; ''Coventry Standard'' 6 January 1871, Friday: 4 [of 4], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000683/18710106/100/0004. Same print title, n.p.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;

=== February ===

==== Birmingham Tennis Court Club Ball ====
1871 February 17, Friday, the &quot;bachelors of the Tennis Court Club&quot; hosted a ball in Birmingham:&lt;blockquote&gt;LEAMINGTON.&lt;p&gt;

B&lt;small&gt;ACHELORS'&lt;/small&gt; B&lt;small&gt;ALL&lt;/small&gt;.&lt;p&gt;— Last night the bachelors of the Tennis Court Club gave a grand ball at the Royal Assembly Rooms, Regent Street. The ball was one of the most brilliant of the season, nearly four hundred of the ''élite'' of the town and neighbourhood having accepted the invitation of the bachelors. The ballroom was specially fitted up for the occasion, and a splendid supper was served in the adjoining rooms, where refreshments were also provided. Coote and Tiney's band was specially engaged for the occasion, and played a selection of the newest and most popular dance music. Amongst the distinguished guests present were — The High Sheriff and Mrs. J. T. Arkwright, Lady Arbuthnott, Lord and Lady Conyers, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mountgarret and the Hon. Miss Butler, Sir John and Lady Blois, Sir Thomas Biddulph, the Hon. Miss Somerville, Sir William and Lady Fairfax, the Hon. Charles L. Butler, Rev. Sir John Rae, General and Mrs. Richmond Jones, Major Eldman, Major and Mrs. James Ashton, Major and Mrs. Boothby, Colonel Ruttie, Colonel Duberly, Colonel and Mrs. Machen, Colonel Rattray, Capt. and Mrs. Kennedy, Capt. W. J. Hall, Capt. Hodge, Capt. and Mrs. Morgan, Capt. and Mrs. Pearse, Capt. Roberts, Capt. Story, Mr. and Mrs. Featherstone Dilke (Maxstoke Castle) and Miss Dixie, Mr. C. M., Miss, and Miss M. A. Caldecott (Holbrooke Grange), Mr. and Mrs. J. Dugdale (Wroxhall Abbey), Mr. E. Greaves, M.P., Mr. and Mrs. C. L. Adderley (Hams Hall), and Capt. and Mrs. Hatherall. Several of the officers from the dragoons and artillery at Coventry and Birmingham were also present. The bachelors who gave the ball were twenty-eight in number.&lt;ref&gt;&quot;Leamington.&quot; &quot;District News.&quot; ''Birmingham Morning News'' 18 February 1871, Saturday: 7 [of 8, print and digital], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005826/18710218/114/0007. Print and digital title are the same.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;Another description of this same event, the Bachelors' Ball at the Leamington Spa:&lt;blockquote&gt;The bachelors’ ball at Leamington Spa, which took place on the 17th inst., was a greater success than ever. It was held as usual in the Assembly Rooms, which, by the bye, might be better adapted to such purposes. Theyare not so bad as far as the ball room goes, but to reach the supper room you have to make a pilgrimage up one of the steepest and most uncomfortable staircases ever seen; still, however difficult the journey, a safe arrival will repay one. The room was very prettily decorated, and most sumptuous fare provided. The following is a list of the bachelors who gave the ball: Mr Neville Bagot, Mr Ramsay Clarke, Mr Erasmus Galton, Mr C. H. Gregg (8th Hussars), Mr Ralph C. Gregg, Mr William Gillett, Mr Thomlinson Grant, Col. Hammond, R.A., Capt. Hull, Mr Wm. Harrison, Mr Pulsford Hobson, Mr Sydney Hobson, Mr F. C. Lister Kay, Viscount St. Lawrence, M.P., Capt. Maxwell Lyte (7th Dragoon Guards), Mr Richard Lant, Mr John Lant, Mr Oswald Milne, Mr W. W. Moore, Mr Thomas Norman, Mr Hamilton Osborne, Capt. John Paynter, Capt. Pullin, Mr George Rennie, Mr Alex. G. Stuart, Mr J. H. Sanders, Mr Edmund Vyner, Captain Vandeleur; and nothing that they could do was wanting to make it a most complete success. The frequenters of the subscription balls could scarcely recognise the rendezvous of their fortnightly meetings. A porch had been erected over the entrance in the parade, and the corridors all round the dancing room carpeted with crimson and prettily decorated. Banks of flowers had been arranged in every available corner of the ball room, and a number of mirrors hung against the wall reflected the gay scene. Coote and Tinney’s band played a charming selection, and dancing was kept up with much spirit to a late hour. The company was a large one, the toilettes exceedingly pretty. Among those present were Lord and Lady Conyers, Sir William and Lady Fairfax, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mount-Garrett, [[Social Victorians/People/Ormonde|Hon. Miss Butler]], Sir John Rae Reid, Hon. Mary Somerville, &amp;c. The whole week was a gay one. Now heat has set in there is an unusual number of public lectures, and Mr and Mrs Howard Paul and the Tyrolese singers have had performances during this week. The Concordia Choir held their third concert on Saturday last, and on Monday there was an amateur performance of &quot;Maritana&quot; and &quot;The Captain is not a-Miss&quot; at the Royal Music Hall, in aid of the volunteer corps. The Warwickshire, North Warwickshire, the Bicester, the Cotherstone, the Pytchley, and Mr W. W. Tailby’s hounds are meeting four or five days in the week each, and having air sport.&lt;ref&gt;&quot;Fashionable Entertainments.&quot; ''The Queen'' 25 February 1871, Saturday: 19 [of 24], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18710225/121/0019. Print title: The Queen, ''The Lady's Newspaper'', p. 133.&lt;/ref&gt;&lt;/blockquote&gt;

=== March ===

=== April ===

==== 18 April 1871 ====
&lt;blockquote&gt;Karl Marx “was commissioned by the General Council of the International to write a pamphlet about the Paris [377–378] Commune.&quot;&lt;ref name=&quot;:3&quot;&gt;Smee, Sebastian. ''Paris in Ruins: Love, War, and the Birth of Impressionism''. W. W. Norton, 2024.&lt;/ref&gt;{{rp|377–378 of 667}}&lt;/blockquote&gt;

===May===

==== 9 May 1871, Tuesday, Queen's Drawing-Room ====
&lt;blockquote&gt;THE QUEEN'S DRAWING-ROOM.

The Queen held a Drawing-room at Buckingham Palace on Tuesday afternoon. The Priuce of Wales, Prince Arthur, Prince Leopold, and Princess Beatrice were present. Her Majesty, accompanied by the Prince of Wales and the other members of the royal family, entered the Throne Room shortly after three o'clock. The Queen wore a black moire antique dress with a train, long white tulle veil with a coronet of diamonds. Her Majesty also wore a necklace of diamonds and amethysts, the Riband and Star of the Order of the Garter, the Orders of Victoria and Albert and Louise of Prussia, and the Saxe Coburg and Gotha Family Order. Princess Beatrice wore a dress of white tulle over a rich white silk petticoat looped up with lilies of the valley and apple blossom; ornaments — pearls and diamonds.

The presentations to Her Majesty were about 280 in number, and included the following:— Mrs Atlay, by the Countess Grey; Miss Backhouse, by her mother, Mrs Backhouse; Miss Charlesworth, by her aunt, Frances Lady Hawke; Miss Backhouse Fox, by her aunt, Mrs Backhouse; [[Social Victorians/People/Abercorn|Lady Caroline Howard]], by her mother, [[Social Victorians/People/Abercorn|the Hon. Mrs Howard]]; the Hon. Gwendoline Fitz-Alan Howard, by the Duchess of Sutherland; [[Social Victorians/People/Abercorn|Lady Alice Howard]], by her mother, Hon. Mrs Howard; [[Social Victorians/People/Abercorn|Lady Louisa Howard]], by her mother, Hon. Mrs Howard; Miss Howard (of Corby), by the Hon. Mrs Philip Stourton; Miss Agnes Howard (of Corby), by the Hon. Mrs Philip Stourton; Sir Henry Ingilby, Bart., by Earl Russell; Mrs Frank Lascelles, by Lady Edward Cavendish; Mrs Gerald Liddell, marriage, by the Countess of Normanby.&lt;ref&gt;&quot;Court and Official News.&quot; ''Yorkshire Post and Leeds Intelligencer'' 11 May 1871, Thursday: 3 [of 4], Col. 4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000686/18710511/074/0003. Same print title and p.n.&lt;/ref&gt;&lt;/blockquote&gt;'''24 May 1871, Wednesday''': Derby Day. Baron Rothschild's Favonius won. The Prince of Wales attended.

June
July
August
September
===October===
'''October 1871'''&lt;blockquote&gt;At Londesborough Lodge near Scarborough, where Lady Londesborough gave a royal house party in October 1871, not only [ 41/42 ] were the bathrooms few but the drains seeped into the drinking water. Several guests, including the Prince [of Wales] and his groom and Lord Chesterfield, contracted typhoid fever. When Chesterfield and the groom died, the doctors abandoned hope for the Prince.&lt;ref name=&quot;:1&quot;&gt;Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973. Print.&lt;/ref&gt;{{rp|41–42}}&lt;/blockquote&gt;

The Prince of Wales recovered on 14 December 1871.

November
December

==1872==
January
February
March
April
===May===
'''29 May 1872, Wednesday''': Derby Day

June
July
===August===
'''August 1872''': The &quot;dance on the cruiser Ariadne&quot; probably occurred in August 1872:&lt;blockquote&gt;When his [the Prince of Wales'] brother, the Duke of Edinburgh, married the attractive Grand Duchess Marie, daughter of Tsar Alexander II of Russia, her family made a fuss because she was not granted precedence above the Princess of Wales. Albert Edward soothed ruffled feelings by inviting the Tsarevitch and his wife Marie Feodorovna (who was Alexandra's sister) to stay for two months and be entertained at Cowes. ...&lt;p&gt;&lt;/p&gt;

... At the dance on the cruiser Ariadne which the Prince gave in honour of the Tsarevitch and his Grand Duchess,&quot; Lord Randolph Churchill met the 19-year-old &quot;Miss Jennie Jerome of New York.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|42–43}}&lt;/blockquote&gt;

September
October
November
December

==1873==

=== January ===

==== 13 January 1873, Monday ====

==== Ball at the Chief Secretary's Lodge ====
On Tuesday, 14 January 1873, the Dublin Evening Telegraph reported that the Marquis of Hartington's ball had taken place the evening before.&lt;blockquote&gt;The Marquis of Hartington gave a ball last evening at the Chief Secretary's Lodge, to their Excellencies the Lord Lieutenant and the Countess Spencer, who were accompanied by the Dowager Countess Spencer, the Ladies Sarah and Victoria Spencer and the Hon Robert Spencer, Lord and Lady Charles Bruce, and Major Stirling, A D C.&lt;p&gt;

The following had the honour of receiving invitations to meet their Excellencies — The Duke of Leinster, the Marquis and Marchioness of Kildare, the Ladies Fitzgerald, the Marquis and Marchioness of Drogheda, the Earl and Countess of Listowel, Lord and Lady Edward Cavendish, the Earl of Charleville, the Lord Chancellor and Lady O'Hagan, Viscount, Viscountess, the Hon Misses, and Hon Henry Monck; the Archbishop of Dublin, the Hon Mrs and the Misses Trench; Lord Talbot de Malahide and the Hon Francis Talbot, Lord and Lady Sandhurst and Captain Bang, A D C; Lady Cloncurry, Hon Emily and Hon Mary Lawless, Viscount, Viscountess, Hon Georgiana, and Hon Beatrice [de?] Vesci; Lord and Lady Kilmaize [?], Hon Gertrude [?] Browze, Lord and Lady Ventry, Hon Norah Westenra, Lord and Lady Athlumney, Lord, Lady, and Hon D Plunket, M P; Viscountess and the Hon. Miss Netterivlle, Capt the Hon Mrs Vesey, Captain and Lady Julia Follett, Sir Arthur and Lady Olive Guiness and the Ladies White, the Hon H W L Corry, Lord and Lady and the Hon Miss O'Neill, Viscount Hawarden, the Hon Florence Maude, the Hon. Clementina Maude, the Hon Jenico and Mrs Preston, the Hon Henry Leeson, Colonel and the Hon Mrs Caulfield, Mr and the Hon Mrs Robert Hobart, Captain, Lady Mary and Miss Lindsay; Mr Ion [?] Trent Hamilton, M P; Mr Bagwell; the Hon Mrs and the Misses Bagwell, and Mr Bagwell; Colonel the Hon L and Mrs Curzon Smyth, Mr, Lady Margaret, and the Misses Stronge [?]; Mr and the Hon Mrs O'Hagan, Hon Charles Bourke, Hon Mrs Alfred and Lady Kathleen Bury, [[Social Victorians/People/Abercorn|Hon Mrs, Lady Alice, and Lady Louisa Howard]]; Captain, the Hon Mrs, and Miss Donaldson; Dr and Miss Bans, Mrs Grattan Bellew, Sir Edward and Miss Borough, Mr Arthur Cane, Sir Dominic, Lady, and Miss Corrigan; Mr Corrigan, Mr and Mrs Gustavus Cornwall and Miss Cornwall, Mr D'Arcy, M P, and Mrs D'Arcy; Mr Baron Dowse [?], and Mrs and Miss Dowse, Mr Baron Deasy and Mrs Deasy, Dr, Mrs, and Miss de Ricci; Dr and Miss Hatchell, Sir George and Lady Hudson, Mr, Mrs, and the Misses Huband; Mr Arthur Huband, Miss Caroline Huband, Mr and Mrs Arthur Hume, Dr Hughes, Mr Henry Jephsen and Miss Jephsen, Mr Kearney and the Misses Kearney, Captain Kearney, A D C; Captain Lascelles, A D C; Mr, Mrs, and Miss Kirwan; Mr Justice Lawson and Mrs Lawson, Mr and Mrs W Le Fanu, Mr, Mrs, and Miss Lentaigne; Sir George L'Estrange and the Misses L'Estrange, the Lord and Lady Mayoress, and the Misses Mackey; the Lord Chief Justice Monahan, Mrs and Miss Monahan; Sir J, Lady, and Miss Power; Mr John Talbot Power, M P; Col, Mrs, and Miss Radcliffe; the Master of the Rolls, Mrs and Miss Sullivan; Capt and Mrs Moorsom, A D C; General Sir Thomas and Lady Steel, Captain and Mrs Brownrigg, A D C, Mr Granville Milner, Capt, Mrs and Miss Talbot, Colonel, Mrs, and the Misses White; Sir John Stewart Wood, Lady and the Misses Wood; Mrs and the Misses Williams, Mr Justice Fitzgerald and the Hon Mrs Fitzgerald, Mr Fitzgerald, Mr Justice Barry and Mrs Barry, Mr Sergeant Sherlock, M P, Mrs and Miss Sherlock; Mr Sheriock, the Right Hon W H Conan, M P, and Mrs Cogan; Mr Justice Keogh and Mrs Keogh, Mr Keogh, Capt Keogh, R N; Lord Chief Baron and Miss Pigott, Dr, Mrs, and Miss Nugent; General Wardlaw, Colonel M'Kerlie, Mr Sergeant and Mrs and Miss Armstrong; Col, Mrs, and the Misses Maude; Col, Mrs, and Miss Hillier; Mr Heron, M P; Mr and Mrs Watters, Col and Mrs Wynyard, Dr and the Misses Kennedy, the Attorney General and Mrs Palles, the Solicitor General and Mrs Law, Col, Mrs, and Miss Lake; Lady and the Misses Butler, Mr Butler, Col and Mrs Colthurst Vesey, and Miss Walton; Mr, Lady Fanny and Miss Lambert; Mr E C Guinness, Mr and Mrs MMorer O'Ferrall, Mr and Mrs Leonard Morrogh, Sir Bernard and Lady Burke, Mr G and Mrs G Brooke and Miss Brooke, Mr and Mrs Roe, Mr Vance, M P, Mrs and Miss Vance; Col and Mrs Primrose, Lieut Col Ferdall [?], Col and Mrs Goodlake and Miss Alexander, Mr Alison, Mr, Mrs, and Miss Barton, Mr Justice Flanagan, Mrs and Miss Flanagan, Mer J. N. Lentaigne, Mr Johnson, Captain Harrison, Mr, Mrs, and the Misses Maturin; Mr Justice Morris and Mrs Morris, Mr and Mrs Mazlere [?] Brady, Major, Mrs, and Miss Wilkinson; Mr, Mrs, and Miss Donnelly; Mr and Mrs Cruise, Mrs Power, Mr Braon Fitzgerald and Mrs Fitzgerald, Mr Henry Yates Thompson, Mr Courtenay Boyle, Colonel Forster, Mr, Mrs, and Miss Taylor, Mr Bland and Mrs Godfrey Bland, Mr and Miss Dillon, Mr and Mrs Wallace, Mr M'Kenna, Mr Cullinane, Mr Armstrong, Mr C E [?] Dobbin, Mr J A Blake, Major and Mrs Papillon, Capt and Mrs Keane, Mr E Pretty, Mr, Mrs John L O Ferrall and Miss O'Ferrall, Mrs and Miss Walsh, Mr and Mrs R Howard Brook, Mrs and Miss Brook, Mrs and the Misses Blake, Mr and Mrs J Warren, Sir John Gray, M P, Lady, and Miss Gray; Colonel and Mrs Frank Chaplin, Mr, Mrs, and Miss Hemphill; Sir R, Lady and Miss Kane, Mrs and Miss Courtenay, Mr Arthur Courtenay, Mr G Courtenay, Mr E Hardtop, A D C; Mr Bellew, Dr and Mrs Nedley, Dr and Mrs Newell, Mr and Mrs Freeman, Mr and Mrs Geale, Captain Hutten, A D C; Mr and Mrs Adair and Miss Wadsworth, Captain and Mrs J M Benthall, Sir R, Lady, and the Misses M'Causlend [?]; Mr, Mrs, and the Misses Newell Barron; Mr Hawkins, Colonel Goodlake and the Officers of the Coldstream Guards; Captain Spain, R N, and the Officers (4) of her Majesty's ship Vanguard; Colonel Radcliffe and Officers (4), Royal Artillery; Colonel Spade and Officers (4) 1st King's Dragoon Guards; Colonel Ainslie and Officers (4), 1st Royal Dragoons; Colonel Thompson and Officers (4), 14th Hussars; Colonel Ross and Officers (4), 4th Battalion Rifle Brigade; Colonel Hawkins and Officers (4), Royal Engineers; Colonel Gloster and Officers (4), 97th Regiment; Lieutenant-Colonel Maunsell and Officers (4), 13th Regiment.&lt;ref&gt;&quot;Fashionable.&quot; ''Dublin Evening Telegraph'' 14 January 1873, Tuesday: 4 [of 4], Col. 7a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002093/18730114/044/0004. Print title ''The Evening Telegraph'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;February
March
April

===May===
'''28 May 1873, Wednesday''': Derby Day

June
July
August
September
October
November
December

==1874==
January
February
March
April
===May===

==== 1874 May, Early ====
&lt;blockquote&gt;As monarchists’ hopes flared, the Catholic Church, too, enjoyed a conspicuous revival. The National Assembly approved a design for a new basilica for Paris. Intended as an act of collective atonement, Sacré-Coeur was to perch atop Montmartre, immediately above where Nadar’s balloons had been launched and where the radicals’ insurrection had broken out. Excavations began in early May 1874 ....

But the focus of the penance the basilica was intended to embody gradually shifted from the moral decline of French society in general to the despicable excesses of the Commune. In 1872 Archbishop Darboy’s successor claimed to have had a vision as he climbed the Butte Montmartre. The clouds dispersed, and he realized that it was there, “where the martyrs” were (he meant the murdered generals Lecomte and Clément-Thomas), that a new church should be built. And when the Assembly voted to proceed with the construction, legislators specified that its purpose was to “expiate the crimes of the Commune.”&lt;ref name=&quot;:3&quot; /&gt; (464 of 667)&lt;/blockquote&gt;

===June===
'''3 June 1874, Wednesday''': Derby Day

June
July
August
September
October
November
===December===
'''8 December 1874, Tuesday''': &quot;CHATSWORTH, Tuesday, December 8th, 1874. — We are come to the last slide of the Chatsworth magic lantern: the Duke of Cambridge and his equerry, a funny little man called Tyrwhitt, of no particular age, in a grey wig; Lord Carlingford and Ly. Waldegrave, the Spencers, Mr. Leveson, Cavendish.&quot;&lt;ref&gt;{{Cite web|url=http://ladylucycavendish.blogspot.com/2010/12/08dec1874-chatsworth-magic-lantern.html|title=Lady Lucy Cavendish: 08Dec1874, The Chatsworth Magic Lantern|last=H|first=Denise|date=2010-12-04|website=Lady Lucy Cavendish|access-date=2025-06-18}}&lt;/ref&gt;

==1875==
Disraeli's progressive legislation for labor rights:&lt;blockquote&gt;In 1875, he passed a series of enlightened acts protecting labor rights, arguing they were as important as property rights. Two of the laws ensured that workers would have the same recourse as employers when contracts were breached, and made peaceful picketing legal, protecting unions from charges of conspiracy.&lt;ref name=&quot;:4&quot; /&gt; (578 of 1203)&lt;/blockquote&gt;After women who owned property were allowed by Parliament to stand for local school-board elections in 1870, &quot;Elizabeth Garrett Anderson, the first woman to qualify as a doctor in Britain — in 1865 — stood and was elected to her local board five years later.&quot;&lt;ref name=&quot;:4&quot; /&gt; (199 of 1203)

The relationship between Swinburne and Lord Houghton:&lt;blockquote&gt;...not all Lord Houghton's children appreciated the catholicity of &quot;Papa's&quot; taste in friends: &quot;Swinburne (in a very excited state) came in in the evening,&quot; wrote Florence Milnes to her brother in 1875: &quot;He is madder than ever, to my astonishment he flopped down on one knee in front of me, &amp; announced that my hair had grown darker. This was rather embarrassing, and he is also so deaf now, which does not make it easier to talk to him.&quot;&lt;ref name=&quot;:2&quot;&gt;Pope-Hennessy Lord Crewe.&lt;/ref&gt;{{rp|5}}&lt;/blockquote&gt;

January
February
March
April
===May===
'''26 May 1875, Wednesday''': Derby Day. The Prince and Princess of Wales attended, as did a number of others of the royal family, including Princess Louise and Lorne.

June
July
===August===
'''August through October 1875''' Richard Monckton Milnes (Lord Houghton) and son Robert Milnes toured the U.S. and Canada:&lt;blockquote&gt;They set off in the steamer s.s Sarmatian from Liverpool in August 1875, stopping at Ireland to pick up the usual load of emigrants bound for the U.S.A. The most interesting among the passengers was 'Mr. Butler, author of Erewhon, who is very amusing and clever though infidel,' but, although he played whist with Samuel Butler, the young man was far more interested in the Eustace Smiths (parents of his friend W. H. Smith), and in a Canadian family named Macpherson, the youngest of whose two daughters, the dark-eyed Isobel, caught his fancy: he saw them afterwards in Toronto, and when they parted she gave him two larger than carte-de-visite photographs of herself, he gave her a smaller one of himself together with the inevitable volume of his father's verse.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|10}}&lt;/blockquote&gt;September
October
November
December

==1876==
Disraeli pushed through the Cruelty to Animals Act in order to please Queen Victoria. This act &quot;forced researchers to demonstrate that any experiments with animals involving pain were absolutely necessary, and ensured they would be anesthetized if so.&quot;&lt;ref name=&quot;:4&quot; /&gt; (679 of 1203)

January
February
March
April
===May===
'''11 May 1876''': In the midst of the Aylesford scandal, the Prince of Wales returned from a journey to Egypt and India, etc.:&lt;blockquote&gt;However harassed and exhausted, the Prince and Princess of Wales would put up a good show. Within an hour of their arrival home they set forth to attend a gala performance at Covent Garden Opera House. It was a brave decision to face the public and allow an immediate opportunity for demonstration. The Prince and Princess were rewarded when the audience rose to its feet to give them a standing ovation before the start of every act, as well as at the end, of Verdi's Ballo in Maschera.&lt;ref name=&quot;:1&quot; /&gt;{{rp|63}}&lt;/blockquote&gt;

'''27 May 1877''': Lily Langtry:&lt;blockquote&gt;Her big moment on May 27, 1877, when Sir Allen Young, the arctic explorer, invited her to late supper in his house, where it had been arranged that the Prince of Wales should meet her after the opera. The result was all that could have been expected. Mrs. Langtry became the Prince's first openly recognised mistress.&lt;ref name=&quot;:1&quot; /&gt;{{rp|69}}&lt;/blockquote&gt;'''31 May 1877, Wednesday''': Derby Day. The Prince and Princess of Wales did not attend, as he was ill.

June
July
August
September
October
November
December

==1877==
&quot;In 1877, unemployment was 4.7 percent; by 1879, it had risen to 11.4 percent.&quot;&lt;ref name=&quot;:4&quot; /&gt; (690 of 1203)

January
February
March
April
===May===
'''30 May 1877, Wednesday''': Derby Day.

June
July
August
September
October
November
===December===
'''15 December 1877'''&lt;blockquote&gt;On Dec. 15, 1877, the Queen honoured Lord Beaconsfield, the Premier, with a visit at Hughenden Manor. Her Majesty, accompanied by Princess Beatrice and attended by General Ponsonby and the Marchioness of Ely, left Windsor at 12.40 and proceeded by special train to High Wycombe, which was reached at 1.15. The Premier received the Queen at the station. A lofty triumphal arch spanned the entrance to the station-yard, and beneath this the royal party drove into the gaily decorated little town. The reception along the route was of the heartiest, and the drive of two miles to Hughenden was one long triumph. Lord Beaconsfield, who had preceded the party, welcomed the Queen at his own door. Lunch was served, and her Majesty remained about two hours. Before leaving she planted a memorial tree.&lt;ref&gt;&quot;The Queen's Glorious Reign.&quot; ''Illustrated London News'' (London, England), Saturday, May 27, 1899; pp. 757–765?; Issue 3136. Queen's Glorious Reign [Supplement]: 762?&lt;/ref&gt;&lt;/blockquote&gt;

==1878==
January
February
March
April
May
===June===
'''5 June 1878, Wednesday''': Derby Day.

July
August
September
October
===November===
'''8 November 1878''': from the journal of George, Duke of Cambridge:&lt;blockquote&gt;''November'' 8. — Gave farewell diner to the Lornes; Louise and Lorne, Augusta, Mary and Francis, Arthur, Leopold, Gleichens, J. Macdonald and self, and played at Nap afterwards. It was a good and nice little dinner.&quot;&lt;ref&gt;Sheppard, Edgar, Ed. ''George, Duke of Cambridge: A Memoir of His Private Life, Based on the Journals and Correspondence of His Royal Highness''. Vol. 2, 1871–1904. New York: Longmans, Green, 1906. http://books.google.com/books?id=dFoMAAAAYAAJ.&lt;/ref&gt;&lt;/blockquote&gt;December

==1879==
===January===
'''12 January 1879'''&lt;blockquote&gt;On 12 January 1879 Robert Milnes came of age, an event celebrated at Fryston by a tenants' ball.&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}}&lt;/blockquote&gt;

'''28 January 1879''': Brett &quot;Harte kicked off his tour at the Crystal Palace in Sydenham on January 28, 1879.&quot;&lt;ref&gt;Nissen, Alex. ''Brett Harte: Prince and Pauper''. Jackson, MS: University Press of Mississippi, 2000.&lt;/ref&gt;{{rp|174}}

February
March

===April===
'''Early April 1879''' or so, probably, Bret Harte got &quot;an invitation to dine the same evening with Arthur Sullivan and the Prince of Wales&quot; as a dinner in Birmingham where Harte met T. Edgar Pemberton.&lt;ref&gt;Scharnhorst, Gary. ''Bret Harte: Opening the American Literary West''. Norman, OK: Univ. of Oklahoma Press, 2000.&lt;/ref&gt;{{rp|152}}
===May===
'''28 May 1879, Wednesday''': Derby Day; the Prince and Princess of Wales attended.

===June===
'''June 1879''', Robert Milnes became engaged to &quot;Sibyl Marcia, a daughter of a North-country baronet, Sir Frederick Graham of Netherby.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}} Parties must have followed.

July
August
September
October
November
===December===
'''28 December 1879''': The Tay Bridge Disaster: The Tay Bridge collapsed with a train on it. The weather was very bad, with gale-force winds and rain.

The ''Times'' reported that the average high temperature for the week ending December 31, 1879, was 53° F. and the low was 20° F.

In his column &quot;What the World Says&quot; in the 21 January 1880 World, Edmund Yates writes the following:&lt;blockquote&gt;How am I to describe better the magnificence of the Earl and Countess of Rosslyn’s ball at Euston Lodge last month, than by calling attention to the fact that M. Carlo, the eminent Knightsbridge coiffeur, arrived early in the day to crimp and powder the lacqueys? My informant adds, however, that the curled darlings were rather the worse for the festivities towards night. Was it not enough to turn their heads in every sense of the word?&lt;ref name=&quot;:0&quot;&gt;Edmund Yates, &quot;What the World Says,&quot; ''The World: A Journal for Men and Women''.&lt;/ref&gt;{{rp|21 Jan. 1880, p. 8, col. b.}}&lt;/blockquote&gt;

'''31 December 1879''': Edmund Yates, editor of The World: A Journal for Men and Women, in his column &quot;What the World Says,&quot; describes a private viewing at the Grosvenor Gallery:&lt;blockquote&gt;The private view at the Grosvenor on the last day of the year gave people something to do on a desperately wet afternoon. The artistic dresses were perhaps in greater force than ever; indeed the faces and the hair and the attitudes pursued me to my bed, and gave me many a nightmare. I suppose the plain woman of all time has had the ambition to be looked at: centuries of failure have at last been crowned with a real success. Besides the Cimabue Browns there was an interesting menagerie of real lions, artistic, literary, and clerical. The artists were numerous, and their host and hostess seemed to enjoy themselves very thoroughly.

Frequenters of the picture private views have a new sensation this winter. Last season they mobbed beauty: now hideously-attired unkempt dowdiness provokes the stare. The prize for the new style seems generally awarded to a rhubarb coloured flannel Ulster and a cart-wheel beaver hat, which pervaded both the private views last week. [2 private views last week, one at the Grosvenor]&lt;ref name=&quot;:0&quot; /&gt;{{rp|7 Jan. 1880, p. 9}}&lt;/blockquote&gt;

The official premiere of ''The Pirates of Penzance'' occurred in New York City on 31 December 1879 at the Fifth Avenue Theatre, to establish international copyright. Gilbert and Sullivan were there with the cast. The performance was a social event: attending were Mrs. Vanderbilt and Mrs. Astor.

==Works Cited==
{{reflist}}</text>
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  </page>
  <page>
    <title>Theme park management</title>
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      <text bytes="2245" sha1="nrgy6ejhfp95gksgyt1yd58d772r8cn" xml:space="preserve">{{Prod|undeveloped page}}
This examination of theme park management considerations is under development by University of Florida, College of Health and Human Performance, Department of Tourism, Hospitality &amp; Event Management (THEM) HFT 4755 Theme Park &amp; Attraction Management undergraduate students. This ongoing coursework initiative started Spring 2021 and is being led by the THEM students at the direction of Brian D. Avery, UF SPM/THEM Faculty member. 

Students will develop a foundation based on consensus defining and outlining theme park and attraction management considerations including history, typology, stakeholders, guest experiences, workforce, security, business continuity, legal, and regulatory issues impacting the theme park and attractions industry. 

'''&lt;u&gt;Learning Objective&lt;/u&gt;'''

# Familiarization with the global theme park and attraction industry, including its characteristics, economic impact, volume, industry structure, and historical considerations; 
# Evaluation of the concept of theming and its role in tourism and hospitality operations; 
# Evaluation of the role authentic/inauthentic environments play in themed settings; 
# Evaluation of the planning phases in the development and design processes of theme parks and attractions; 
# Analysis of the managerial aspects of theme park and attractions; and, 
# Familiarization of the selection/sourcing process of the theme park product offerings. 

'''&lt;u&gt;Topics&lt;/u&gt;'''

* History and introduction of theme parks and attractions;
* Typology of parks and attractions;
* Industry stakeholders, organizational structures, management cultures;
* Destination management considerations (travel/tourism - lodging, transportation, etc.)
* Evolution of industry regulations, standards, and practices;
* Modern industry regulations, standards, and practices;
* Guest experiences and expectations / Theming and design characteristics;
* Consumables, environmental impact;
* Theme park metrics / Associations / Marketing and sales 
* Workforce (engineering, maintenance, operations, talent, govt. controls, accounting);
* Crowd management and control / Security and loss prevention; and,
* Emergency preparedness and response; and, 
* Legal considerations and exposures.</text>
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  <page>
    <title>C language in plain view</title>
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      <comment>/* Applications */</comment>
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      <text bytes="9878" sha1="l27079ceys7vfylaz5epsin3odgl64f" 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.20260709.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>l27079ceys7vfylaz5epsin3odgl64f</sha1>
    </revision>
  </page>
  <page>
    <title>Module:Params</title>
    <ns>828</ns>
    <id>308015</id>
    <revision>
      <id>2818013</id>
      <parentid>2816625</parentid>
      <timestamp>2026-07-09T19:54:57Z</timestamp>
      <contributor>
        <username>Grufo</username>
        <id>1192007</id>
      </contributor>
      <comment>[[:mw:Module:Params|Upstream]] updates</comment>
      <origin>2818013</origin>
      <model>Scribunto</model>
      <format>text/plain</format>
      <text bytes="82130" sha1="bx8a2av95b6jpca3zvf7xnia4bz7hsf" xml:space="preserve">require[[strict]]


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

Meanings of the columns:

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

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

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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


-- Parse optional user arguments of type `...|[let]|[...][number of additional
-- parameters]|[parameter 1]|[parameter 2]|[...]`
local function load_child_opts (src, start_from, append_after)
	local tbl, pin = {}, start_from
	local tnamed
	while src[pin] ~= nil and src[pin]:match'^%s*let%s*$' and
		src[pin + 1] ~= nil and src[pin + 2] ~= nil
	do
		tbl[get_parameter_name(src[pin + 1])] = src[pin + 2]
		pin = pin + 3
	end
	if pin ~= start_from then tnamed, tbl = tbl, {} end
	local tmp = tonumber(src[pin])
	if tmp ~= nil and math.floor(tmp) == tmp then
		if tmp &lt; 0 then tmp = -1 end
		local shf = append_after - pin
		for idx = pin + 1, pin + tmp do tbl[idx + shf] = src[idx] end
		pin = pin + tmp + 1
	end
	if tnamed ~= nil then
		for key, val in pairs(tnamed) do tbl[key] = val end
	end
	return tbl, pin
end


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


-- Syntax:  #invoke:params|filling_the_gaps|pipe to
library.filling_the_gaps = function (ctx)
	local tbl, tmp, nmin, nmax, nnums = ctx.params, {}, 1, nil, -1
	for key, val in pairs(tbl) do
		if type(key) == 'number' then
			if nmax == nil then
				if key &lt; nmin then nmin = key end
				nmax = key
			elseif key &gt; nmax then nmax = key
			elseif key &lt; nmin then nmin = key end
			nnums = nnums + 1
			tmp[key] = val
		end
	end
	if nmax ~= nil and nmax - nmin &gt; nnums then
		ctx.n_available = ctx.n_available + nmin + nnums - nmax
		if ctx.n_available &lt; 0 then error(modulename ..
			', ‘filling_the_gaps’: It is possible to fill at most ' ..
			tostring(maxfill) .. ' parameters', 0) end
		for idx = nmin, nmax, 1 do tbl[idx] = '' end
		for key, val in pairs(tmp) do tbl[key] = val end
	end
	return context_iterate(ctx, 1)
end


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


-- Syntax:  #invoke:params|combining|new parameter name|[sort order]|setting
--            directives|...|pipe to
library.combining = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return key .. kvs .. val end,
		'combining'
	) + 1)
end


-- Syntax:  #invoke:params|combining_values|new parameter name|[sort
--            order]|setting directives|...|pipe to
library.combining_values = function (ctx)
	-- NOTE: `ctx.params` might be the original metatable! As a modifier,
	-- this function MUST create a copy of it before returning
	return context_iterate(ctx, combine_parameters(
		ctx,
		function (key, val, kvs) return val end,
		'combining_values'
	) + 1)
end


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

	local ret, nss, ccs = {}, 0, ctx.itersep or ''
	flush_params(
		ctx,
		function (key, val)
			opts[1] = key
			opts[2] = val
			ret[nss + 1] = ccs
			ret[nss + 2] = mfunc(ctx.frame:newChild(model))
			nss = nss + 2
		end
	)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


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


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


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


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


-- Syntax:  #invoke:params|call_for_each_group|template name|[append 1]|[append
--            2]|[...]|[append n]|[named param 1=value 1]|[...]|[named param
--            n=value n]|[...]
library.call_for_each_group = function (ctx)
	-- NOTE: `ctx.pipe` and `ctx.params` might be the original metatables!
	local tmp
	if ctx.pipe[1] ~= nil then tmp = ctx.pipe[1]:match'^%s*(.*%S)' end
	if tmp == nil then error(modulename ..
		', ‘call_for_each_group’: No template name was provided', 0) end
	local model = { title = tmp }
	local opts, ret, nss, ccs = {}, {}, 0, ctx.itersep or ''
	for key, val in pairs(ctx.pipe) do
		if type(key) == 'number' then opts[key - 1] = val
		else opts[key] = val end
	end
	ctx.pipe = opts
	ctx.params = make_groups(ctx.params)
	flush_params(
		ctx,
		function (gid, group)
			for key, val in pairs(opts) do group[key] = val end
			group[0] = gid
			model.args = group
			ret[nss + 1] = ccs
			ret[nss + 2] = ctx.frame:expandTemplate(model)
			nss = nss + 2
		end
	)
	finalize_and_return_concatenated_list(ctx, ret, nss, 2)
	return false
end


	--[[ First-position-only modifiers ]]--
	---------------------------------------


-- Syntax:  #invoke:params|new|pipe to
static_iface.new = function (child_frame)
	local ctx = context_new(child_frame)
	ctx.pipe = copy_or_ref_table(ctx.opipe, false)
	ctx.params = {}
	main_loop(ctx, context_iterate(ctx, 1))
	return ctx.text
end


	--[[ First-position-only functions ]]--
	---------------------------------------


-- Syntax:  #invoke:params|self
static_iface.self = function (frame)
	return frame:getParent():getTitle()
end


	--[[ Public metatable of functions ]]--
	---------------------------------------


return setmetatable({}, {
	__index = function (_, query)
		local fname = query:match'^%s*(.*%S)'
		if fname == nil then error(modulename ..
			': You must specify a function to call', 0) end
		local func = static_iface[fname]
		if func ~= nil then return func end
		func = library[fname]
		if func == nil then error(modulename ..
			': The function ‘' .. fname .. '’ does not exist', 0) end
		return function (child_frame)
			local ctx = context_new(child_frame)
			ctx.pipe = copy_or_ref_table(ctx.opipe, refpipe[fname])
			ctx.params = copy_or_ref_table(ctx.oparams, refparams[fname])
			main_loop(ctx, func)
			return ctx.text
		end
	end
})</text>
      <sha1>bx8a2av95b6jpca3zvf7xnia4bz7hsf</sha1>
    </revision>
  </page>
  <page>
    <title>User:Tommy Kronkvist</title>
    <ns>2</ns>
    <id>320737</id>
    <revision>
      <id>2818053</id>
      <parentid>2817866</parentid>
      <timestamp>2026-07-10T09:45:56Z</timestamp>
      <contributor>
        <username>Tommy Kronkvist</username>
        <id>31941</id>
      </contributor>
      <comment>User statistics.</comment>
      <origin>2818053</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2646" sha1="kyhjndw7jgpev76tn4qlk7gwn0xae3m" xml:space="preserve">&lt;div style=&quot;margin: 0 0 1em 0;&quot;&gt;{{userpage}}&lt;/div&gt;
{{Userboxtop|toptext=Babel:}}
{{#babel:sv|en-4|de-2|la-1}}
{{Userboxbottom}}
[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|310px|The intracanopy of a Wild Service Tree, i.e. &lt;small&gt;''Torminalis glaberrima'' (Gand.) Sennikov &amp; Kurtto, ''Memoranda Soc. Fauna Fl. Fenn.'' 93: 32 (2017).&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (July 10, 2026), I've made just over 396,100 edits to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]].

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

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
{{User committed identity|a6edd6d2fdbf82621f0cda4e5525c71f8da9b5dfd308242c3c63365e998c32c5406b75448380903265a5403edffd1a0435b61ac943f3c65870db9250f8b884a9|SHA-512|background=#e0e8ff|border=e0e8ff}}</text>
      <sha1>kyhjndw7jgpev76tn4qlk7gwn0xae3m</sha1>
    </revision>
  </page>
  <page>
    <title>Probability Dilation Theory</title>
    <ns>0</ns>
    <id>321584</id>
    <revision>
      <id>2818032</id>
      <parentid>2816275</parentid>
      <timestamp>2026-07-10T01:33:01Z</timestamp>
      <contributor>
        <username>Howie2024</username>
        <id>2995240</id>
      </contributor>
      <comment>/* Subpages */ Dilation Vector Field new subpage</comment>
      <origin>2818032</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38445" sha1="ha6ocuz6yegu03tw9dqc4alhzmt9zii" xml:space="preserve">{{Research project}}
{{Original research}}
{{To be peer reviewed}}

{{subst:proofread}}

== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential PDT transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

Not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established.

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

Whether a consistent discrete formulation of probability geometry can be constructed.

Whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles.

Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions.

== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

== Subpages ==

The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT).

* [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]]
  – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories.

* [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]]
  – develops the log-odds representation of probability elements and exponential PDT flows.

* [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]]
  – investigates invariant measures, attractors, and stability of iterative PDT transformations.

* [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]]
  – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution.

* [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]]
  – examines Shannon entropy under repeated probability dilation.

* [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]]
  – explores distances between probability measures and convergence in measure space.

* [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]]
  – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions.

* [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]]
  – investigates continuous probability flows and Euler approximations of PDT.

* [[Probability Dilation Theory/Worked Example]]
  – canonical binary example illustrating PDT transformations and geometry.

[[Probability Dilation Theory/Decoherence Analogy and Simulation]]

[[Probability Dilation Theory / Dilation Vector Field]]

== Notation ==

Throughout PDT, the following notation is used:

{| class=&quot;wikitable&quot;
! Symbol
! Meaning
|-
| &lt;math&gt;P&lt;/math&gt;
| Probability measure
|-
| &lt;math&gt;P_n&lt;/math&gt;
| nth iterate of PDT
|-
| &lt;math&gt;T_D&lt;/math&gt;
| Probability dilation operator
|-
| &lt;math&gt;D(x)&lt;/math&gt;
| Dilation field
|-
| &lt;math&gt;Z(P,D)&lt;/math&gt;
| Normalization factor
|-
| &lt;math&gt;H(P)&lt;/math&gt;
| Shannon entropy
|-
| &lt;math&gt;d_F&lt;/math&gt;
| Fisher-Rao distance
|-
| &lt;math&gt;W_p&lt;/math&gt;
| Wasserstein distance
|-
| &lt;math&gt;\ell&lt;/math&gt;
| Logit coordinate
|-
| &lt;math&gt;PE&lt;/math&gt;
| Probability Element
|}

== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.

* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

Text and original figures © Howard Richardson.

Original contributions by the author. Content is available under the Creative Commons Attribution-ShareAlike license (CC BY-SA) as required by Wikiversity.

Reuse permitted with attribution.</text>
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== Research abstract ==
'''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values.

The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics.

At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions.
PDT treats a probability measure as the primary mathematical object and investigates:
* invariant identities induced by reweighting,
* composition and iteration of dilations,
* fixed points and near-fixed behavior,
* whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations).

PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions.

== Overview ==
PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is:
* mathematically well-defined (positivity and normalization),
* composable under iteration,
* analyzable for invariants and fixed points.
=== Conceptual interpretation ===

A simplified conceptual flow of the PDT framework is:

&lt;pre&gt;
Baseline probability measure P
            ↓
Positive dilation field D(x)
            ↓
Reweighted probability measure P~
            ↓
Observable statistical changes
&lt;/pre&gt;
Repeated dilation may qualitatively behave as:

&lt;pre&gt;
Broad initial distribution
        ↓
Localized reweighting
        ↓
Probability concentration
        ↓
Emergent multiscale structure
&lt;/pre&gt;

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.
In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field.

Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as:

&lt;pre&gt;
P₀
 ↓ D₁
P₁
 ↓ D₂
P₂
 ↓ D₃
P₃
 ↓ ⋯
&lt;/pre&gt;

where each dilation field reweights the probability structure generated by the previous step.

Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics.

= Mathematical framework =

== Definitions and notation ==
Let &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt; be a measurable space.

* &lt;math&gt;P&lt;/math&gt; denotes a probability measure on &lt;math&gt;(\Omega,\Sigma)&lt;/math&gt;.
* If &lt;math&gt;P&lt;/math&gt; has a density &lt;math&gt;p&lt;/math&gt; with respect to a reference measure &lt;math&gt;\mu&lt;/math&gt;, then &lt;math&gt;dP=p\,d\mu&lt;/math&gt;.
* &lt;math&gt;D:\Omega\to(0,\infty)&lt;/math&gt; is a measurable '''dilation field''' (a positive weight function).
* &lt;math&gt;Z(P,D)&lt;/math&gt; is the normalization constant:
.&lt;math&gt;
Z(P,D)=\int_\Omega D\,dP
&lt;/math&gt;
* For an observable &lt;math&gt;f:\Omega\to\mathbb{R}&lt;/math&gt; integrable under the relevant measure,
&lt;math&gt;
\mathbb{E}_P[f]
=
\int_\Omega f\,dP
&lt;/math&gt;.

== PDT transformation (probability reweighting) ==

Given &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;D&lt;/math&gt; with &lt;math&gt;0&lt;Z(P,D)&lt;\infty&lt;/math&gt;, define the '''PDT transform''' &lt;math&gt;\widetilde{P}=\mathrm{PDT}(P;D)&lt;/math&gt; by:

&lt;math&gt;
\widetilde{P}(A)
=
\frac{
\int_A D\,dP
}{
\int_\Omega D\,dP
}
\quad\text{for all }A\in\Sigma
&lt;/math&gt;

If &lt;math&gt;dP=p\,d\mu&lt;/math&gt;, then &lt;math&gt;d\widetilde{P}=\widetilde{p}\,d\mu&lt;/math&gt;, where

&lt;math&gt;
\widetilde{p}(x)
=
\frac{D(x)\,p(x)}{Z}
&lt;/math&gt;

and

&lt;math&gt;
Z
=
\int_\Omega D(x)\,p(x)\,d\mu
&lt;/math&gt;

'''Interpretation:''' the dilation field &lt;math&gt;D&lt;/math&gt; shifts probability mass toward regions where &lt;math&gt;D&lt;/math&gt; is larger, while renormalization keeps total probability equal to 1.

PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures.

Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations.
A familiar physical example of a strictly positive factor is the Lorentz factor:

&lt;math&gt;
\gamma(v)
=
\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
&lt;/math&gt;

for

&lt;math&gt;
|v|&lt;c
&lt;/math&gt;

Lorentz contraction for a rod of rest length &lt;math&gt;L_0&lt;/math&gt; moving at speed &lt;math&gt;v&lt;/math&gt; is:

&lt;math&gt;
L(v)=\frac{L_0}{\gamma(v)}
&lt;/math&gt;

To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on &lt;math&gt;\gamma&lt;/math&gt;.

== Worked finite example ==

Consider a finite probability space:

&lt;math&gt;
\Omega=\{a,b,c\}
&lt;/math&gt;

with baseline probabilities:

&lt;math&gt;
P(a)=0.2,\quad
P(b)=0.3,\quad
P(c)=0.5
&lt;/math&gt;

Define a positive dilation field:

&lt;math&gt;
D(a)=1,\quad
D(b)=2,\quad
D(c)=4
&lt;/math&gt;

The normalization constant is:

&lt;math&gt;
Z=\sum_x D(x)P(x)
&lt;/math&gt;

giving:

&lt;math&gt;
Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8
&lt;/math&gt;

The PDT-transformed probabilities become:

&lt;math&gt;
\widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214
&lt;/math&gt;

&lt;math&gt;
\widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714
&lt;/math&gt;

This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization.

== Composition of dilations ==

An important structural property of sequential PDT transformations is that compose multiplicatively.

Suppose two positive dilation fields:

&lt;math&gt;
D_1(x)&gt;0
&lt;/math&gt;

and

&lt;math&gt;
D_2(x)&gt;0
&lt;/math&gt;

are applied successively to a baseline probability measure &lt;math&gt;P&lt;/math&gt;.

The first dilation produces:

&lt;math&gt;
\widetilde{P}_1(A)
=
\frac{\int_A D_1\,dP}
{\int_\Omega D_1\,dP}
&lt;/math&gt;

Applying the second dilation field to &lt;math&gt;\widetilde{P}_1&lt;/math&gt; gives:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{\int_A D_2\,d\widetilde{P}_1}
{\int_\Omega D_2\,d\widetilde{P}_1}
&lt;/math&gt;

Substituting the first transformation into the second yields:

&lt;math&gt;
\widetilde{P}_2(A)
=
\frac{
\int_A D_2D_1\,dP
}{
\int_\Omega D_2D_1\,dP
}
&lt;/math&gt;

This shows that sequential PDT transformations compose through multiplication of the dilation fields.

This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application.

== Fixed points and iterative dynamics ==

An important question in PDT concerns the long-term behavior of repeated PDT transformations.

Given an initial probability measure:

&lt;math&gt;
P_0
&lt;/math&gt;

and a sequence of positive dilation fields:

&lt;math&gt;
D_1,D_2,D_3,\dots
&lt;/math&gt;

successive PDT transformations generate a sequence of measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow
P_3
\rightarrow \cdots
&lt;/math&gt;

where each transformed measure is obtained by reweighting the previous one.

A measure &lt;math&gt;P&lt;/math&gt; is called a fixed point of a dilation field &lt;math&gt;D&lt;/math&gt; if:

&lt;math&gt;
\widetilde{P}=P
&lt;/math&gt;

under the PDT transformation.

In the simplest case, this requires the dilation field to be constant almost everywhere with respect to &lt;math&gt;P&lt;/math&gt;. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization.

More generally, repeated compositions of nontrivial dilation fields may generate:

* hierarchical probability structure;
* multiscale statistical behavior;
* attractor-like distributions;
* approximately stable transformed measures.

These questions connect PDT to broader areas of:

* dynamical systems;
* stochastic processes;
* iterative renormalization methods;
* probabilistic geometry.

At present these iterative properties remain largely unexplored within the PDT framework.

== Entropy and iterative probability flow ==

Repeated PDT transformations may alter the entropy structure of a probability measure.

For a discrete probability distribution:

&lt;math&gt;
P=\{p_i\}
&lt;/math&gt;

the Shannon entropy is:

&lt;math&gt;
H(P)
=
-\sum_i p_i \log p_i
&lt;/math&gt;

Under iterative PDT transformation, successive transformed measures:

&lt;math&gt;
P_0
\rightarrow
P_1
\rightarrow
P_2
\rightarrow \cdots
&lt;/math&gt;

may exhibit changing entropy behavior depending on the structure of the dilation fields.

For example:
* strongly localized dilation fields may concentrate probability mass and reduce entropy;
* broader or smoothing dilation fields may distribute probability more evenly and increase entropy;
* iterative compositions may generate approximately stable entropy profiles.

These questions connect PDT to:
* information theory,
* statistical mechanics,
* stochastic dynamics,
* and renormalization-style iterative systems.

At present the entropy behavior of iterative PDT transformations remains an open area for investigation.

== Toy experiment: entropy under repeated dilation ==

A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution.

Let the initial probability distribution be:

&lt;math&gt;
P_0=(0.2,0.2,0.2,0.2,0.2)
&lt;/math&gt;

and define a positive dilation field:

&lt;math&gt;
D=(1,1,2,4,8)
&lt;/math&gt;

At each step, apply the PDT update:

&lt;math&gt;
P_{n+1}(i)
=
\frac{D(i)P_n(i)}
{\sum_j D(j)P_n(j)}
&lt;/math&gt;

The Shannon entropy is:

&lt;math&gt;
H(P_n)
=
-\sum_i P_n(i)\log P_n(i)
&lt;/math&gt;

In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately:

&lt;math&gt;
H(P_0)\approx1.6094
&lt;/math&gt;

to:

&lt;math&gt;
H(P_{10})\approx0.00775
&lt;/math&gt;

The final distribution is approximately:

&lt;math&gt;
P_{10}
\approx
(0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437)
&lt;/math&gt;

This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior.
== Mathematical context ==
PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature.
In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis.
The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations.
=== Example entropy evolution ===

{| class=&quot;wikitable&quot;
! Iteration !! Shannon entropy
|-
| 0 || 1.6094
|-
| 1 || 1.2990
|-
| 2 || 0.7790
|-
| 3 || 0.4399
|-
| 5 || 0.1500
|-
| 10 || 0.0078
|}
Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states.
=== Localized dilation fields ===

A useful class of PDT transformations is generated by localized positive dilation fields.

Consider a one-dimensional finite configuration space with states indexed by:

&lt;math&gt;
x=0,1,2,\dots,N
&lt;/math&gt;

and define a localized dilation field centered at &lt;math&gt;x_0&lt;/math&gt;:

&lt;math&gt;
D(x)
=
\exp\!\left(
\lambda
\exp\!\left(
-\frac{(x-x_0)^2}{2\sigma^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\sigma&lt;/math&gt; controls the spatial width of the localized field.

Narrow values of &lt;math&gt;\sigma&lt;/math&gt; produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space.

Under iterative PDT dynamics:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

the probability distribution may progressively concentrate near the center of the dilation field.

=== Example entropy evolution for localized fields ===

Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior:

{| class=&quot;wikitable&quot;
! Field width &lt;math&gt;\sigma&lt;/math&gt;
! Final entropy after 10 iterations
! Maximum probability after 10 iterations
|-
| 1.5 || 0.0352 || 0.9950
|-
| 3.0 || 0.8162 || 0.7141
|-
| 6.0 || 1.5367 || 0.3595
|}
[[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]]
[[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]]
These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction.

== Comparative entropy-flow experiments ==

The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions.
{| class=&quot;wikitable&quot;
|+ Comparative entropy-flow behavior under PDT field classes
! Field class
! Final entropy
! Entropy decrease
! Final max probability
! Qualitative behavior
|-
| Localized
| 0.3104
| 3.4032
| 0.9275
| Strong probability concentration
|-
| Oscillatory
| 1.5779
| 2.1357
| 0.3418
| Distributed oscillatory structure
|-
| Multi-peak
| 0.2851
| 3.4284
| 0.9425
| Multiple concentration regions
|-
| Stochastic
| 0.7744
| 2.9392
| 0.7413
| Fluctuating concentration behavior
|}
These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example.
In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space.
[[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]]
The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.

=== Oscillatory dilation fields ===

Another useful class of PDT transformations is generated by oscillatory positive dilation fields.

One example is:

&lt;math&gt;
D(x)
=
\exp(\lambda\sin(kx))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the oscillatory amplification;
* &lt;math&gt;k&lt;/math&gt; controls the spatial frequency of the oscillation.

Because the exponential is always positive, the dilation field remains strictly positive for all states.

Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space.

Under repeated PDT transformation:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor.

=== Example oscillatory-field experiment ===

A finite-state experiment was performed using:

* 41 discrete states;
* an initially uniform probability distribution;
* a positive oscillatory dilation field with three spatial oscillation cycles;
* 10 successive PDT iterations.

Representative entropy behavior was:

{| class=&quot;wikitable&quot;
! Iteration
! Shannon entropy
|-
| 0 || 3.7136
|-
| 2 || 2.8699
|-
| 5 || 2.3018
|-
| 10 || 1.9335
|}

Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space.

After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state.

This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures.

The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence.
=== Multi-peak localized dilation fields ===

A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space.

One example is:

&lt;math&gt;
D(x)
=
\exp\!\left(
\sum_k
\lambda_k
\exp\!\left(
-\frac{(x-x_k)^2}{2\sigma_k^2}
\right)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;x_k&lt;/math&gt; are the locations of the dilation peaks;
* &lt;math&gt;\lambda_k&gt;0&lt;/math&gt; control the amplification strength of each peak;
* &lt;math&gt;\sigma_k&lt;/math&gt; control the spatial width of each localized region.

This construction generates a positive multimodal dilation landscape containing several competing amplification regions.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D(x)P_n(x)
}{
\sum_y D(y)P_n(y)
}
&lt;/math&gt;

probability mass may evolve toward multiple partially localized concentration regions.

Unlike single localized dilation fields, multi-peak fields may generate:

* competing attractor-like regions;
* hierarchical probability concentration;
* partially stabilized multimodal distributions;
* multiscale probability structure.

Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor:
* dominance by a single peak;
* coexistence of several concentration regions;
* or slowly evolving metastable probability structures.

=== Conceptual interpretation ===

A qualitative iterative evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Multiple localized amplifications
        ↓
Competing concentration regions
        ↓
Emergent multimodal probability structure
&lt;/pre&gt;

This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone.

At present these behaviors remain exploratory computational observations within finite-state toy models.
=== Random and stochastic dilation fields ===

Another important class of PDT transformations arises when the dilation field itself varies stochastically.

A simple stochastic dilation field may be written schematically as:

&lt;math&gt;
D_n(x)
=
\exp\!\left(
\sigma \eta_n(x)
\right)
&lt;/math&gt;

where:

* &lt;math&gt;\eta_n(x)&lt;/math&gt; is a random field or stochastic fluctuation at iteration &lt;math&gt;n&lt;/math&gt;;
* &lt;math&gt;\sigma&gt;0&lt;/math&gt; controls the strength of the stochastic variation.

Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process.

Under repeated PDT iteration:

&lt;math&gt;
P_{n+1}(x)
=
\frac{
D_n(x)P_n(x)
}{
\sum_y D_n(y)P_n(y)
}
&lt;/math&gt;

the probability landscape itself fluctuates dynamically from one iteration to the next.

Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate:

* fluctuating concentration regions;
* transient attractor-like structures;
* noise-driven entropy evolution;
* intermittent probability concentration;
* metastable probabilistic configurations.

=== Conceptual interpretation ===

A qualitative stochastic evolution may be visualized as:

&lt;pre&gt;
Broad initial distribution
        ↓
Random localized amplification
        ↓
Fluctuating concentration regions
        ↓
Dynamic probabilistic structure
&lt;/pre&gt;

Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit:
* partial concentration,
* persistent fluctuations,
* stochastic stabilization,
* or continuously evolving probabilistic structure.

These ideas connect PDT to broader areas of:
* stochastic processes;
* random multiplicative systems;
* statistical mechanics;
* noise-driven dynamical systems;
* probabilistic geometry.

At present these behaviors remain exploratory computational possibilities within finite-state toy models.

== Qualitative classes of iterative PDT behavior ==

Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation.

The following table summarizes several representative classes explored within finite-state toy models.

{| class=&quot;wikitable&quot;
! Dilation-field class
! Typical iterative behavior
! Representative qualitative structure
|-
| Localized fields
| Strong entropy reduction and concentration toward a dominant region
| Single attractor-like concentration
|-
| Oscillatory fields
| Distributed amplification with slower entropy reduction
| Patterned multimodal structure
|-
| Multi-peak localized fields
| Competition between several concentration regions
| Hierarchical or metastable probability structure
|-
| Random and stochastic fields
| Fluctuating amplification and noise-driven evolution
| Dynamic probabilistic landscapes
|}

These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field.

Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself.

At present these qualitative behaviors remain exploratory computational observations within finite-state toy models.

== Numerical simulation and iterative models ==

=== Simulation model description ===

In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points.

Two equivalent discrete implementations are common:

* '''weighted evaluation''': retain all points and assign weights proportional to &lt;math&gt;D&lt;/math&gt;;
* '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to &lt;math&gt;D&lt;/math&gt;.

=== Demonstration: reweighting mock galaxy catalogs ===

A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box.

The demonstration pipeline is:

# generate a baseline mock catalog;
# define a positive dilation field over the configuration space;
# perform PDT-style importance resampling;
# compute the resulting two-point correlation function &lt;math&gt;\xi(r)&lt;/math&gt;;
# compare transformed and baseline catalogs.

One example dilation field is:

&lt;math&gt;
D(x)=\exp(\lambda\phi(x))
&lt;/math&gt;

where:

* &lt;math&gt;\lambda&gt;0&lt;/math&gt; controls the strength of the dilation;
* &lt;math&gt;\phi(x)\ge0&lt;/math&gt; is a nonnegative configuration-space field.

An example seed-field construction is:

&lt;math&gt;
\phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right)
&lt;/math&gt;

where &lt;math&gt;s_k&lt;/math&gt; are seed locations and &lt;math&gt;\sigma&lt;/math&gt; controls the width of the seed influence.

The two-point correlation function may be estimated using the normalized Landy–Szalay estimator:

&lt;math&gt;
\xi(r)
=
\frac{DD(r)-2DR(r)+RR(r)}{RR(r)}
&lt;/math&gt;

where &lt;math&gt;DD&lt;/math&gt;, &lt;math&gt;DR&lt;/math&gt;, and &lt;math&gt;RR&lt;/math&gt; are normalized pair counts.

{{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}}

When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration.

=== Computational demonstrations ===

Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages.

{{collapse top|Python demonstration placeholder}}

&lt;syntaxhighlight lang=&quot;python&quot;&gt;
# Example implementations may be maintained separately
# on GitHub, OSF, or supplementary Wikiversity pages.
&lt;/syntaxhighlight&gt;

{{collapse bottom}}


== Scope and Limitations ==

PDT is a mathematical framework for measure transformations. It does not claim:

* a replacement theory for General Relativity or Quantum Mechanics;
* empirical confirmation without explicit predictions and tests;
* observational validation without independently reproducible analysis.

The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations.

== Speculative Extensions and Geometric Renormalization ==

''This section is speculative and exploratory in nature.''

Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry.

Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects.
Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation.
At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations.

== Future directions ==

* develop canonical families of dilation fields and invariants;
* clarify “structure-from-measure” diagnostics;
* publish reproducible simulation notebooks and parameter sweeps;
* compare multiple dilation families under shared evaluation criteria;
* investigate connections between probabilistic geometry and curvature-dependent statistical measures.
== Future Directions: Probability Element (PE) ==

A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics.

The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution.

This can be expressed in terms of a dimensionless ratio:

&lt;math&gt;\eta = \frac{\sigma_P}{\sigma}&lt;/math&gt;

where:

&lt;math&gt;\sigma_P&lt;/math&gt; is a hypothesized minimal probability-resolution scale,
&lt;math&gt;\sigma&lt;/math&gt; is an effective distinguishability scale in probability-state space.

=== Conceptual motivation ===

Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry.

=== Illustrative toy model (not derived physics) ===

As a heuristic example, one may consider a modification to special relativistic time dilation of the form:

&lt;math&gt;d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}&lt;/math&gt;

where:

&lt;math&gt;v&lt;/math&gt; is velocity,
&lt;math&gt;c&lt;/math&gt; is the speed of light,
&lt;math&gt;\eta = \sigma_P / \sigma&lt;/math&gt; encodes a proposed probability-resolution scale.

This expression is constructed such that standard special relativity is recovered exactly in the limit &lt;math&gt;\eta \to 0&lt;/math&gt;.

=== Status ===

The Probability Element concept is:

Not part of standard Fisher information geometry
not derived from quantum mechanics or general relativity
not currently empirically established.

It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale.

=== Open questions ===

Key open research directions include:

Whether a consistent discrete formulation of probability geometry can be constructed.

Whether a fundamental probability-resolution scale &lt;math&gt;\sigma_P&lt;/math&gt; can be derived from known physical principles.

Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions.

== Convergence behavior ==

Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations.

=== Qualitative convergence classes ===

Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior:

* '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy.

* '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration.

* '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions.

* '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior.

=== Entropy and convergence ===

In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time.

The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations.

=== Attractor-like behavior ===

Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense.

Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks.

== Current limitations ==
PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation.
Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms.
Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks.

== See also ==

* [[w:Buffon's needle problem|Buffon's needle problem]]
* [[w:Probability measure|Probability measure]]
* [[w:Importance sampling|Importance sampling]]
* [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]]
* [[w:Dynamical system|Dynamical systems]]

* [[w:Entropy (information theory)|Entropy]]
* [[w:Information theory|Information theory]]
* [[w:Measure theory|Measure theory]]
* [[w:Geometric probability|Geometric probability]]

* [[w:Shannon entropy|Shannon entropy]]
* [[w:Stochastic process|Stochastic process]]
* [[w:Fixed point (mathematics)|Fixed point]]
* [[w:Convergence (mathematics)|Convergence]]

== Subpages ==

The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT).

* [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]]
  – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories.

* [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]]
  – develops the log-odds representation of probability elements and exponential PDT flows.

* [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]]
  – investigates invariant measures, attractors, and stability of iterative PDT transformations.

* [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]]
  – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution.

* [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]]
  – examines Shannon entropy under repeated probability dilation.

* [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]]
  – explores distances between probability measures and convergence in measure space.

* [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]]
  – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions.

* [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]]
  – investigates continuous probability flows and Euler approximations of PDT.

* [[Probability Dilation Theory/Worked Example]]
  – canonical binary example illustrating PDT transformations and geometry.

[[Probability Dilation Theory/Decoherence Analogy and Simulation]]

[[Probability Dilation Theory / Dilation Vector Field]]

[[Probability Dilation Theory / Dilation Flows]]

== Notation ==

Throughout PDT, the following notation is used:

{| class=&quot;wikitable&quot;
! Symbol
! Meaning
|-
| &lt;math&gt;P&lt;/math&gt;
| Probability measure
|-
| &lt;math&gt;P_n&lt;/math&gt;
| nth iterate of PDT
|-
| &lt;math&gt;T_D&lt;/math&gt;
| Probability dilation operator
|-
| &lt;math&gt;D(x)&lt;/math&gt;
| Dilation field
|-
| &lt;math&gt;Z(P,D)&lt;/math&gt;
| Normalization factor
|-
| &lt;math&gt;H(P)&lt;/math&gt;
| Shannon entropy
|-
| &lt;math&gt;d_F&lt;/math&gt;
| Fisher-Rao distance
|-
| &lt;math&gt;W_p&lt;/math&gt;
| Wasserstein distance
|-
| &lt;math&gt;\ell&lt;/math&gt;
| Logit coordinate
|-
| &lt;math&gt;PE&lt;/math&gt;
| Probability Element
|}

== Related probabilistic and geometric literature ==

Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works:

* Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014.

* Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997.

* Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005.

* Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007.

* Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656.

Text and original figures © Howard Richardson.

Original contributions by the author. Content is available under the Creative Commons Attribution-ShareAlike license (CC BY-SA) as required by Wikiversity.

Reuse permitted with attribution.</text>
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      <text bytes="27778" sha1="rvbk2gvl1yizaz08tgtzlmieoe43qgc" xml:space="preserve">{{Prod|undeveloped, possibly AI-generated as well}}
= Machine Learning Fundamentals =
Machine learning (ML) enables computers to learn patterns and make predictions or decisions from data. Modern ML spans multiple paradigms: '''supervised learning''' (learning from labeled examples), '''unsupervised learning''' (finding structure in unlabeled data), '''semi-supervised learning''' (combining a small amount of labeled data with large unlabeled data), and '''reinforcement learning''' (learning via trial-and-error interaction with an environment). Deep learning is a subfield of ML using multi-layer neural networks to learn hierarchical representations. This course is organized in modules covering core ML concepts, mathematics, algorithms, evaluation methods, and practical implementation. Each module includes learning outcomes to guide study.

== Module 1: Introduction to Machine Learning ==
'''Learning Outcomes:''' Students will be able to describe the main categories of machine learning, explain the difference between supervised and unsupervised methods, and understand the role of data and models.

Machine learning systems are typically built from data and models. In '''supervised learning''', each training instance includes input features ''x'' and a known label or output ''y'' (for example, class labels or target values). The goal is to learn a function that maps inputs to outputs and can generalize to new data. In '''regression''', the output is continuous (e.g. predicting house prices), whereas in '''classification''' the output is discrete (e.g. spam vs. not spam). By contrast, '''unsupervised learning''' deals with unlabeled data: it seeks to uncover hidden patterns or groupings solely from input features. For example, clustering algorithms group similar data points without using class labels. '''Semi-supervised learning''' lies between supervised and unsupervised: it uses a mix of labeled and unlabeled data to improve learning, leveraging large unlabeled datasets when labels are scarce. '''Reinforcement learning''' involves an agent interacting with an environment to maximize cumulative rewards; it learns by trial-and-error without explicit labeled outputs.

Overall, ML finds applications across fields (vision, NLP, robotics, etc.) by learning from data rather than requiring hand-coded rules. The modules that follow will dive into each of these categories, the underlying mathematics, key algorithms, and how to implement them.

== Module 2: Mathematical Foundations ==
'''Learning Outcomes:''' Students will gain proficiency in linear algebra, calculus, probability, and optimization as they apply to machine learning; understand how these areas underpin model formulation and learning algorithms.

Machine learning relies on mathematical tools to describe models and optimize them. '''Linear algebra''' provides the language of vectors and matrices for data representation and operations (e.g. vector spaces, matrix decompositions, eigenvalues). For example, many models involve operations on data matrices (such as computing projections or decompositions). '''Calculus''' (especially multivariable calculus) underlies optimization algorithms: gradients and derivatives are used to adjust model parameters by minimizing loss functions (e.g. gradient descent). '''Probability and statistics''' form the basis for modeling uncertainty and data distributions. Concepts like random variables, probability distributions, expectation, Bayes’ theorem, and statistical estimators are essential for understanding models such as Naive Bayes classifiers or Bayesian methods. '''Optimization theory''' ties these together: many ML models are trained by solving optimization problems, often convex (for linear models) or non-convex (for neural networks).

Modern deep learning texts emphasize this background: for example, Goodfellow ''et al.'' note that deep learning techniques require understanding linear algebra, probability theory, information theory, and numerical optimization. In practice, students should be comfortable with vector dot products, matrix factorization (e.g. Singular Value Decomposition), computing derivatives of functions, and solving optimization problems (including using Lagrange multipliers for constrained problems or gradient-based methods for large models). These foundations will be applied in later modules when deriving learning algorithms and understanding their properties.

== Module 3: Supervised Learning ==
'''Learning Outcomes:''' Students will learn to implement and apply supervised learning algorithms, including regression and classification techniques; understand model training, loss functions, and evaluation on labeled data.

Supervised learning algorithms build predictive models from labeled examples. '''Regression''' models predict continuous outcomes. A common example is ''linear regression'', which fits a linear function to data by minimizing the mean squared error (MSE) between predictions and targets. The model parameters (weights) are typically found via ordinary least squares or gradient descent. In the example below, scikit-learn’s &lt;code&gt;LinearRegression&lt;/code&gt; is trained on a toy diabetes dataset and evaluated by MSE and R² score:
 &lt;code&gt;from sklearn.linear_model import LinearRegression
 from sklearn.metrics import mean_squared_error
 # X_train, y_train, X_test, y_test prepared beforehand
 model = LinearRegression()
 model.fit(X_train, y_train)                      # Train on training set
 predictions = model.predict(X_test)              # Predict on test set
 mse = mean_squared_error(y_test, predictions)
 print(&quot;Mean squared error:&quot;, mse)&lt;/code&gt;
This illustrates that linear regression learns coefficients ''w'' to best fit data, and performance is measured by error metrics (lower MSE is better).

In '''classification''', models predict discrete class labels. For binary classification, techniques include logistic regression, which uses a sigmoid function on a linear combination of features; and support vector machines (SVMs), which find a hyperplane that maximizes the margin between classes. In multiclass settings, one can extend binary methods or use inherently multi-class models (e.g. multinomial logistic regression). Decision trees are another supervised method: they recursively split the feature space using simple rules to predict the label. For example, scikit-learn explains that “Decision Trees (DTs) are a non-parametric supervised learning method used for classification and regression. The goal is to create a model that predicts the value of a target variable by learning simple decision rules inferred from the data features”.

Other supervised algorithms include '''k-nearest neighbors''' (classifying based on nearest training samples) and '''ensemble methods'''. Random forests build many decision trees and average their outputs; boosting methods (like XGBoost or AdaBoost) sequentially train trees to correct predecessor errors. These methods will be covered with examples. Every supervised model relies on a '''loss function''' (e.g. squared error for regression, cross-entropy for classification) and an optimization method to minimize that loss. Finally, model performance is evaluated using metrics such as accuracy for classification (the proportion of correct predictions) or F1-score (the harmonic mean of precision and recall), and metrics like MSE or R² for regression. We will also discuss issues like overfitting (when a model fits training data too closely) and underfitting, and techniques such as regularization (adding a penalty on model complexity) to mitigate them.

== Module 4: Unsupervised Learning ==
'''Learning Outcomes:''' Students will understand techniques for finding structure in unlabeled data, including clustering and dimensionality reduction; learn to apply common unsupervised algorithms to exploratory data analysis.

Unsupervised learning discovers patterns without labeled outputs. Common tasks are '''clustering''' and '''dimensionality reduction'''. In clustering, the algorithm groups similar data points together. A classic method is ''k''-means clustering: it partitions data into ''k'' clusters by minimizing the within-cluster sum of squares (inertia). Specifically, scikit-learn notes that “The KMeans algorithm clusters data by trying to separate samples in ''n'' groups of equal variance, minimizing a criterion known as the inertia or within-cluster sum-of-squares”. In practice, one chooses ''k'', initializes centroids, and iteratively assigns points to nearest centroids and re-computes centroids until convergence. Other clustering approaches include hierarchical clustering (agglomerative or divisive linking) and density-based methods like DBSCAN (which forms clusters of arbitrary shape based on density thresholds). These algorithms are useful for tasks like market segmentation or anomaly detection.

Dimensionality reduction transforms data into a lower-dimensional space, often for visualization or preprocessing. '''Principal Component Analysis (PCA)''' is a linear method that finds orthogonal directions (principal components) capturing maximal variance. As scikit-learn documents, “Principal component analysis (PCA). Linear dimensionality reduction using Singular Value Decomposition of the data to project it to a lower dimensional space”. In other words, PCA computes an SVD of the (centered) data matrix and projects points onto the top ''d'' components. Non-linear techniques include ''t''-SNE: a probabilistic embedding that “converts similarities between data points to joint probabilities and tries to minimize the Kullback–Leibler divergence between the joint probabilities of the low-dimensional embedding and the high-dimensional data”. t-SNE is often used for 2D/3D visualization of high-dimensional data (e.g. images or word embeddings). Other manifold learning methods like UMAP or Isomap can also reduce dimensionality by preserving local distances. In summary, unsupervised learning explores data structure through clustering and by creating compact representations; these methods are widely used in exploratory data analysis and feature engineering.

== Module 5: Semi-Supervised Learning ==
'''Learning Outcomes:''' Students will learn how semi-supervised approaches leverage both labeled and unlabeled data; understand common algorithms for semi-supervised classification.

Semi-supervised learning uses a small amount of labeled data together with a larger pool of unlabeled examples to improve learning performance. Conceptually, it sits between supervised and unsupervised learning. As van Engelen and Hoos explain, “Semi-supervised learning is the branch of machine learning concerned with using labelled as well as unlabelled data to perform certain learning tasks”. Common settings include cases where labels are costly to obtain: using unlabeled data can help capture the underlying data structure and improve the model. Techniques include ''self-training'' (iteratively labeling most confident unlabeled points), ''co-training'' (using multiple views of data to label examples for each other), and graph-based label propagation. For example, label propagation builds a graph of data points (edges weighted by similarity) and spreads label information across this graph. Deep learning has also embraced semi-supervised methods, such as using autoencoders or generative models to learn from unlabeled data, then fine-tuning on the small labeled set. Recent surveys show growing interest in semi-supervised methods driven by neural architectures. Students should understand when semi-supervised learning is appropriate and how to apply basic algorithms (often available in libraries) to datasets with few labels.

== Module 6: Reinforcement Learning ==
'''Learning Outcomes:''' Students will grasp the basics of reinforcement learning (RL), including Markov decision processes, value functions, and policy-based methods; implement simple RL algorithms.

Reinforcement learning deals with sequential decision-making: an '''agent''' interacts with an '''environment''', observes states, takes actions, and receives rewards. The objective is to learn a policy (mapping from states to actions) that maximizes cumulative reward over time. A standard formalism is the Markov Decision Process (MDP), which models the probability transitions and rewards. Unlike supervised learning, RL does not use labeled input/output pairs. As IBM explains, “Reinforcement learning (RL) is a type of machine learning process that focuses on decision making by autonomous agents... the agent learns to perform a task by trial and error in the absence of any guidance from a human user”. In practice, RL algorithms include ''value-based'' methods (e.g. Q-learning or Deep Q-Networks), where the agent estimates the value (expected return) of state-action pairs, and ''policy-based'' methods (e.g. REINFORCE, Actor-Critic), where the agent directly optimizes a policy parameterization. Modern RL often combines deep neural networks (Deep RL) to handle high-dimensional states (for example, playing Atari games or Go). Key concepts to learn include exploration vs. exploitation (trying new actions vs. using known good actions), reward shaping, and common benchmarks. Example applications include game playing (AlphaGo, AlphaZero), robotics, and resource management. We will cover algorithmic basics and use libraries (such as OpenAI Gym for environments and stable-baselines or PyTorch for implementation).

== Module 7: Neural Networks and Deep Learning ==
'''Learning Outcomes:''' Students will learn to design, train, and evaluate neural networks; understand deep learning architectures and gain experience with frameworks like TensorFlow and PyTorch.

Neural networks are a family of models inspired by the brain, consisting of layers of interconnected “neurons” that compute weighted sums of inputs and apply nonlinear activation functions. Deep learning refers to using ''deep'' (multi-layer) neural networks. An intuitive description is: '''“Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts”'''. By stacking layers, deep networks can learn very complex functions. For example, a ''multi-layer perceptron (MLP)'' is a feedforward network with one or more hidden layers. As scikit-learn’s documentation notes:&lt;blockquote&gt;“Multi-layer Perceptron (MLP) is a supervised learning algorithm that learns a function  by training on a dataset… it can learn a non-linear function approximator for either classification or regression. It is different from logistic regression, in that between the input and output layer, there can be one or more non-linear layers, called hidden layers”.&lt;/blockquote&gt;Neural networks are trained by backpropagating errors (using gradient descent) through the network to update weights. Modern networks often have specialized architectures: convolutional neural networks (CNNs) for images, recurrent or LSTM networks for sequences, and Transformer models with self-attention for language. For example, Vaswani ''et al.'' introduced the Transformer, “a model architecture eschewing recurrence and instead relying entirely on an attention mechanism to draw global dependencies between input and output”. This idea underpins large language models (LLMs) like BERT and GPT.

''Figure: A simple artificial neural network with an input layer, one hidden layer, and an output layer. Neural networks are trained by adjusting the connection weights to minimize a loss function.''

Popular frameworks make neural network implementation convenient. In '''TensorFlow''', one typically uses the &lt;code&gt;tf.keras&lt;/code&gt; API to build models. For example, a simple image classifier might be defined with layers as follows:
 &lt;code&gt;import tensorflow as tf
 model = tf.keras.Sequential([
     tf.keras.layers.Flatten(input_shape=(28,28)),         # flatten 28x28 image to vector
     tf.keras.layers.Dense(128, activation='relu'),       # hidden layer
     tf.keras.layers.Dense(10, activation='softmax')      # 10 output classes
 ])
 model.compile(optimizer='adam', loss='sparse_categorical_crossentropy',
               metrics=['accuracy'])
 model.fit(train_images, train_labels, epochs=5)          # train on data&lt;/code&gt;
TensorFlow’s documentation highlights that &lt;code&gt;tf.keras&lt;/code&gt; is a ''“high-level API to build and train models in TensorFlow.”''. In '''PyTorch''', one defines networks by subclassing &lt;code&gt;torch.nn.Module&lt;/code&gt; and composing layers. For example:
 &lt;code&gt;import torch.nn as nn
 class Net(nn.Module):
     def __init__(self):
         super(Net, self).__init__()
         self.fc = nn.Linear(10, 1)    # linear layer 
     def forward(self, x):
         return torch.sigmoid(self.fc(x))
 model = Net()&lt;/code&gt;
PyTorch emphasizes the use of &lt;code&gt;Tensor&lt;/code&gt; objects (like NumPy arrays) and automatic differentiation to compute gradients. Both frameworks support GPU acceleration for large models. Throughout this course, we will provide examples in Python using scikit-learn for smaller models and TensorFlow/PyTorch for deep learning.

== Module 8: Decision Trees and Support Vector Machines ==
'''Learning Outcomes:''' Students will understand how decision trees and SVMs work, implement them, and compare their strengths and weaknesses.

'''Decision Trees''' build models as tree structures of decisions. At each internal node, the tree splits on a feature condition (e.g., ''feature_i &lt;= threshold''), and leaves give class or value predictions. As cited earlier, decision trees “predict the value of a target variable by learning simple decision rules inferred from the data features”. Trees are intuitive, handle categorical and numerical data, and require little feature preprocessing. However, single trees can overfit; ensemble methods like random forests or gradient boosting mitigate this by averaging many trees.

'''Support Vector Machines (SVMs)''' are powerful classifiers (and regressors) that find a hyperplane maximizing the margin between classes. Scikit-learn describes SVMs as “a set of supervised learning methods used for classification, regression and outliers detection”. The key idea is to maximize the distance (margin) from the closest points (support vectors) to the decision boundary. With kernels, SVMs can handle non-linear boundaries by mapping data into higher-dimensional spaces. SVMs tend to be effective in high-dimensional spaces, though they can be sensitive to the choice of kernel and parameters (like the regularization cost ''C'').

For both trees and SVMs, scikit-learn provides easy interfaces (&lt;code&gt;DecisionTreeClassifier&lt;/code&gt;, &lt;code&gt;SVC&lt;/code&gt;). Students should practice fitting these models to data, visualizing decision boundaries, and tuning hyperparameters (e.g. tree depth, SVM kernel settings). Evaluate these models using cross-validation (see Module 10) to avoid overfitting. Relevant documentation includes the SVM user guide and decision tree guide for further algorithm details.

== Module 9: Clustering and Dimensionality Reduction ==
'''Learning Outcomes:''' Students will apply clustering and dimensionality reduction techniques to datasets, using them for data analysis and preprocessing.

This module dives into unsupervised methods in more detail. '''Clustering algorithms''' partition data without labels. We already introduced ''k''-means. Other important methods include hierarchical clustering (agglomerative or divisive), which builds a tree of clusters using distance/linkage criteria, and density-based clustering (DBSCAN) that identifies arbitrarily shaped clusters based on density reachability. Gaussian Mixture Models (GMMs) are a soft clustering approach using a mixture of Gaussians to model data. In practice, one should know how to use scikit-learn’s &lt;code&gt;KMeans&lt;/code&gt;, &lt;code&gt;AgglomerativeClustering&lt;/code&gt;, and &lt;code&gt;DBSCAN&lt;/code&gt; classes, and how to choose parameters (like ''k'' or distance thresholds). Clustering quality can be evaluated by silhouette score or domain-specific validation.

'''Dimensionality Reduction''' reduces feature dimensions. We introduced PCA and t-SNE. Other tools include ''kernel PCA'', ''Isomap'', ''UMAP'', and ''Truncated SVD'' (for sparse data). For example, scikit-learn’s PCA implements the linear SVD-based reduction. Students should practice applying PCA to visualize high-dimensional data (e.g. projecting images or embeddings to 2D) and understand how much variance the components explain. Techniques like t-SNE or UMAP can produce informative visualizations of clusters. Dimensionality reduction is also used to speed up learning (by removing noisy features) and to combat the “curse of dimensionality.” In assignments, we will provide datasets (e.g. digit images) to experiment with these algorithms, and require linking to datasets (e.g. MNIST, CIFAR, or the UCI repository) for practice.

== Module 10: Model Evaluation and Selection ==
'''Learning Outcomes:''' Students will learn to evaluate models using appropriate metrics, perform model selection, and apply cross-validation and regularization to improve generalization.

Proper evaluation is crucial. For '''classification''', we use the confusion matrix outcomes (true positives TP, false positives FP, etc.) to compute metrics. '''Accuracy''' = (TP+TN)/(TP+TN+FP+FN) measures overall correctness. '''Precision''' = TP/(TP+FP) indicates correctness of positive predictions, while '''recall (sensitivity)''' = TP/(TP+FN) reflects coverage of actual positives. The '''F1-score''' is the harmonic mean of precision and recall: ''F1 = 2·TP/(2·TP + FP + FN)'', balancing both. Scikit-learn’s documentation also notes that F1 can be seen as a harmonic mean. These metrics are important especially for imbalanced classes (where accuracy can be misleading). We will cover ROC curves and AUC for binary classifiers as well. For '''regression''', typical metrics are '''Mean Squared Error (MSE)''' and '''Mean Absolute Error (MAE)'''. As one survey explains, MSE = ∑(prediction – truth)² and MAE = ∑|prediction – truth|, with smaller values indicating better fit.

'''Cross-validation''' is used to estimate a model’s generalization performance. In ''k''-fold CV, the data is split into ''k'' folds; the model is trained on ''k–1'' folds and tested on the held-out fold, repeated for each fold. The reported performance is the average across folds. As scikit-learn notes, this “does not waste too much data (as is the case when fixing an arbitrary validation set)” and gives a more reliable estimate. We will use utilities like &lt;code&gt;cross_val_score&lt;/code&gt; and &lt;code&gt;GridSearchCV&lt;/code&gt; to tune hyperparameters and select models.

We will also study '''bias-variance tradeoff''': how model complexity and training data affect overfitting/underfitting, and use techniques like regularization (L1/L2 penalties), early stopping, and learning curves. Students should gain hands-on experience by comparing models using these metrics and validation strategies.

== Module 11: Implementation in Python – Libraries and Examples ==
'''Learning Outcomes:''' Students will become proficient in using Python libraries for ML and deep learning (scikit-learn, TensorFlow, PyTorch), and will complete coding exercises implementing ML workflows.

Throughout the course, we emphasize practical implementation. '''Scikit-learn''' provides a rich suite of algorithms with consistent interfaces. For example, to train a classifier and evaluate it:
 &lt;code&gt;from sklearn.ensemble import RandomForestClassifier
 from sklearn.model_selection import cross_val_score
 clf = RandomForestClassifier(n_estimators=100)
 scores = cross_val_score(clf, X, y, cv=5, scoring='accuracy')
 print(&quot;CV accuracy:&quot;, scores.mean())&lt;/code&gt;
This uses a 5-fold CV to report average accuracy. Scikit-learn’s online examples illustrate standard usage, such as fitting linear models and computing metrics.

For deep learning, students should use frameworks as described above. TensorFlow tutorials (e.g. the Fashion MNIST classifier) serve as references. PyTorch examples (like the “Learning PyTorch with Examples” tutorial) introduce core concepts: PyTorch provides '''tensors''' (n-dimensional arrays) and '''autograd''' for gradient computation. We will link to official documentation and encourage students to experiment with sample notebooks. External tools (e.g. Jupyter notebooks, Google Colab) and libraries for data (NumPy, Pandas, Matplotlib) will be used. For datasets, we’ll point to public sources: e.g. the UCI repository (with hundreds of datasets for ML), Kaggle datasets, or built-in loaders (scikit-learn’s datasets). By course end, students will have written and cited code for key algorithms, following best practices from official docs and tutorials.

== Module 12: Advanced Topics and Latest Developments ==
'''Learning Outcomes:''' Students will survey current trends in ML, such as deep architectures, large-scale models, and self-supervised methods; understand the research frontier and resources for further study.

The field of machine learning is advancing rapidly. Notable recent developments include '''self-supervised learning (SSL)''': techniques that leverage unlabeled data by creating surrogate tasks. In SSL, a model generates its own labels from the data (for instance, predicting missing parts of inputs). IBM notes SSL is “a machine learning technique that uses unsupervised learning for tasks that conventionally require supervised learning,” generating implicit labels from unstructured data. State-of-the-art models like BERT and GPT use self-supervised pretraining on massive text corpora. SSL is also used in vision (e.g. contrastive learning with SimCLR).

Large Language Models (LLMs) are a major breakthrough. For example, GPT-4 (2023) is described as “a large-scale, multimodal model which can accept image and text inputs and produce text outputs”. These models, trained on vast data, achieve near human-level performance on many tasks. Students should be aware of the Transformer architecture (self-attention-based) behind LLMs. Generative models are also prominent: ''Generative Adversarial Networks'' (GANs) and ''Variational Autoencoders'' (VAEs) for images, and ''Diffusion Models'' (like Stable Diffusion) for photo-realistic image generation. While detailed study of these is beyond an introductory course, we will give an overview and pointers to resources. Ethical considerations (bias, fairness, privacy) and best practices for deploying ML systems are also discussed in the context of this course.

'''External Resources:''' Throughout the course, authoritative sources will be provided for further reading. Textbooks include ''Pattern Recognition and Machine Learning'' (Bishop 2006) and ''Deep Learning'' (Goodfellow ''et al.'' 2016). Peer-reviewed surveys and papers (e.g. on semi-supervised learning) will be cited. Official documentation for tools (scikit-learn, TensorFlow, PyTorch) will be linked for in-depth guidance. Open datasets (e.g. from UCI, Kaggle) and tools (e.g. WEKA, TensorBoard) will be referenced.

This course is designed to give students a comprehensive foundation in machine learning. By completing the modules and hands-on projects, students will be well-equipped to apply ML techniques in research or industry, and to stay abreast of new developments in this rapidly evolving field.

'''References:''' Authoritative sources are cited throughout, including textbooks, peer-reviewed articles, and official documentation, among others. These provide the basis for the definitions and methods presented above.</text>
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      <text bytes="108195" sha1="m5kky5jdqmskuj7kqu71jx9w1q743mm" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

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

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

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

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

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

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

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

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

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

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

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

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

In this project, we aim to test 4 hypothesis : 

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

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

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

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

We also have 2 assumptions :

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Recommandations : Contradictions are allowed in Wikidata. 

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

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

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

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

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

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

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

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

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

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

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

{{Wikidata entity link|Q30241068}}

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

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

{{Wikidata entity link|Q99527517}}

{{Wikidata entity link|Q30241068}}

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

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

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

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

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

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

=== Filter statements ===

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

=== Properties visualisation ===

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

=== Concepts visualisation ===

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

=== Mapping sources consensus ===

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

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

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

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

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

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

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

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

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


== References ==
{{References}}</text>
      <sha1>m5kky5jdqmskuj7kqu71jx9w1q743mm</sha1>
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    <revision>
      <id>2818022</id>
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      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Future steps */ adding future step ideas</comment>
      <origin>2818022</origin>
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      <format>text/x-wiki</format>
      <text bytes="108316" sha1="dbn7738amcfb6nsegsq6brzunusms3g" xml:space="preserve">== Utiliser Wikidata pour mettre en œuvre une méthode de revue de littérature vivante, Conférence pour les méthodes pour les sciences sociales et les humanités, 9 et 10 Juillet 2026 (Aubervilliers, France) ==

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

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

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

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

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

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

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

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

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

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

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

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

In this project, we aim to test 4 hypothesis : 

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

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

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

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

We also have 2 assumptions :

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Recommandations : Contradictions are allowed in Wikidata. 

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

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

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

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

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

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

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

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

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

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

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

{{Wikidata entity link|Q30241068}}

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

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

{{Wikidata entity link|Q99527517}}

{{Wikidata entity link|Q30241068}}

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

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

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

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

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

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

=== Filter statements ===

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

=== Properties visualisation ===

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

=== Concepts visualisation ===

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

=== Mapping sources consensus ===

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

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

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

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

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

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

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

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

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
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== References ==
{{References}}</text>
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    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
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        <username>Dc.samizdat</username>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="86447" sha1="9n31u747kqzhxij2h1xjqsepaz8upvv" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32 skew&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In every 180° of isoclinic rotation each vertex circles a skew triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct skew triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

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

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

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

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

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

...  

{{Clear}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Clear}}

== The 5-cell 4-simplex ==

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

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

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

...

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

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

... 

{{Clear}}

== Conclusions ==

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

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>9n31u747kqzhxij2h1xjqsepaz8upvv</sha1>
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        <username>Dc.samizdat</username>
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      <text bytes="86486" sha1="8vxgrwo912uven7i4u6sqpixk39wz80" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32 skew&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the ''characteristic right rotation of the 24-cell'', also Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In every 180° of isoclinic rotation each vertex circles a skew triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct skew triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

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

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

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

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

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

...  

{{Clear}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{Clear}}

== The 5-cell 4-simplex ==

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

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

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

...

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

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

... 

{{Clear}}

== Conclusions ==

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

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="297" sha1="2h0vvjdlpir3nimrc59ca9jjhcqgpix" xml:space="preserve">https://en.wikiversity.org/wiki/Logic-Physics-Mind-Ontology

https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; neutrino gradient induced gravitation

https://en.wikiversity.org/wiki/Talk:Consciousness#everything_is_made_of_and_causal_with_consciousness_that_has_always_existed_everywhere</text>
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{{User talk:Mu301/Archive Index}}&lt;/noinclude&gt;

== Time to heal ==
As many of you know, I am an educator at Brown. I'd like to inform the community that the [[w:2025 Brown University shooting|2025 Brown University shooting]] occurred in my office building. I've given lectures in the auditorium where the tragedy happened. I'm unable to respond to on-wiki communication at this time. I need to provide and receive support from my colleagues and students who are struggling to cope with this horrific event. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:07, 16 December 2025 (UTC)

:We just prayed the ''[[w:janazah|janazah]]'' prayer for Mukhammad Aziz Umurzokov here in Virginia. Very heartbreaking. My thoughts are with you, the victims, and the Brown University community. Please take your time and heal, and I'm hoping that the preparator of this senseless violence is apprehend as soon as possible. All the best, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 01:24, 17 December 2025 (UTC)

::I greatly appreciate your support and concern. Thank you for reaching out to me in this difficult time. It is deeply meaningful to me. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:23, 17 December 2025 (UTC)</text>
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  <page>
    <title>Wikiversity talk:Inactivity policy</title>
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      <text bytes="4590" sha1="sjc836dz42t6pen268u59x4pweamujg" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)</text>
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      <text bytes="4616" sha1="qg8ev5ui3tcc5d0kzeekha1lekks2ni" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)</text>
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      <text bytes="40227" sha1="1dw706hzd04ptqeq90y7e6z7wf8s4mb" xml:space="preserve">:''This discusses a 2026-07-02 interview with [[w:Robert Corn-Revere|Robert Corn-Revere]] about the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]] (FIRE), including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-07-11 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;

&lt;!--[[File:2026-07-02 interview withRobert Corn-Revere about the Foundation for Individual Rights and Expression.webm|thumb|2026-07-02 interview with [[w:Robert Corn-Revere|Robert Corn-Revere]] about the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]].]]--&gt; 
&lt;!--[[File:2026-07-02 interview withRobert Corn-Revere about the Foundation for Individual Rights and Expression.ogg|thumb|29:00 mm:ss excerpts from the 2026-07-02 interview with [[w:Robert Corn-Revere|Robert Corn-Revere]] about the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]].]]--&gt; 

[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel at the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]] (FIRE), discusses their mission and activities and his recommendations for changes in media law to improve democracy. 

Corn-Revere has appeared on multiple lists of best lawyers since 2017. Some of his expertise is summarized in his (2021) ''The Mind of the Censor and the Eye of the Beholder : the First Amendment and the Censor's Dilemma'', building on his extensive research and experience including three earlier books.&lt;ref&gt;Corn-Revere, ed. (1997), Corn-Revere (1998), Zuckman et al. (1999).&lt;/ref&gt; 

==Maintaining principle in a time of polarization== 

Corn-Revere's 2026-03-11 keynote&lt;ref&gt;Corn-Revere (2026).&lt;/ref&gt; for the Delaware Inns of Court&lt;ref&gt;The [[w:American Inns of Court|American Inns of Court]] includes several in Delaware. They held a joint meeting [https://inns.innsofcourt.org/for-members/inns/the-randy-j-holland-delaware-workers-compensation-american-inn-of-court/meetings-2025-2026/ 2026-03-11 with a&quot;Free CLE programming followed by Keynote Speaker Robert Corn-Revere, Chief Counsel of Foundation of Individual Rights and Expression&quot;.]&lt;/ref&gt; described, &quot;Maintaining principle in a time of polarization&quot;. He began with, 

:''We live in interesting times.''

=== Interesting times === 

He then noted that the &quot;ancient Chinese curse, '[[w:May you live in interesting times|May you live in interesting times]]'&quot; is ''not'' Chinese: It has been traced to [[w:Austen Chamberlain|Austen Chamberlain]], brother of the British Prime Minister, [[w:Neville Chamberlain|Neville Chamberlain]]. Corn-Revere mentioned a vaguely related Chinese aphorism, that it is &quot;Better to be a dog in times of tranquility than a human in times of chaos.&quot; He then said, &quot;Some days, when I look at our two [[w:Australian Shepherd|Australian Shepherd]]s, [[w:Lenny Bruce|Lenny and Bruce]], and think I wouldn't mind trading places.&quot;&lt;ref&gt;Should we be surprised that a [[w:First Amendment to the United States Constitution|First Amendment]] lawyer would name his dogs after a comedian famously prosecuted for obscenity?&lt;/ref&gt; 

=== Learned Hand: &quot;Liberty lies in the hearts of men and women&quot; ===

He praised [[w:Learned Hand|Learned Hand]], who issued more than 3,000 legal opinions in his 52 years as a US federal judge, which as of 2014 had reportedly been quoted more often than those of any other US judge who had not served on the Supreme Court. He rose to fame outside the legal profession after describing &quot;The Spirit of Liberty&quot; to a crowd of 150,000 newly naturalized US citizens 1944-05-21. Hand said, {{quote|

Liberty lies in the hearts of men and women; when it dies there, no constitution, no law, no court can save it; no constitution, no law, no court can even do much to help it.}}

=== A culture of free expression === 

Corn-Revere said we need a culture of free expression. {{quote|

[I]t is essential to provide breathing space for all points of view, even for those we detest. Perhaps ''especially'' for those we detest.}}

He quoted Justice [[w:Oliver Wendell Holmes Jr.|Holmes]] from a 1929 dissent that supported the &quot;principle of free thought — not free thought for those who agree with us but freedom for the thought that we hate.&quot; 

=== FIRE's business Model === 

Corn-Revere continued that FIRE's president, [[w:Greg Lukianoff|Greg Lukianoff]], has been told they have &quot;a terrible business model, because it guarantees we are going to annoy every faction in American politics at one point or another. ... We take it as a point of pride. ... The corollary to our motto {{quote|

If it is protected, we will defend it}}

is this: {{quote|

If you are a censor, we will fight you.}}

=== Free speech is the best and perhaps only way to resolve disputes without violence or coercion ===

Corn-Revere continued, claiming that 

:[[q:Robert Corn-Revere#2026|'''''free speech is the best and perhaps only way to resolve disputes without violence or coercion.''''']]

We try &quot;to remind people that there is a possibility they might not be right about everything.&quot; Judge Hand said that the spirit of liberty “is not too sure that it is right.&quot; Former Justice [[w:Anthony Kennedy|Anthony Kennedy]] described the flip-side of this concept. “Self-assurance,” he wrote, “has always been the hallmark of a censor.” A mind filled with certainty is a closed mind. 

Corn-Revere continued, {{quote|

I think for most issues, you are more likely to be persuasive if your objective isn’t to explain to someone why they are dead wrong. Maybe it is enough just to learn a little more about why you disagree. ... [D]ialogue is the beginning of finding a peaceful resolution of any problem. ...

I believe the era we are living through now will make us stronger still. ... [W]e went through the [[w:First Red Scare|Red Scare of 1919]], World War II, the [[w:McCarthyism|McCarthy Era]], the [[w:Civil rights movement|civil rights]] and [[w:Opposition to United States involvement in the Vietnam War|antiwar demonstrations]] of the 60s and 70s, and since then an upheaval in the technology of how we communicate. Each was seen at the time as an existential crisis. ...

Is success guaranteed? No, of course not, and it never was. What is certain is that history is written by those who stand up, not by those who bend the knee to temporary power. ... 

[F]or those who find themselves confronted with frivolous litigation brought by figures of great influence, it means not settling what you know to be meritless claims. Doing so may at the time seem expedient and even rational. But ... it represents short-term thinking.

If you doubt this, just ask the law firms targeted with executive orders who settled with the administration, as distinguished from those who stood on their rights.}}

== Highlights ==

:''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.''

When asked for the most important things he would like to communicate to this audience, Corn-Revere said, {{quote|

I think the most important thing is to understand and respect and use your First Amendment rights. Surveys over the years have shown that most people don't really understand what the First Amendment is, or what role it plays. They know of it as sort of the freedom of speech amendment, but in fact the First Amendment protects five freedoms: 
# freedom of religion, 
# freedom of speech, 
# freedom of the press, 
# freedom of assembly, and 
# freedom to petition the government. 

Those five freedoms are central to who we are as Americans and how our system works, so it's important for people to understand that and to exercise their First Amendment rights.}}

Graves asked, &quot;What about the freedom to hear contrary perspectives?&quot; Corn-Revere  replied, &quot;That's implied by the freedom of speech, because, of course, if you don't have an audience, then the freedom of speech means little. So a corollary to those five freedoms is the freedom to listen to those competing views and to make up your mind as to which ones you're going to subscribe to.&quot;

=== What's FIRE ===

Graves asked, &quot;Give us a brief overview of the Foundation for Individual Rights and Expression.&quot; Corn-Revere replied, {{quote|

FIRE, the Foundation for Individual Rights and Expression, started out as an organization devoted exclusively to freedom of speech and due process issues on American college campuses. Originally, we were known as the &quot;Foundation for Individual Rights in Education&quot;. Over time the mission of FIRE was expanded beyond campus to cover free speech issues in general throughout society. We kept the same acronym, FIRE, but with a broadened mission. For those first 22 or 23 years, FIRE focused on free speech in higher education. 

During that time, when I was still in private practice, FIRE was one of my clients. We litigated a number of campus free speech cases on behalf of FIRE. I joined FIRE in 2023 after the mission was expanded to help with that and to focus on free speech issues throughout society. 

FIRE is expressly nonprofit and nonpartisan. We take First Amendment issues regardless of the point of view of the speaker, because that's the point of the First Amendment. The glory of the First Amendment is its neutrality between what your views are, or what your religion is, for that matter. If you're talking about the freedom of religion aspects of the First Amendment, it is to protect the freedom of the human mind.}}

=== The freedom to say unpopular things === 

Graves said, &quot;In the late 60s, I remember the [[w:American Civil Liberties Union|American Civil Liberties Union]] saying that in any country in the world you're free to say whatever you want -- as long as it's popular with those in power. It's only the freedom to say unpopular things that needs to be protected. Your comment?&quot; 

Corn-Revere agreed: &quot;I think that's always been true, that people don't object to your speech, and people in power in the government don't object to your speech, as long as you fall in line and parrot the government's message. But if you are a dissenter, if you are someone who is out of step, if you are an unpopular minority, that's when you need the protections of our Constitution, and in particular the protections of the First Amendment.&quot;

=== Funding === 

Graves asked, &quot;Where do you get your funding?&quot; Corn-Revere replied, We get most of our funding from individual donors: 75%. We also get foundation funding and other some large donors, but most of our funding comes from individuals.&quot;

Graves added, &quot;And from libertarians and the left ... .&quot; Corn-Revere agreed, &quot;across the political spectrum.&quot;

=== How does FIRE decide what to work on ? ===

Graves  asked, &quot;How does FIRE decide what what issues to focus on?&quot; Corn-Revere said, {{quote|

There's a lot of internal discussion about that. We try to deal not just with legal protections for the First Amendment, but we try and promote what we call a culture of free expression, and that is the willingness of people to accept the idea that freedom means people are going to disagree with you and to promote the idea that that kind of civil public discussion is really what this country is supposed to be about. 

We have various missions. 
* We have a ''litigation'' mission, and that's primarily where I work in taking cases where we think that the government in one way or another has infringed First Amendment rights. 
* We also have a ''public advocacy'' section that deals with trying to bring public pressure and attention on what we see as violations of the First Amendment. 
* We also have a ''campus rights advocacy'' group that works with specific restrictions on college campuses, harkening back to FIRE's origins, very much doing the same thing that FIRE had always done in that space. 
* And we also have a ''legislative'' group that works with trying to not only oppose legislation that we think is ill advised and contrary to our constitutional values, but also to promote legislative solutions that we think are more effective and better ways of dealing with those problems. 

So, across the board, in all aspects of society, we try and come up with speech protective solutions that deal both with legal issues and with public perceptions.}} 

=== JAWBONE Act===

Graves asked about legislation they are promoting. Corn-Revere said, {{quote|

We support something called the JAWBONE Act that has just been introduced.&lt;ref&gt;Cruz and Ron Wyden (2026a, b), Iodice (2026).&lt;/ref&gt; This is a law that would give real protections to prevent efforts by those in government from whichever party that try and use the power of leverage and the power of threats to try and suppress speech. A classic example that just may resonate with many of your listeners is the current chairman of the [[w:Federal Communications Commission|FCC]], [[w:Brendan Carr|Brendan Carr]], making threats against the [[w:American Broadcasting Company|ABC network]] because he didn't like something in a [[w:Jimmy Kimmel|Jimmy Kimmel]] monolog, and telling the network, do something about Kimmel, or there'll be more work for the FCC, the Federal Communications Commission, to do, 

:and we can do this the easy way or the hard way. 

I haven't heard language like that since I saw ''[[w:The Godfather|The Godfather]]''. 

The anti-JAWBONE act is designed to get at those kinds of regulatory threats and transgressions that we see as obvious First Amendment violations. This legislation, if it passes, will provide a federal cause of action for those who are the recipients of threats like that. It will also require those in government to be transparent. They can't use these sort of back room threats and pressure tactics. They have to be reported, which then could be the subject of litigation. 

We also are monitoring legislative efforts in the area of regulating social media and in regulating [[w:Artificial intelligence|AI]] and trying to make sure that the sort of techno-panic&lt;ref name=tp&gt;[https://www.urbandictionary.com/define.php?term=Techno-Panic Tecno-Panic]&lt;/ref&gt; that surrounds many of those technologies doesn't lead to more unwise legislation. ...}}

=== Jury decisions against Meta and Google === 

Graves asked, &quot;Comment on the recent jury trial decision against Meta in New Mexico&lt;ref&gt;McQue (2026).&lt;/ref&gt; and against Meta and Google in Los Angeles.&quot;&lt;ref&gt;Allyn (2026).&lt;/ref&gt; Corn-Revere said, &quot;I really don't know enough to speak intelligently about those decisions.&quot; 

Graves said, &quot;They were sued not for speech but for the impact.&quot;&lt;ref&gt;More on this is available in the 2026-05-06 interview in this series with [[w:Yael Eisenstat|Yael Eisenstat]] on [[Online platforms' effects on public health, safety and democracy]], accessed 2026-07-09.&lt;/ref&gt; Corn-Revere replied, {{quote|

Well, yeah, you can say that it's not for speech, and that it's for impact. But this is in line with when I talked about techno-panic earlier.&lt;ref name=tp/&gt; You can name a technology throughout history, and you will see the same kinds of arguments being made: It's addictive. The same argument was made about video games. The arguments that have been marshaled in the past have not been, I think, particularly persuasive usually. If you look back, people wonder what was all that fuss about on that technology. 

=== Social media === 

With social media, we're dealing with a new phenomenon that everyone has access to. I've often said with the internet in general, not just social media in particular, that 

:''the best thing about the internet is exactly the same thing as the worst thing about the internet, and that is that it allows instant access to a global audience for everybody.'' 

There's a lot of good that comes with that, and we forget how much good comes from it. And there's bad stuff that comes from it too, which all requires us to be more vigilant as free humans to try and make wise choices. 

The problem is that many people think that the first solution is to find someone to blame and bring a lawsuit, or to have government regulated in your name. 

The problem is, the more powerful the technology, the more important it is ''not'' to let the government have control over it, because ultimately that is control over all of us. Just because these issues come in a new package is not a reason to deviate from the principles that have protected us. 

That being said, there has been a lot of talk about young people and social media these days, but a lot of it is driven by really a one-sided narrative. So, for example, many people in the United States Court of Appeals for the Sixth Circuit just pointed to this in a recent decision,&lt;ref&gt;Marchese (2024).&lt;/ref&gt; pointed to a 2024 advisory of the Surgeon General, saying that social media can have deleterious effects for young people.&lt;ref&gt;Chapman (2024).&lt;/ref&gt; The problem with that is, if you actually read the Surgeon General's advisory,&lt;ref&gt;U. S. Surgeon General (2023).&lt;/ref&gt; you'll find that it says that they don't know what the evidence is, they have no definitive answer, and that what they do find this is far more nuanced: Social media has beneficial effects for young people, and it can have detrimental effects for some subset of young people if there's excessive use or whatever. But those issues and those problems are usually tied to some other underlying question. 

The benefits of social media have included things like making marginalized kids feel like they're not alone ... . So to say that we're going to impose either through a jury verdict or through legislation sort of a one size fits all solution ignores that aspect of what makes those technologies beneficial in the first place. 

A better solution than either trying to sue someone you want to blame for a problem or to find a legislative solution is to educate the public and educate kids. Certain states, New Jersey, for example, has passed a K-12 curriculum on media education, talking talks about online safety, and so on.&lt;ref&gt;More on media literacy education is included in the 2026-05-29 interview in this series with [[w:Nolan Higdon|Nolan Higdon]] on &quot;[[Let's agree to disagree and seek common ground]]&quot;.&lt;/ref&gt; 

So it begins with understanding the media that we use. That's especially important the more powerful the media, whether or not you're talking about online media and social media or AI: It begins with education and is going to be an inherently less restrictive means than handing over the reins of power to government. 

People forget that these technologies have so many more ways to give families control over how they want to consume the media in their households. ... You preserve freedom and you preserve people's choices if you empower them through knowledge and through technical tools to be able to handle these powerful technologies. The ability to determine what you want your kids to do experience online begins with you. There are protections built into the technologies, both at the network level and the individual device level. ... It does require people to actively understand them and use them. That's a feature, not a bug. That means that you need to be more proactive in dealing with these technologies. 

The real rap on kids and social media is that it is a distraction: It diverts their attention span. It makes them stay up too late. Two things: 

# One is there's no reason why anyone needs to have their phone at their desk in school. Schools can have control over that kind of thing by having phone lockers or having non-phone policies in class, so that people pay attention to actually what's happening in class. 
# As a parent ... if you think your kids are staying up too late, there's no reason why the phone needs to go with them to the bedroom. Make a decision about how you want to parent, and most of your problems will be solved.}}

=== Shouting ''fire'' in a crowded theater===

Graves said, &quot;Talk about '[[w:Shouting fire in a crowded theater|shouting ''fire'' in a crowded theater.]]'&quot; Corn-Revere replied, {{quote| 

The expression has become almost American folklore, &quot;You can't shout fire in a crowded theater.&quot; 

Well, in fact, you can. That aphorism comes from Justice [[w:Oliver Wendell Holmes Jr.|Oliver Wendell Holmes]], who was influential in the formation of American First Amendment law. In one of the decisions in which he actually had written in favor of government restrictions on speech, when during World War I there were very stringent restrictions on the ability to criticize the war effort through the [[w:Sedition Act of 1918|Sedition Act]] and the [[w:Espionage Act of 1917|Espionage Act]]. 

There were some 2000 prosecutions of American citizens during that period. Those kinds of restrictions on speech led to the formation of the [[w:American Civil Liberties Union|ACLU]]. 

In one of the early Supreme Court decisions, I believe it was the ''[[w:Schenck v. United States|Schenck]]'' decision upholding the restrictions, Holmes wrote that no one would argue that you can shout that free speech protects shouting fire in a crowded theater. The actual quote is to say 

:you cannot shout fire ''falsely'' in a theater in a crowded theater and cause a panic. 

Those elements of what he was writing about are part of our law to this day, but the protections have been expanded, so that you have to have that element of ''falsity'' and you have to have the element of ''immediacy'', for example, incitement to crime, which was one of the issues that came up in these cases. You cannot incite a crowd by inciting them to commit a crime immediately ''and under circumstances where that is likely to happen.''&lt;ref&gt;Corn-Revere mentioned three elements that need to be present for someone to be punished for &quot;shouting fire in a crowded theater&quot;: 
# Falsity,
# Immediacy, and 
# Circumstances that will likely bring about &quot;substantive evils that Congress has a right to prevent&quot;, [[w:Shouting fire in a crowded theater#Decision|in the words of Justice Holmes in the ''Schenck'' decision.]]&lt;/ref&gt;  

The first amendment, the right to speak is protected, but there are very narrow exceptions that require the government to clear very high hurdles before you are able to sanction that speech.  

But that aphorism, &quot;You can't shout fire any crowded theater&quot;, is one of those things that it is partially understood and often misused.}} 

=== Suing or being sued by major media? ===

Graves claimed, &quot;There are cases to be made that the increasing concentration of ownership of the major media shifts the terms of the debate. So I might want to sue the major media for stampeding, for example, the US into invading Iraq in 2003 on grounds that the senior executives should have known at the time were at best questionable and likely fraudulent. I was wondering if I could sue them for shouting fire at a crowded theater over that?&quot;   

Corn-Revere replied, {{quote|

That's a lesson that goes back to the founding of the country. It goes back to the [[w:Alien and Sedition Acts#Sedition Act|Sedition Act of 1798]] where Congress, with John Adams as president, passed a law that basically prohibited anyone from telling falsehoods about the president or about Congress, not the vice president, because that was at the time was Thomas Jefferson, and he was in opposition to all of that. That was one of the lessons of history that led to ultimately the way the First Amendment was interpreted and has developed. 

We saw that as our first big mistake under the new Constitution. When Thomas Jefferson succeeded Adams as president, one of the first acts was to let the Sedition Act lapse. It became a nullity, and all of the people who had been convicted under it were pardoned, and their fines were remitted.&lt;ref name=sedition&gt;Corn-Revere (2021, p. 181).&lt;/ref&gt; 

Jefferson later wrote to [[w:Abigail Adams|Abigail Adams]] that he considered that to be as palpable a constitutional violation as if the government had called forced people to fall down and worship a golden image.&lt;ref name=sedition/&gt;

That was one of the lessons of history that then got carried into the development of First Amendment law during the 20th century. If you go to the seminal decision of ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' in 1964, the Supreme Court recounted that example of the Sedition Act, and said, &quot;This is why we can't have the government being the arbiter of truth. We can't allow it through government legislation, and we can't allow it through private lawsuits.&quot; 

At the time the southern states were using defamation suits against the northern newspapers, and in particular ''[[w:The New York Times|The New York Times]]'', because they wanted to blunt the coverage of the civil rights movement. And they were being very successful at it. ''The New York Times'' were being sued for over $6 million. There were a large number of efforts beyond just the ''Times'', and ultimately the issue went to the Supreme Court. That's one of the reasons why our law of defamation and libel developed the way it did. ... We were not going to let the government become the arbiter of truth. ...}}

=== SLAPP ===

Graves asked about [[w:Strategic lawsuit against public participation|SLAPP lawsuits]]: &quot;For example, if I put together a mock trial like the [[w:Russell Tribunal|Russell Tribunal]] of the 1960s, under what circumstances could I be hit with a SLAPP lawsuit, strategic lawsuit against public participation?&quot; 

Corn-Revere replied, {{quote|

The acronym, SLAPP, is &quot;Strategic Litigation Against Public Participation&quot;. They are designed specifically like the campaign against ''The New York Times'' that I mentioned. They are designed specifically to silence the speaker and to generally use dubious legal arguments to be able to do that. 

In the 60s, when there was the campaign against ''The New York Times'' using defamation cases, libel law was very robust, and there were few constitutional limits on the ability to bring a libel case. Those constitutional limits have provided a measure of protection, but beyond that, most states now have passed what are called anti-SLAPP laws. They're designed to discourage the ability to bring frivolous cases in court. The anti-SLAPP movement is important. For many years, there's been an attempt to get a federal anti-SLAPP law, but that hasn't been successful so far. 

The case by President Trump against [[w:Ann Selzer|Ann Selzer]] and ''[[w:The Des Moines Register|The Des Moines Register]]'' is another example of a SLAPP suit, ... that is not just dubious but frivolous. There's absolutely no legal basis for for this case, and because it was brought in a state where there was not up until recently an anti-SLAPP law, it's being evaluated under normal legal standards. But under any standards the case is frivolous and should be dismissed.&lt;ref&gt;More on this case is available in the section on &quot;[[w:Ann Selzer#2024 presidential election|2024 presidential election]] in the Wikipedia article on [[w:Ann Selzer|Ann Selzer]], accessed 2026-07-08.&lt;/ref&gt; It is a significant problem. 

Your question reminds me of a question I often got when I was in private practice: My client would ask me, &quot;Can I get sued if I do x?&quot; The answer to that question is always, &quot;Yes. You can get sued under any circumstances.&quot; The important question is, &quot;If I get sued, will I lose?&quot;}}

Graves noted that, {{quote|

[[w:Lawrence Lessig|Lawrence Lessig]] in his book on ''[[w:Free Culture (book)|Free Culture]]'' described a college student in the early 2000s that had improved a search engine that was used internally to a university, and the [[w:Recording Industry Association of America|Recording Industry Association of America]] sued him. He had an uncle who was an attorney who told him that it'll cost you a million bucks to defend yourself, and even if you win, you will not recover your legal fees. So, they asked him, &quot;How much money do you have?&quot; And he said he'd save $12,000 for his college education.&lt;ref&gt;In the interview, Graves said it was $17,000, when it was actually only $12,000.&lt;/ref&gt; They took his savings and settled the case. Your comment?}}

Corn-Revere replied, {{quote|

That is the problem of SLAPP suits, and it's one of the reasons why most states have adopted protections against that. What those laws tend to do is to provide early dismissal of those kinds of cases. They can also provide a way to avoid [[w:Discovery (law)|discovery]], which is often the most important and expensive part of litigation, and get an early evaluation and a right of immediate appeal. 

If you don't automatically have that case at the outset dismissed, you can also recover legal fees in much of that legislation, and it's one of the reasons why it would be a very good idea to adopt federal legislation giving a federal anti-SLAPP law. ... 

Anti-SLAPP laws are not themselves the First Amendment, but they support First Amendment values: They provide an important procedural protection that reinforces those constitutional protections that the Constitution doesn't automatically provide on its own. 

[[w:Section 230|Section 230]], which many people have heard about, of the [[w:Communications Decency Act|Communications Decency Act]], is similar. I compare it often to an anti-SLAPP law in that it insulates the platforms that host third-party speech from being responsible legally for the speech posted by third parties. The reason it's important is that the internet and the platforms that we use to access the internet allow us, for the first time in human history, to access a global audience, and for that global audience to speak. 

The problem is, if those platforms that host millions and millions of users are responsible for everything that is said on their platforms, they're not going to host anybody. So that thing that is ultimately really beneficial also has its downsides. But the benefit of being able to speak to a global audience, to hear from a global audience, to have the world's knowledge at your fingertips, that depends on not saddling the platforms that host that material with an impossible burden of trying to vet everything that is on their platforms. Otherwise they're simply not going to take the chance on hosting it. 

Section 230 was one of the things that Congress actually got right, even if not entirely consciously, for how that would be important for internet speech.}}

=== The contributions of comedy to free speech and a First Amendment martyr ===

:''A portion of the video (37:20 - 42:07 mm:ss) that are not in the 29:00 mm:ss podcast discuss the contributions of two comedians to First Amendment law.''

Corn-Revere continued, {{quote| 

Section 230 was part of the [[w:Communications Decency Act|Communications Decency Act]], which was part of the [[w:Telecommunications Act of 1996|Telecommunications Act of 1996]]. The reason it's called the Communications Decency Act is that it was an amendment to another part of the Telecommunications Act that regulated the internet: It regulated the language that could be used online. The part of that legislation that was known as the Communications Decency Act was the part that tried to impose FCC indecency rules on the internet, those rules that came from the [[w:George Carlin|George Carlin]] monolog, &quot;[[w:Seven dirty words|The Seven Dirty Words]]&quot;, which [[w:Pacifica Foundation|Pacifica]] listeners should be well familiar with, because it was the [[w:FCC v. Pacifica Foundation|Pacifica case]] that said that those seven words could be regulated on broadcasting.  

It is the only time in history I will say that we had a legal standard developed by a stand-up comic. George Carlin had a bit called &quot;Filthy Words&quot;. He originally had various variations on it, but he originally started with the seven dirty words. That bit was used, broadcasted on [[w:WBAI|WBAI Pacifica]] in 1975. That became the subject of the test case that the FCC used to justify its indecency standard. It was very narrowly upheld to regulate broadcasting. Congress, when it was legislating for the first time with the internet in the mid 1990s said, &quot;Let's use that standard that we use for broadcasting. We're going to restrict what can be said on the internet.&quot;  

The other part, Section 230, got added as an amendment to that. The indecency regulations were struck down by the Supreme Court in ''[[w:Reno v. American Civil Liberties Union|Reno versus ACLU]]'' in 1997, but section 230 remained. And that is essentially the anti-SLAPP provision that gives you early dismissal of cases trying to hold platforms responsible for the third party speech. 

Very long answer, I know, to a very simple question, but that is the kind of legislation that can help and support the First Amendment, like anti-SLAPP legislation.}}

Graves then asked, &quot;You mentioned George Carlin. I read ... you have two dogs named 'Lenny' and 'Bruce'. Talk about that.&quot;

Corn-Revere laughed and replied, {{quote|

I'll start with the dogs. My dogs, Lenny and Bruce, are two [[w:Australian Shepherds|Australian Shepherds]]. They're about a year and a half old. We love them very much. They are two brothers from the same litter. We were trying to think of names for them, and then it just seemed just natural to call them 'Lenny' and 'Bruce'. 

Part of that is because one of the things that I've done during my career was to write a petition to [[w:George Pataki|Governor Pataki of New York]] to grant a posthumous pardon to the late, great comedian [[w:Lenny Bruce|Lenny Bruce]], who not so voluntarily became a First Amendment martyr, because he was prosecuted around the country in San Francisco, Los Angeles, Chicago, and New York of all places for obscenity for late night comedy routines in adults only night clubs. 

That's how much the law has changed since the mid 1960s: Now the language for many of Lenny Bruce's routines wouldn't really raise an eyebrow, but at the time it was shocking. Some tried to say obscene, and even under the developing law at the time, simply made no sense. 

But Lenny was convicted in New York, and that conviction remained on the books. And the authors of a book, ''The Trials of Lenny Bruce'',&lt;ref&gt;Collins and Skover (2002).&lt;/ref&gt; approached me -- they're friends of mine -- and asked if I'd be interested in getting involved in a drive to get a posthumous pardon in New York. In 2004 that was granted. We were kind of floored by that; we had no idea it would succeed. 

So when we got our dogs last year, we decided, in honor of Lenny Bruce, to name them Lenny and Bruce.}} 

=== In sum ===

Graves then noted that we were out of time and asked for parting comments. Corn-Revere replied, {{quote|

Just keep standing up for your rights. We are often told that right now we're going through a threatening period, whether or not we're going to be able to maintain our ability as a republic to protect our rights. We are sliding into authoritarianism. 

There are reasons to be concerned. 

But having gone through periods like this in our past, I try and take a longer view and see what we're going through in this moment as something that will strengthen us as a society. If you look through the 20th century until now, those things that have strengthened our ability to protect free speech the most are those things that seemed like existential threats at the time. I mentioned the prosecutions during World War I,&lt;ref&gt;prosecutions under the [[w:Espionage Act of 1917|Espionage Act of 1917]] and the [[w:Sedition Act of 1918|Sedition Act of 1918]], discussed above with the quotes from Justice Holmes.&lt;/ref&gt; for example. Or you can think about the [[w:McCarthyism|McCarthy era]], or the [[w:Civil rights movement|Civil Rights era from the 60s]], or the [[w:Opposition to United States involvement in the Vietnam War|anti-war movement]]. All of those crises led to what became our critical, pivotal protections for freedom of speech.}} 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Bobby Allyn (2026-03-25) &quot;Jury finds Meta and Google negligent in social media harms trial--&gt;{{cite Q|Q139572103}}
* &lt;!--Michelle Chapman (2024-06-17) &quot;Tobacco-like warning label for social media sought by US surgeon general who asks Congress to act--&gt;{{cite Q|Q140473041}}
* &lt;!--Ronald K.L. Collins and David M. Skover (2002) The trials of lenny bruce--&gt;{{cite Q|Q140472611}}
* &lt;!--Robert Corn-Revere, ed. (1997) Rationales &amp; rationalizations : regulating the electronic media--&gt;{{cite Q|Q140322854|editor=Robert Corn-Revere}}
* &lt;!--Robert Corn-Revere (1998-03-01) Unreasonable Access: Another Turn of the Regulatory Spin Cycle--&gt;{{cite Q|Q140323237}}
* &lt;!--Robert Corn-Revere (2021) The Mind of the Censor and the Eye of the Beholder : the First Amendment and the Censor's Dilemma--&gt;{{cite Q|Q140323606}}
* &lt;!--Robert Corn-Revere (2026-03-11) &quot;Maintaining principle in a time of polarization&quot;--&gt;{{cite Q|Q140325219}}
* &lt;!--Ted Cruz and Ron Wyden (2026-06-11a) Cruz, Wyden Introduce Legislation to Guard First Amendment Speech Rights Against Government Jawboning--&gt;{{cite Q|Q140467468|date=2026a}}
* &lt;!--Ted Cruz and Ron Wyden (2026-06-11b)  S. 4749 To amend the Communications Act of 1934 to create a Federal cause of action to address jawboning, and for other purposes--&gt;{{cite Q|Q140467733|date=2026b}} 
* &lt;!--Carolyn Iodice (2026-06-11)  FIRE backs JAWBONE Act to end backdoor censorship--&gt;{{cite Q|Q140469215}} 
* &lt;!--Lawrence Lessig (2004-03-25) Free Culture--&gt;{{cite Q|Q830526}}
* &lt;!--Chris Marchese (2024-01-05) NetChoice v. Yost (Ohio)--&gt;{{cite Q|Q140473137}}
* &lt;!--Katie McQue (2026-04-24) &quot; Meta ordered to pay $375m after being found liable in child exploitation case--&gt;{{cite Q|Q139572337}}
* &lt;!--U. S. Surgeon General (2023-05-23) Social Media and Youth Mental Health: The U.S. Surgeon General’s Advisory--&gt;{{cite Q|Q118978725|author=U. S. Surgeon General}}
* &lt;!--Harvey Zuckman, Robert Corn-Revere, Rob Frieden, Charles H. Kennedy (1999-01-01) Modern Communication Law--&gt;{{cite Q|Q140323259}}

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Constitutional Law]]
[[Category:Media reform to improve democracy]]
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      <text bytes="40147" sha1="ti1i053zjlz0scfhbgobwmbaqpo2yfg" xml:space="preserve">:''This discusses a 2026-07-02 interview with [[w:Robert Corn-Revere|Robert Corn-Revere]] about the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]] (FIRE), including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-07-11 to the fortnightly &quot;Media &amp; Democracy&quot; show&lt;ref name=M&amp;D&gt;&lt;!--Media &amp; Democracy--&gt;{{cite Q|Q127839818}}&lt;/ref&gt; syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]&lt;ref&gt;&lt;!--Pacifica Radio Network--&gt;{{cite Q|Q2045587}}&lt;/ref&gt; Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''&lt;ref&gt;&lt;!--list of Pacifica Radio stations and affiliates--&gt;{{cite Q|Q6593294}}&lt;/ref&gt;

:''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV&gt;The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.&lt;/ref&gt; and treating others with respect.''&lt;ref name=AGF&gt;[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|&quot;Don't assume things; be skeptical about everything.&quot;]] That's wise. However, we should still treat others with respect while being skeptical.&lt;/ref&gt;

[[File:Interview with Robert Corn-Revere about the Foundation for Individual Rights and Expression.webm|thumb|2026-07-02 interview with [[w:Robert Corn-Revere|Robert Corn-Revere]] about the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]].]]
[[File:Foundation for Individual Rights and Expression.ogg|thumb|29:00 mm:ss excepted from the 2026-07-02 interview with [[w:Robert Corn-Revere|Robert Corn-Revere]] about the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]].]]

[[w:Robert Corn-Revere|Robert Corn-Revere]], Chief Counsel at the [[w:Foundation for Individual Rights and Expression|Foundation for Individual Rights and Expression]] (FIRE), discusses their mission and activities and his recommendations for changes in media law to improve democracy. 

Corn-Revere has appeared on multiple lists of best lawyers since 2017. Some of his expertise is summarized in his (2021) ''The Mind of the Censor and the Eye of the Beholder : the First Amendment and the Censor's Dilemma'', building on his extensive research and experience including three earlier books.&lt;ref&gt;Corn-Revere, ed. (1997), Corn-Revere (1998), Zuckman et al. (1999).&lt;/ref&gt; 

==Maintaining principle in a time of polarization== 

Corn-Revere's 2026-03-11 keynote&lt;ref&gt;Corn-Revere (2026).&lt;/ref&gt; for the Delaware Inns of Court&lt;ref&gt;The [[w:American Inns of Court|American Inns of Court]] includes several in Delaware. They held a joint meeting [https://inns.innsofcourt.org/for-members/inns/the-randy-j-holland-delaware-workers-compensation-american-inn-of-court/meetings-2025-2026/ 2026-03-11 with a&quot;Free CLE programming followed by Keynote Speaker Robert Corn-Revere, Chief Counsel of Foundation of Individual Rights and Expression&quot;.]&lt;/ref&gt; described, &quot;Maintaining principle in a time of polarization&quot;. He began with, 

:''We live in interesting times.''

=== Interesting times === 

He then noted that the &quot;ancient Chinese curse, '[[w:May you live in interesting times|May you live in interesting times]]'&quot; is ''not'' Chinese: It has been traced to [[w:Austen Chamberlain|Austen Chamberlain]], brother of the British Prime Minister, [[w:Neville Chamberlain|Neville Chamberlain]]. Corn-Revere mentioned a vaguely related Chinese aphorism, that it is &quot;Better to be a dog in times of tranquility than a human in times of chaos.&quot; He then said, &quot;Some days, when I look at our two [[w:Australian Shepherd|Australian Shepherd]]s, [[w:Lenny Bruce|Lenny and Bruce]], and think I wouldn't mind trading places.&quot;&lt;ref&gt;Should we be surprised that a [[w:First Amendment to the United States Constitution|First Amendment]] lawyer would name his dogs after a comedian famously prosecuted for obscenity?&lt;/ref&gt; 

=== Learned Hand: &quot;Liberty lies in the hearts of men and women&quot; ===

He praised [[w:Learned Hand|Learned Hand]], who issued more than 3,000 legal opinions in his 52 years as a US federal judge, which as of 2014 had reportedly been quoted more often than those of any other US judge who had not served on the Supreme Court. He rose to fame outside the legal profession after describing &quot;The Spirit of Liberty&quot; to a crowd of 150,000 newly naturalized US citizens 1944-05-21. Hand said, {{quote|

Liberty lies in the hearts of men and women; when it dies there, no constitution, no law, no court can save it; no constitution, no law, no court can even do much to help it.}}

=== A culture of free expression === 

Corn-Revere said we need a culture of free expression. {{quote|

[I]t is essential to provide breathing space for all points of view, even for those we detest. Perhaps ''especially'' for those we detest.}}

He quoted Justice [[w:Oliver Wendell Holmes Jr.|Holmes]] from a 1929 dissent that supported the &quot;principle of free thought — not free thought for those who agree with us but freedom for the thought that we hate.&quot; 

=== FIRE's business Model === 

Corn-Revere continued that FIRE's president, [[w:Greg Lukianoff|Greg Lukianoff]], has been told they have &quot;a terrible business model, because it guarantees we are going to annoy every faction in American politics at one point or another. ... We take it as a point of pride. ... The corollary to our motto {{quote|

If it is protected, we will defend it}}

is this: {{quote|

If you are a censor, we will fight you.}}

=== Free speech is the best and perhaps only way to resolve disputes without violence or coercion ===

Corn-Revere continued, claiming that 

:[[q:Robert Corn-Revere#2026|'''''free speech is the best and perhaps only way to resolve disputes without violence or coercion.''''']]

We try &quot;to remind people that there is a possibility they might not be right about everything.&quot; Judge Hand said that the spirit of liberty “is not too sure that it is right.&quot; Former Justice [[w:Anthony Kennedy|Anthony Kennedy]] described the flip-side of this concept. “Self-assurance,” he wrote, “has always been the hallmark of a censor.” A mind filled with certainty is a closed mind. 

Corn-Revere continued, {{quote|

I think for most issues, you are more likely to be persuasive if your objective isn’t to explain to someone why they are dead wrong. Maybe it is enough just to learn a little more about why you disagree. ... [D]ialogue is the beginning of finding a peaceful resolution of any problem. ...

I believe the era we are living through now will make us stronger still. ... [W]e went through the [[w:First Red Scare|Red Scare of 1919]], World War II, the [[w:McCarthyism|McCarthy Era]], the [[w:Civil rights movement|civil rights]] and [[w:Opposition to United States involvement in the Vietnam War|antiwar demonstrations]] of the 60s and 70s, and since then an upheaval in the technology of how we communicate. Each was seen at the time as an existential crisis. ...

Is success guaranteed? No, of course not, and it never was. What is certain is that history is written by those who stand up, not by those who bend the knee to temporary power. ... 

[F]or those who find themselves confronted with frivolous litigation brought by figures of great influence, it means not settling what you know to be meritless claims. Doing so may at the time seem expedient and even rational. But ... it represents short-term thinking.

If you doubt this, just ask the law firms targeted with executive orders who settled with the administration, as distinguished from those who stood on their rights.}}

== Highlights ==

:''These excerpts from the interview are rushed, lightly edited for readability, and may not be in final form. The ultimate authority on what was said is the accompanying video.''

When asked for the most important things he would like to communicate to this audience, Corn-Revere said, {{quote|

I think the most important thing is to understand and respect and use your First Amendment rights. Surveys over the years have shown that most people don't really understand what the First Amendment is, or what role it plays. They know of it as sort of the freedom of speech amendment, but in fact the First Amendment protects five freedoms: 
# freedom of religion, 
# freedom of speech, 
# freedom of the press, 
# freedom of assembly, and 
# freedom to petition the government. 

Those five freedoms are central to who we are as Americans and how our system works, so it's important for people to understand that and to exercise their First Amendment rights.}}

Graves asked, &quot;What about the freedom to hear contrary perspectives?&quot; Corn-Revere  replied, &quot;That's implied by the freedom of speech, because, of course, if you don't have an audience, then the freedom of speech means little. So a corollary to those five freedoms is the freedom to listen to those competing views and to make up your mind as to which ones you're going to subscribe to.&quot;

=== What's FIRE ===

Graves asked, &quot;Give us a brief overview of the Foundation for Individual Rights and Expression.&quot; Corn-Revere replied, {{quote|

FIRE, the Foundation for Individual Rights and Expression, started out as an organization devoted exclusively to freedom of speech and due process issues on American college campuses. Originally, we were known as the &quot;Foundation for Individual Rights in Education&quot;. Over time the mission of FIRE was expanded beyond campus to cover free speech issues in general throughout society. We kept the same acronym, FIRE, but with a broadened mission. For those first 22 or 23 years, FIRE focused on free speech in higher education. 

During that time, when I was still in private practice, FIRE was one of my clients. We litigated a number of campus free speech cases on behalf of FIRE. I joined FIRE in 2023 after the mission was expanded to help with that and to focus on free speech issues throughout society. 

FIRE is expressly nonprofit and nonpartisan. We take First Amendment issues regardless of the point of view of the speaker, because that's the point of the First Amendment. The glory of the First Amendment is its neutrality between what your views are, or what your religion is, for that matter. If you're talking about the freedom of religion aspects of the First Amendment, it is to protect the freedom of the human mind.}}

=== The freedom to say unpopular things === 

Graves said, &quot;In the late 60s, I remember the [[w:American Civil Liberties Union|American Civil Liberties Union]] saying that in any country in the world you're free to say whatever you want -- as long as it's popular with those in power. It's only the freedom to say unpopular things that needs to be protected. Your comment?&quot; 

Corn-Revere agreed: &quot;I think that's always been true, that people don't object to your speech, and people in power in the government don't object to your speech, as long as you fall in line and parrot the government's message. But if you are a dissenter, if you are someone who is out of step, if you are an unpopular minority, that's when you need the protections of our Constitution, and in particular the protections of the First Amendment.&quot;

=== Funding === 

Graves asked, &quot;Where do you get your funding?&quot; Corn-Revere replied, We get most of our funding from individual donors: 75%. We also get foundation funding and other some large donors, but most of our funding comes from individuals.&quot;

Graves added, &quot;And from libertarians and the left ... .&quot; Corn-Revere agreed, &quot;across the political spectrum.&quot;

=== How does FIRE decide what to work on ? ===

Graves  asked, &quot;How does FIRE decide what what issues to focus on?&quot; Corn-Revere said, {{quote|

There's a lot of internal discussion about that. We try to deal not just with legal protections for the First Amendment, but we try and promote what we call a culture of free expression, and that is the willingness of people to accept the idea that freedom means people are going to disagree with you and to promote the idea that that kind of civil public discussion is really what this country is supposed to be about. 

We have various missions. 
* We have a ''litigation'' mission, and that's primarily where I work in taking cases where we think that the government in one way or another has infringed First Amendment rights. 
* We also have a ''public advocacy'' section that deals with trying to bring public pressure and attention on what we see as violations of the First Amendment. 
* We also have a ''campus rights advocacy'' group that works with specific restrictions on college campuses, harkening back to FIRE's origins, very much doing the same thing that FIRE had always done in that space. 
* And we also have a ''legislative'' group that works with trying to not only oppose legislation that we think is ill advised and contrary to our constitutional values, but also to promote legislative solutions that we think are more effective and better ways of dealing with those problems. 

So, across the board, in all aspects of society, we try and come up with speech protective solutions that deal both with legal issues and with public perceptions.}} 

=== JAWBONE Act===

Graves asked about legislation they are promoting. Corn-Revere said, {{quote|

We support something called the JAWBONE Act that has just been introduced.&lt;ref&gt;Cruz and Ron Wyden (2026a, b), Iodice (2026).&lt;/ref&gt; This is a law that would give real protections to prevent efforts by those in government from whichever party that try and use the power of leverage and the power of threats to try and suppress speech. A classic example that just may resonate with many of your listeners is the current chairman of the [[w:Federal Communications Commission|FCC]], [[w:Brendan Carr|Brendan Carr]], making threats against the [[w:American Broadcasting Company|ABC network]] because he didn't like something in a [[w:Jimmy Kimmel|Jimmy Kimmel]] monolog, and telling the network, do something about Kimmel, or there'll be more work for the FCC, the Federal Communications Commission, to do, 

:and we can do this the easy way or the hard way. 

I haven't heard language like that since I saw ''[[w:The Godfather|The Godfather]]''. 

The anti-JAWBONE act is designed to get at those kinds of regulatory threats and transgressions that we see as obvious First Amendment violations. This legislation, if it passes, will provide a federal cause of action for those who are the recipients of threats like that. It will also require those in government to be transparent. They can't use these sort of back room threats and pressure tactics. They have to be reported, which then could be the subject of litigation. 

We also are monitoring legislative efforts in the area of regulating social media and in regulating [[w:Artificial intelligence|AI]] and trying to make sure that the sort of techno-panic&lt;ref name=tp&gt;[https://www.urbandictionary.com/define.php?term=Techno-Panic Tecno-Panic]&lt;/ref&gt; that surrounds many of those technologies doesn't lead to more unwise legislation. ...}}

=== Jury decisions against Meta and Google === 

Graves asked, &quot;Comment on the recent jury trial decision against Meta in New Mexico&lt;ref&gt;McQue (2026).&lt;/ref&gt; and against Meta and Google in Los Angeles.&quot;&lt;ref&gt;Allyn (2026).&lt;/ref&gt; Corn-Revere said, &quot;I really don't know enough to speak intelligently about those decisions.&quot; 

Graves said, &quot;They were sued not for speech but for the impact.&quot;&lt;ref&gt;More on this is available in the 2026-05-06 interview in this series with [[w:Yael Eisenstat|Yael Eisenstat]] on [[Online platforms' effects on public health, safety and democracy]], accessed 2026-07-09.&lt;/ref&gt; Corn-Revere replied, {{quote|

Well, yeah, you can say that it's not for speech, and that it's for impact. But this is in line with when I talked about techno-panic earlier.&lt;ref name=tp/&gt; You can name a technology throughout history, and you will see the same kinds of arguments being made: It's addictive. The same argument was made about video games. The arguments that have been marshaled in the past have not been, I think, particularly persuasive usually. If you look back, people wonder what was all that fuss about on that technology. 

=== Social media === 

With social media, we're dealing with a new phenomenon that everyone has access to. I've often said with the internet in general, not just social media in particular, that 

:''the best thing about the internet is exactly the same thing as the worst thing about the internet, and that is that it allows instant access to a global audience for everybody.'' 

There's a lot of good that comes with that, and we forget how much good comes from it. And there's bad stuff that comes from it too, which all requires us to be more vigilant as free humans to try and make wise choices. 

The problem is that many people think that the first solution is to find someone to blame and bring a lawsuit, or to have government regulated in your name. 

The problem is, the more powerful the technology, the more important it is ''not'' to let the government have control over it, because ultimately that is control over all of us. Just because these issues come in a new package is not a reason to deviate from the principles that have protected us. 

That being said, there has been a lot of talk about young people and social media these days, but a lot of it is driven by really a one-sided narrative. So, for example, many people in the United States Court of Appeals for the Sixth Circuit just pointed to this in a recent decision,&lt;ref&gt;Marchese (2024).&lt;/ref&gt; pointed to a 2024 advisory of the Surgeon General, saying that social media can have deleterious effects for young people.&lt;ref&gt;Chapman (2024).&lt;/ref&gt; The problem with that is, if you actually read the Surgeon General's advisory,&lt;ref&gt;U. S. Surgeon General (2023).&lt;/ref&gt; you'll find that it says that they don't know what the evidence is, they have no definitive answer, and that what they do find this is far more nuanced: Social media has beneficial effects for young people, and it can have detrimental effects for some subset of young people if there's excessive use or whatever. But those issues and those problems are usually tied to some other underlying question. 

The benefits of social media have included things like making marginalized kids feel like they're not alone ... . So to say that we're going to impose either through a jury verdict or through legislation sort of a one size fits all solution ignores that aspect of what makes those technologies beneficial in the first place. 

A better solution than either trying to sue someone you want to blame for a problem or to find a legislative solution is to educate the public and educate kids. Certain states, New Jersey, for example, has passed a K-12 curriculum on media education, talking talks about online safety, and so on.&lt;ref&gt;More on media literacy education is included in the 2026-05-29 interview in this series with [[w:Nolan Higdon|Nolan Higdon]] on &quot;[[Let's agree to disagree and seek common ground]]&quot;.&lt;/ref&gt; 

So it begins with understanding the media that we use. That's especially important the more powerful the media, whether or not you're talking about online media and social media or AI: It begins with education and is going to be an inherently less restrictive means than handing over the reins of power to government. 

People forget that these technologies have so many more ways to give families control over how they want to consume the media in their households. ... You preserve freedom and you preserve people's choices if you empower them through knowledge and through technical tools to be able to handle these powerful technologies. The ability to determine what you want your kids to do experience online begins with you. There are protections built into the technologies, both at the network level and the individual device level. ... It does require people to actively understand them and use them. That's a feature, not a bug. That means that you need to be more proactive in dealing with these technologies. 

The real rap on kids and social media is that it is a distraction: It diverts their attention span. It makes them stay up too late. Two things: 

# One is there's no reason why anyone needs to have their phone at their desk in school. Schools can have control over that kind of thing by having phone lockers or having non-phone policies in class, so that people pay attention to actually what's happening in class. 
# As a parent ... if you think your kids are staying up too late, there's no reason why the phone needs to go with them to the bedroom. Make a decision about how you want to parent, and most of your problems will be solved.}}

=== Shouting ''fire'' in a crowded theater===

Graves said, &quot;Talk about '[[w:Shouting fire in a crowded theater|shouting ''fire'' in a crowded theater.]]'&quot; Corn-Revere replied, {{quote| 

The expression has become almost American folklore, &quot;You can't shout fire in a crowded theater.&quot; 

Well, in fact, you can. That aphorism comes from Justice [[w:Oliver Wendell Holmes Jr.|Oliver Wendell Holmes]], who was influential in the formation of American First Amendment law. In one of the decisions in which he actually had written in favor of government restrictions on speech, when during World War I there were very stringent restrictions on the ability to criticize the war effort through the [[w:Sedition Act of 1918|Sedition Act]] and the [[w:Espionage Act of 1917|Espionage Act]]. 

There were some 2000 prosecutions of American citizens during that period. Those kinds of restrictions on speech led to the formation of the [[w:American Civil Liberties Union|ACLU]]. 

In one of the early Supreme Court decisions, I believe it was the ''[[w:Schenck v. United States|Schenck]]'' decision upholding the restrictions, Holmes wrote that no one would argue that you can shout that free speech protects shouting fire in a crowded theater. The actual quote is to say 

:you cannot shout fire ''falsely'' in a theater in a crowded theater and cause a panic. 

Those elements of what he was writing about are part of our law to this day, but the protections have been expanded, so that you have to have that element of ''falsity'' and you have to have the element of ''immediacy'', for example, incitement to crime, which was one of the issues that came up in these cases. You cannot incite a crowd by inciting them to commit a crime immediately ''and under circumstances where that is likely to happen.''&lt;ref&gt;Corn-Revere mentioned three elements that need to be present for someone to be punished for &quot;shouting fire in a crowded theater&quot;: 
# Falsity,
# Immediacy, and 
# Circumstances that will likely bring about &quot;substantive evils that Congress has a right to prevent&quot;, [[w:Shouting fire in a crowded theater#Decision|in the words of Justice Holmes in the ''Schenck'' decision.]]&lt;/ref&gt;  

The first amendment, the right to speak is protected, but there are very narrow exceptions that require the government to clear very high hurdles before you are able to sanction that speech.  

But that aphorism, &quot;You can't shout fire any crowded theater&quot;, is one of those things that it is partially understood and often misused.}} 

=== Suing or being sued by major media? ===

Graves claimed, &quot;There are cases to be made that the increasing concentration of ownership of the major media shifts the terms of the debate. So I might want to sue the major media for stampeding, for example, the US into invading Iraq in 2003 on grounds that the senior executives should have known at the time were at best questionable and likely fraudulent. I was wondering if I could sue them for shouting fire at a crowded theater over that?&quot;   

Corn-Revere replied, {{quote|

That's a lesson that goes back to the founding of the country. It goes back to the [[w:Alien and Sedition Acts#Sedition Act|Sedition Act of 1798]] where Congress, with John Adams as president, passed a law that basically prohibited anyone from telling falsehoods about the president or about Congress, not the vice president, because that was at the time was Thomas Jefferson, and he was in opposition to all of that. That was one of the lessons of history that led to ultimately the way the First Amendment was interpreted and has developed. 

We saw that as our first big mistake under the new Constitution. When Thomas Jefferson succeeded Adams as president, one of the first acts was to let the Sedition Act lapse. It became a nullity, and all of the people who had been convicted under it were pardoned, and their fines were remitted.&lt;ref name=sedition&gt;Corn-Revere (2021, p. 181).&lt;/ref&gt; 

Jefferson later wrote to [[w:Abigail Adams|Abigail Adams]] that he considered that to be as palpable a constitutional violation as if the government had called forced people to fall down and worship a golden image.&lt;ref name=sedition/&gt;

That was one of the lessons of history that then got carried into the development of First Amendment law during the 20th century. If you go to the seminal decision of ''[[w:New York Times Co. v. Sullivan|New York Times v. Sullivan]]'' in 1964, the Supreme Court recounted that example of the Sedition Act, and said, &quot;This is why we can't have the government being the arbiter of truth. We can't allow it through government legislation, and we can't allow it through private lawsuits.&quot; 

At the time the southern states were using defamation suits against the northern newspapers, and in particular ''[[w:The New York Times|The New York Times]]'', because they wanted to blunt the coverage of the civil rights movement. And they were being very successful at it. ''The New York Times'' were being sued for over $6 million. There were a large number of efforts beyond just the ''Times'', and ultimately the issue went to the Supreme Court. That's one of the reasons why our law of defamation and libel developed the way it did. ... We were not going to let the government become the arbiter of truth. ...}}

=== SLAPP ===

Graves asked about [[w:Strategic lawsuit against public participation|SLAPP lawsuits]]: &quot;For example, if I put together a mock trial like the [[w:Russell Tribunal|Russell Tribunal]] of the 1960s, under what circumstances could I be hit with a SLAPP lawsuit, strategic lawsuit against public participation?&quot; 

Corn-Revere replied, {{quote|

The acronym, SLAPP, is &quot;Strategic Litigation Against Public Participation&quot;. They are designed specifically like the campaign against ''The New York Times'' that I mentioned. They are designed specifically to silence the speaker and to generally use dubious legal arguments to be able to do that. 

In the 60s, when there was the campaign against ''The New York Times'' using defamation cases, libel law was very robust, and there were few constitutional limits on the ability to bring a libel case. Those constitutional limits have provided a measure of protection, but beyond that, most states now have passed what are called anti-SLAPP laws. They're designed to discourage the ability to bring frivolous cases in court. The anti-SLAPP movement is important. For many years, there's been an attempt to get a federal anti-SLAPP law, but that hasn't been successful so far. 

The case by President Trump against [[w:Ann Selzer|Ann Selzer]] and ''[[w:The Des Moines Register|The Des Moines Register]]'' is another example of a SLAPP suit, ... that is not just dubious but frivolous. There's absolutely no legal basis for for this case, and because it was brought in a state where there was not up until recently an anti-SLAPP law, it's being evaluated under normal legal standards. But under any standards the case is frivolous and should be dismissed.&lt;ref&gt;More on this case is available in the section on &quot;[[w:Ann Selzer#2024 presidential election|2024 presidential election]] in the Wikipedia article on [[w:Ann Selzer|Ann Selzer]], accessed 2026-07-08.&lt;/ref&gt; It is a significant problem. 

Your question reminds me of a question I often got when I was in private practice: My client would ask me, &quot;Can I get sued if I do x?&quot; The answer to that question is always, &quot;Yes. You can get sued under any circumstances.&quot; The important question is, &quot;If I get sued, will I lose?&quot;}}

Graves noted that, {{quote|

[[w:Lawrence Lessig|Lawrence Lessig]] in his book on ''[[w:Free Culture (book)|Free Culture]]'' described a college student in the early 2000s that had improved a search engine that was used internally to a university, and the [[w:Recording Industry Association of America|Recording Industry Association of America]] sued him. He had an uncle who was an attorney who told him that it'll cost you a million bucks to defend yourself, and even if you win, you will not recover your legal fees. So, they asked him, &quot;How much money do you have?&quot; And he said he'd save $12,000 for his college education.&lt;ref&gt;In the interview, Graves said it was $17,000, when it was actually only $12,000.&lt;/ref&gt; They took his savings and settled the case. Your comment?}}

Corn-Revere replied, {{quote|

That is the problem of SLAPP suits, and it's one of the reasons why most states have adopted protections against that. What those laws tend to do is to provide early dismissal of those kinds of cases. They can also provide a way to avoid [[w:Discovery (law)|discovery]], which is often the most important and expensive part of litigation, and get an early evaluation and a right of immediate appeal. 

If you don't automatically have that case at the outset dismissed, you can also recover legal fees in much of that legislation, and it's one of the reasons why it would be a very good idea to adopt federal legislation giving a federal anti-SLAPP law. ... 

Anti-SLAPP laws are not themselves the First Amendment, but they support First Amendment values: They provide an important procedural protection that reinforces those constitutional protections that the Constitution doesn't automatically provide on its own. 

[[w:Section 230|Section 230]], which many people have heard about, of the [[w:Communications Decency Act|Communications Decency Act]], is similar. I compare it often to an anti-SLAPP law in that it insulates the platforms that host third-party speech from being responsible legally for the speech posted by third parties. The reason it's important is that the internet and the platforms that we use to access the internet allow us, for the first time in human history, to access a global audience, and for that global audience to speak. 

The problem is, if those platforms that host millions and millions of users are responsible for everything that is said on their platforms, they're not going to host anybody. So that thing that is ultimately really beneficial also has its downsides. But the benefit of being able to speak to a global audience, to hear from a global audience, to have the world's knowledge at your fingertips, that depends on not saddling the platforms that host that material with an impossible burden of trying to vet everything that is on their platforms. Otherwise they're simply not going to take the chance on hosting it. 

Section 230 was one of the things that Congress actually got right, even if not entirely consciously, for how that would be important for internet speech.}}

=== The contributions of comedy to free speech and a First Amendment martyr ===

:''A portion of the video (37:20 - 42:07 mm:ss) that are not in the 29:00 mm:ss podcast discuss the contributions of two comedians to First Amendment law.''

Corn-Revere continued, {{quote| 

Section 230 was part of the [[w:Communications Decency Act|Communications Decency Act]], which was part of the [[w:Telecommunications Act of 1996|Telecommunications Act of 1996]]. The reason it's called the Communications Decency Act is that it was an amendment to another part of the Telecommunications Act that regulated the internet: It regulated the language that could be used online. The part of that legislation that was known as the Communications Decency Act was the part that tried to impose FCC indecency rules on the internet, those rules that came from the [[w:George Carlin|George Carlin]] monolog, &quot;[[w:Seven dirty words|The Seven Dirty Words]]&quot;, which [[w:Pacifica Foundation|Pacifica]] listeners should be well familiar with, because it was the [[w:FCC v. Pacifica Foundation|Pacifica case]] that said that those seven words could be regulated on broadcasting.  

It is the only time in history I will say that we had a legal standard developed by a stand-up comic. George Carlin had a bit called &quot;Filthy Words&quot;. He originally had various variations on it, but he originally started with the seven dirty words. That bit was used, broadcasted on [[w:WBAI|WBAI Pacifica]] in 1975. That became the subject of the test case that the FCC used to justify its indecency standard. It was very narrowly upheld to regulate broadcasting. Congress, when it was legislating for the first time with the internet in the mid 1990s said, &quot;Let's use that standard that we use for broadcasting. We're going to restrict what can be said on the internet.&quot;  

The other part, Section 230, got added as an amendment to that. The indecency regulations were struck down by the Supreme Court in ''[[w:Reno v. American Civil Liberties Union|Reno versus ACLU]]'' in 1997, but section 230 remained. And that is essentially the anti-SLAPP provision that gives you early dismissal of cases trying to hold platforms responsible for the third party speech. 

Very long answer, I know, to a very simple question, but that is the kind of legislation that can help and support the First Amendment, like anti-SLAPP legislation.}}

Graves then asked, &quot;You mentioned George Carlin. I read ... you have two dogs named 'Lenny' and 'Bruce'. Talk about that.&quot;

Corn-Revere laughed and replied, {{quote|

I'll start with the dogs. My dogs, Lenny and Bruce, are two [[w:Australian Shepherds|Australian Shepherds]]. They're about a year and a half old. We love them very much. They are two brothers from the same litter. We were trying to think of names for them, and then it just seemed just natural to call them 'Lenny' and 'Bruce'. 

Part of that is because one of the things that I've done during my career was to write a petition to [[w:George Pataki|Governor Pataki of New York]] to grant a posthumous pardon to the late, great comedian [[w:Lenny Bruce|Lenny Bruce]], who not so voluntarily became a First Amendment martyr, because he was prosecuted around the country in San Francisco, Los Angeles, Chicago, and New York of all places for obscenity for late night comedy routines in adults only night clubs. 

That's how much the law has changed since the mid 1960s: Now the language for many of Lenny Bruce's routines wouldn't really raise an eyebrow, but at the time it was shocking. Some tried to say obscene, and even under the developing law at the time, simply made no sense. 

But Lenny was convicted in New York, and that conviction remained on the books. And the authors of a book, ''The Trials of Lenny Bruce'',&lt;ref&gt;Collins and Skover (2002).&lt;/ref&gt; approached me -- they're friends of mine -- and asked if I'd be interested in getting involved in a drive to get a posthumous pardon in New York. In 2004 that was granted. We were kind of floored by that; we had no idea it would succeed. 

So when we got our dogs last year, we decided, in honor of Lenny Bruce, to name them Lenny and Bruce.}} 

=== In sum ===

Graves then noted that we were out of time and asked for parting comments. Corn-Revere replied, {{quote|

Just keep standing up for your rights. We are often told that right now we're going through a threatening period, whether or not we're going to be able to maintain our ability as a republic to protect our rights. We are sliding into authoritarianism. 

There are reasons to be concerned. 

But having gone through periods like this in our past, I try and take a longer view and see what we're going through in this moment as something that will strengthen us as a society. If you look through the 20th century until now, those things that have strengthened our ability to protect free speech the most are those things that seemed like existential threats at the time. I mentioned the prosecutions during World War I,&lt;ref&gt;prosecutions under the [[w:Espionage Act of 1917|Espionage Act of 1917]] and the [[w:Sedition Act of 1918|Sedition Act of 1918]], discussed above with the quotes from Justice Holmes.&lt;/ref&gt; for example. Or you can think about the [[w:McCarthyism|McCarthy era]], or the [[w:Civil rights movement|Civil Rights era from the 60s]], or the [[w:Opposition to United States involvement in the Vietnam War|anti-war movement]]. All of those crises led to what became our critical, pivotal protections for freedom of speech.}} 

== The need for media reform to improve democracy == 

This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media &amp; Democracy lessons for the future]]. 

==Discussion ==

:''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]&lt;ref name=NPOV/&gt; and treating others with respect.&lt;ref name=AGF/&gt;]''

== Notes ==
{{reflist}}

== Bibliography ==

* &lt;!--Bobby Allyn (2026-03-25) &quot;Jury finds Meta and Google negligent in social media harms trial--&gt;{{cite Q|Q139572103}}
* &lt;!--Michelle Chapman (2024-06-17) &quot;Tobacco-like warning label for social media sought by US surgeon general who asks Congress to act--&gt;{{cite Q|Q140473041}}
* &lt;!--Ronald K.L. Collins and David M. Skover (2002) The trials of lenny bruce--&gt;{{cite Q|Q140472611}}
* &lt;!--Robert Corn-Revere, ed. (1997) Rationales &amp; rationalizations : regulating the electronic media--&gt;{{cite Q|Q140322854|editor=Robert Corn-Revere}}
* &lt;!--Robert Corn-Revere (1998-03-01) Unreasonable Access: Another Turn of the Regulatory Spin Cycle--&gt;{{cite Q|Q140323237}}
* &lt;!--Robert Corn-Revere (2021) The Mind of the Censor and the Eye of the Beholder : the First Amendment and the Censor's Dilemma--&gt;{{cite Q|Q140323606}}
* &lt;!--Robert Corn-Revere (2026-03-11) &quot;Maintaining principle in a time of polarization&quot;--&gt;{{cite Q|Q140325219}}
* &lt;!--Ted Cruz and Ron Wyden (2026-06-11a) Cruz, Wyden Introduce Legislation to Guard First Amendment Speech Rights Against Government Jawboning--&gt;{{cite Q|Q140467468|date=2026a}}
* &lt;!--Ted Cruz and Ron Wyden (2026-06-11b)  S. 4749 To amend the Communications Act of 1934 to create a Federal cause of action to address jawboning, and for other purposes--&gt;{{cite Q|Q140467733|date=2026b}} 
* &lt;!--Carolyn Iodice (2026-06-11)  FIRE backs JAWBONE Act to end backdoor censorship--&gt;{{cite Q|Q140469215}} 
* &lt;!--Lawrence Lessig (2004-03-25) Free Culture--&gt;{{cite Q|Q830526}}
* &lt;!--Chris Marchese (2024-01-05) NetChoice v. Yost (Ohio)--&gt;{{cite Q|Q140473137}}
* &lt;!--Katie McQue (2026-04-24) &quot; Meta ordered to pay $375m after being found liable in child exploitation case--&gt;{{cite Q|Q139572337}}
* &lt;!--U. S. Surgeon General (2023-05-23) Social Media and Youth Mental Health: The U.S. Surgeon General’s Advisory--&gt;{{cite Q|Q118978725|author=U. S. Surgeon General}}
* &lt;!--Harvey Zuckman, Robert Corn-Revere, Rob Frieden, Charles H. Kennedy (1999-01-01) Modern Communication Law--&gt;{{cite Q|Q140323259}}

[[Category:Media]]
[[Category:News]]
[[Category:Democracy]]
[[Category:Politics]]
[[Category:Media literacy]]
[[Category:Constitutional Law]]
[[Category:Media reform to improve democracy]]
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      <text bytes="53288" sha1="sj17eduvfafxs8r7ogn8uh0rzcxmxvh" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Things Yet to Do ====

# Look at structures of other wiki courses
## School vs Course vs Lesson vs Activity?
## Multiple pages per course? Multiple pages per lesson?
# Gather Wikimedia photos for use

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Add more about legal obligation of bartenders (and a source for students to look up their area) under Safety
# Add a section about not taking abuse or being a punching bag under Safety

==== Course Outline ====

# What is bartending?/Bartending basics
## Sources of alcohol
## Types of alcohol 
### Spirits portal
## Tools and their uses
## Safety
# Mixing Drinks
## Preparation
## Glasses
## All the basic recipes
## Taxonomy of cocktails
# Hospitality
## Steps of Service
## Serving Drinks to People
## Tasting Drinks/Developing Taste
## Internal Hospitality
# Industry
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
# References
## [[wikipedia:Pub|Public House]] page
## [[wikipedia:Portal:Liquor|Liquor Portal]]
## https://en.wikibooks.org/wiki/Bartending

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for anyone wanting to learn the trade of bartending, whether that be for a job or at home. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality and Industry

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10oz of liquid.)
#Bottle Spout
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]
In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools and Their Uses
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine.

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====

==== Soju ====

==== Liqueurs ====

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Preparation and Skills
# Glasses
# All the basic recipes
# Taxonomy of cocktails
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks for uploading this picture though.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed overtop the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). 
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

== Glassware ==
The last bit of knowledge you need before making cocktails is about the glasses they come in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses. When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. That being said, 20 is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple.

=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

{{RoundBoxTop|theme=7}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' lemon twist

'''Ingredients'''

1 ½ oz Citron Vodka

⅓ oz fresh lime juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz cranberry juice cocktail

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
{{RoundBoxBottom}}

=== Margarita ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

1/2 oz Agave Nectar

1 oz Lime Juice

1/2 oz Triple Sec

2 oz Tequilla

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
{{RoundBoxBottom}}

=== Paloma ===
{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

1/2 oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
{{RoundBoxTop|theme=8}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

1 oz Lemon Juice

1/2 tsp. Sugar or Simple Syrup

2 oz Whiskey

'''Shake''' ingredients with ice. Strain. Garnish.
{{RoundBoxBottom}}

=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz vodka

1/2 oz fresh lime juice

3 oz ginger beer

'''Combine''' all ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
{{RoundBoxBottom}}

=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=8}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

5 to 6 mint leaves

1 oz fresh lime juice

2 tsp sugar

1 1/2 oz white rum

Soda water

'''Muddle''' the mint, lime juice and sugar until well mixed but not mushy. Add the rum and fill the glass with ice. Top with soda water. Garnish.
{{RoundBoxBottom}}

=== Daiquiri ===
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'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 1/2 oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' all ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
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'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

1/2 oz Lemon Juice

1/2 oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
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'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 1/2 oz Gin

1/2 oz Lime Juice

1/2 oz Simple Syrup

'''Shake''' Gin, Lime Juice and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 1/2 oz Rye Whiskey

1/2 oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

1/2 oz Orange Curaçao

1/2 oz Orgeat

1/2 oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
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'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 1/2 oz Coconut Cream

1 oz Pineapple Juice

1/2 oz Lime Juice

1 1/2 oz Malibu Rum

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
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'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

1/2 Lemon, juiced

5 oz Tomato Juice

2 oz Vodka

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

2 oz Whiskey

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
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'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

1/2 oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

1/2 oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

1/2 oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill'' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
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= Hospitality =

# Steps of Service
# Serving Drinks to People
# Tasting Drinks/Developing Taste
# Internal Hospitality</text>
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      <text bytes="53700" sha1="m5iu69yq7lcbjlcr1xvz0yi66sl4wxg" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Things Yet to Do ====

# Look at structures of other wiki courses
## School vs Course vs Lesson vs Activity?
## Multiple pages per course? Multiple pages per lesson?
# Gather Wikimedia photos for use

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Add more about legal obligation of bartenders (and a source for students to look up their area) under Safety
# Add a section about not taking abuse or being a punching bag under Safety
# Add course objectives to Introduction

==== Course Outline ====

# What is bartending?/Bartending basics
## Sources of alcohol
## Types of alcohol 
### Spirits portal
## Tools and their uses
## Safety
# Mixing Drinks
## Preparation
## Glasses
## All the basic recipes
## Taxonomy of cocktails
# Hospitality
## Steps of Service
## Serving Drinks to People
## Tasting Drinks/Developing Taste
## Internal Hospitality
# Industry
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
# References
## [[wikipedia:Pub|Public House]] page
## [[wikipedia:Portal:Liquor|Liquor Portal]]
## https://en.wikibooks.org/wiki/Bartending

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults wanting to start bartending in a professional establishment. Adults wanting to learn about bartending as a trade or to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools and Their Uses
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine.

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====

==== Soju ====

==== Liqueurs ====

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Preparation and Skills
# Glasses
# All the basic recipes
# Taxonomy of cocktails
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks for uploading this picture though.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed overtop the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). 
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

== Glassware ==
The last bit of knowledge you need before making cocktails is about the glasses they come in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses. When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. That being said, 20 is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple.

=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

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'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' lemon twist

'''Ingredients'''

1 ½ oz Citron Vodka

⅓ oz fresh lime juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz cranberry juice cocktail

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

1/2 oz Agave Nectar

1 oz Lime Juice

1/2 oz Triple Sec

2 oz Tequilla

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
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'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

1/2 oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
{{RoundBoxTop|theme=8}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

1 oz Lemon Juice

1/2 tsp. Sugar or Simple Syrup

2 oz Whiskey

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz vodka

1/2 oz fresh lime juice

3 oz ginger beer

'''Combine''' all ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=8}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

5 to 6 mint leaves

1 oz fresh lime juice

2 tsp sugar

1 1/2 oz white rum

Soda water

'''Muddle''' the mint, lime juice and sugar until well mixed but not mushy. Add the rum and fill the glass with ice. Top with soda water. Garnish.
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=== Daiquiri ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 1/2 oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' all ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

1/2 oz Lemon Juice

1/2 oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
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'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 1/2 oz Gin

1/2 oz Lime Juice

1/2 oz Simple Syrup

'''Shake''' Gin, Lime Juice and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
{{RoundBoxTop|theme=12}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 1/2 oz Rye Whiskey

1/2 oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

1/2 oz Orange Curaçao

1/2 oz Orgeat

1/2 oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
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'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 1/2 oz Coconut Cream

1 oz Pineapple Juice

1/2 oz Lime Juice

1 1/2 oz Malibu Rum

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
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'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

1/2 Lemon, juiced

5 oz Tomato Juice

2 oz Vodka

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

2 oz Whiskey

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
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'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

1/2 oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

1/2 oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

1/2 oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill'' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
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= Hospitality =

# Steps of Service
# Serving Drinks to People
# Tasting Drinks/Developing Taste
# Internal Hospitality</text>
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      <text bytes="53271" sha1="l6ooz7zgi8zw40il7bh53r9dgimqa1m" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Things Yet to Do ====

# Look at structures of other wiki courses
## School vs Course vs Lesson vs Activity?
## Multiple pages per course? Multiple pages per lesson?
# Gather Wikimedia photos for use

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Add more about legal obligation of bartenders (and a source for students to look up their area) under Safety
# Add a section about not taking abuse or being a punching bag under Safety
# Add course objectives to Introduction

==== Course Outline ====

# What is bartending?/Bartending basics
## Sources of alcohol
## Types of alcohol 
### Spirits portal
## Tools and their uses
## Safety
# Mixing Drinks
## Preparation
## Glasses
## All the basic recipes
## Taxonomy of cocktails
# Hospitality
## Steps of Service
## Serving Drinks to People
## Tasting Drinks/Developing Taste
## Internal Hospitality
# Industry
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
# References
## [[wikipedia:Pub|Public House]] page
## [[wikipedia:Portal:Liquor|Liquor Portal]]
## https://en.wikibooks.org/wiki/Bartending

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults wanting to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending as a trade or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools and Their Uses
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine.

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====

==== Soju ====

==== Liqueurs ====

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Preparation and Skills
# Glasses
# All the basic recipes
# Taxonomy of cocktails
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks for uploading this picture though.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed overtop the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). 
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

== Glassware ==
The last bit of knowledge you need before making cocktails is about the glasses they come in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses. When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Typology of Cocktails (Cockail Families) ==

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. That being said, 20 is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple.

=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

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'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
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'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' lemon twist

'''Ingredients'''

1 ½ oz Citron Vodka

⅓ oz fresh lime juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz cranberry juice cocktail

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

1/2 oz Agave Nectar

1 oz Lime Juice

1/2 oz Triple Sec

2 oz Tequilla

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

1/2 oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
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'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

1 oz Lemon Juice

1/2 tsp. Sugar or Simple Syrup

2 oz Whiskey

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz vodka

1/2 oz fresh lime juice

3 oz ginger beer

'''Combine''' all ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=8}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

5 to 6 mint leaves

1 oz fresh lime juice

2 tsp sugar

1 1/2 oz white rum

Soda water

'''Muddle''' the mint, lime juice and sugar until well mixed but not mushy. Add the rum and fill the glass with ice. Top with soda water. Garnish.
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=== Daiquiri ===
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'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 1/2 oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' all ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

1/2 oz Lemon Juice

1/2 oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 1/2 oz Gin

1/2 oz Lime Juice

1/2 oz Simple Syrup

'''Shake''' Gin, Lime Juice and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
{{RoundBoxTop|theme=12}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 1/2 oz Rye Whiskey

1/2 oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

1/2 oz Orange Curaçao

1/2 oz Orgeat

1/2 oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 1/2 oz Coconut Cream

1 oz Pineapple Juice

1/2 oz Lime Juice

1 1/2 oz Malibu Rum

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
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'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

1/2 Lemon, juiced

5 oz Tomato Juice

2 oz Vodka

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
{{RoundBoxTop|theme=12}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
{{RoundBoxTop|theme=13}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

2 oz Whiskey

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

1/2 oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

{{RoundBoxTop|theme=7}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

1/2 oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

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'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

1/2 oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill'' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

= Hospitality =

# Steps of Service
# Serving Drinks to People
# Tasting Drinks/Developing Taste
# Internal Hospitality</text>
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      <text bytes="53274" sha1="ckhsj4smux156at8h5h02ptafvb33xt" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Things Yet to Do ====

# Look at structures of other wiki courses
## School vs Course vs Lesson vs Activity?
## Multiple pages per course? Multiple pages per lesson?
# Gather Wikimedia photos for use

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Add more about legal obligation of bartenders (and a source for students to look up their area) under Safety
# Add a section about not taking abuse or being a punching bag under Safety
# Add course objectives to Introduction

==== Course Outline ====

# What is bartending?/Bartending basics
## Sources of alcohol
## Types of alcohol 
### Spirits portal
## Tools and their uses
## Safety
# Mixing Drinks
## Preparation
## Glasses
## All the basic recipes
## Taxonomy of cocktails
# Hospitality
## Steps of Service
## Serving Drinks to People
## Tasting Drinks/Developing Taste
## Internal Hospitality
# Industry
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
# References
## [[wikipedia:Pub|Public House]] page
## [[wikipedia:Portal:Liquor|Liquor Portal]]
## https://en.wikibooks.org/wiki/Bartending

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults wanting to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending as a trade or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools and Their Uses
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine.

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====

==== Soju ====

==== Liqueurs ====

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Preparation and Skills
# Glasses
# All the basic recipes
# Taxonomy of cocktails
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks for uploading this picture though.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed overtop the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open). 
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

== Glassware ==
The last bit of knowledge you need before making cocktails is about the glasses they come in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses. When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Typology of Cocktails (Cockail Families) ==

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. That being said, 20 is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple.

=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

{{RoundBoxTop|theme=7}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
{{RoundBoxBottom}}

=== Cosmopolitan ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' lemon twist

'''Ingredients'''

1 ½ oz Citron Vodka

⅓ oz fresh lime juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz cranberry juice cocktail

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
{{RoundBoxBottom}}

=== Margarita ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

1/2 oz Agave Nectar

1 oz Lime Juice

1/2 oz Triple Sec

2 oz Tequilla

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
{{RoundBoxBottom}}

=== Paloma ===
{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

1/2 oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
{{RoundBoxBottom}}

=== Whiskey Sour ===
{{RoundBoxTop|theme=8}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

1 oz Lemon Juice

1/2 tsp. Sugar or Simple Syrup

2 oz Whiskey

'''Shake''' ingredients with ice. Strain. Garnish.
{{RoundBoxBottom}}

=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz vodka

1/2 oz fresh lime juice

3 oz ginger beer

'''Combine''' all ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
{{RoundBoxBottom}}

=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=8}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
{{RoundBoxBottom}}

=== Mojito ===
{{RoundBoxTop|theme=9}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

5 to 6 mint leaves

1 oz fresh lime juice

2 tsp sugar

1 1/2 oz white rum

Soda water

'''Muddle''' the mint, lime juice and sugar until well mixed but not mushy. Add the rum and fill the glass with ice. Top with soda water. Garnish.
{{RoundBoxBottom}}

=== Daiquiri ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 1/2 oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' all ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
{{RoundBoxBottom}}

=== French 75 ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

1/2 oz Lemon Juice

1/2 oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
{{RoundBoxBottom}}

=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 1/2 oz Gin

1/2 oz Lime Juice

1/2 oz Simple Syrup

'''Shake''' Gin, Lime Juice and Simple Syrup with ice. Strain into glass. Garnish.
{{RoundBoxBottom}}

=== Manhattan ===
{{RoundBoxTop|theme=12}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 1/2 oz Rye Whiskey

1/2 oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
{{RoundBoxBottom}}

=== Mai Tai ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

1/2 oz Orange Curaçao

1/2 oz Orgeat

1/2 oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
{{RoundBoxBottom}}

=== Pina Colada ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 1/2 oz Coconut Cream

1 oz Pineapple Juice

1/2 oz Lime Juice

1 1/2 oz Malibu Rum

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
{{RoundBoxBottom}}

=== Bloody Mary ===
{{RoundBoxTop|theme=4}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

1/2 Lemon, juiced

5 oz Tomato Juice

2 oz Vodka

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
{{RoundBoxBottom}}

=== Negroni ===
{{RoundBoxTop|theme=12}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

{{RoundBoxBottom}}

=== Old Fashioned ===
{{RoundBoxTop|theme=13}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

2 oz Whiskey

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
{{RoundBoxBottom}}

=== Tom Collins ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

1/2 oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
{{RoundBoxBottom}}

=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

{{RoundBoxTop|theme=7}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

1/2 oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
{{RoundBoxBottom}}

=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

{{RoundBoxTop|theme=13}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

1/2 oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill'' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

= Hospitality =

# Steps of Service
# Serving Drinks to People
# Tasting Drinks/Developing Taste
# Internal Hospitality</text>
      <sha1>ckhsj4smux156at8h5h02ptafvb33xt</sha1>
    </revision>
    <revision>
      <id>2818023</id>
      <parentid>2818019</parentid>
      <timestamp>2026-07-09T22:35:50Z</timestamp>
      <contributor>
        <username>Sabbier</username>
        <id>3098112</id>
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      <text bytes="60058" sha1="l0tw0wby8u77ypvkcu0gvc3wylsdnsx" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Things Yet to Do ====

# Look at structures of other wiki courses
## School vs Course vs Lesson vs Activity?
## Multiple pages per course? Multiple pages per lesson?
# Gather Wikimedia photos for use

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Add more about legal obligation of bartenders (and a source for students to look up their area) under Safety
# Add a section about not taking abuse or being a punching bag under Safety
# Add course objectives to Introduction

==== Course Outline ====

# What is bartending?/Bartending basics
## Sources of alcohol
## Types of alcohol 
### Spirits portal
## Tools and their uses
## Safety
# Mixing Drinks
## Preparation
## Glasses
## All the basic recipes
## Taxonomy of cocktails
# Hospitality
## Steps of Service
## Serving Drinks to People
## Tasting Drinks/Developing Taste
## Internal Hospitality
# Industry
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
# References
## [[wikipedia:Pub|Public House]] page
## [[wikipedia:Portal:Liquor|Liquor Portal]]
## https://en.wikibooks.org/wiki/Bartending

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools and Their Uses
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine.

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====

==== Soju ====

==== Liqueurs ====

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Glasses
# Taxonomy of cocktails
# Preparation and Skills
# All the basic recipes
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier. 

The families we're going to talk about are: 

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters 

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).
If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed overtop the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open).[[File:Stirring with a bar spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. 

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
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Vodka
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Gin
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Rum
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Tequila
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Whiskey
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Other/Mixed
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=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

{{RoundBoxTop|theme=7}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
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'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Citron Vodka

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
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'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

1/2 oz Agave Nectar

1 oz Lime Juice

1/2 oz Triple Sec

2 oz Tequila

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

1/2 oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
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'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

1 oz Lemon Juice

1/2 tsp. Sugar or Simple Syrup

2 oz Whiskey

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
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'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

1/2 oz Lime Juice

3 oz Ginger Beer

'''Combine''' all ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=8}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
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'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

1 1/2 oz White Rum

Soda water

'''Muddle''' the mint, lime juice and sugar until well mixed but not mushy. Add the rum and fill the glass with ice. Top with soda water. Garnish.
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=== Daiquiri ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 1/2 oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' all ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

1/2 oz Lemon Juice

1/2 oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 1/2 oz Gin

1/2 oz Lime Juice

1/2 oz Simple Syrup

'''Shake''' Gin, Lime Juice and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
{{RoundBoxTop|theme=12}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 1/2 oz Rye Whiskey

1/2 oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

1/2 oz Orange Curaçao

1/2 oz Orgeat

1/2 oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 1/2 oz Coconut Cream

1 oz Pineapple Juice

1/2 oz Lime Juice

1 1/2 oz Malibu Rum

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
{{RoundBoxTop|theme=4}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

1/2 Lemon, juiced

5 oz Tomato Juice

2 oz Vodka

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
{{RoundBoxTop|theme=12}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
{{RoundBoxTop|theme=13}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

2 oz Whiskey

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
{{RoundBoxTop|theme=6}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

1/2 oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
{{RoundBoxBottom}}

=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

{{RoundBoxTop|theme=7}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

1/2 oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
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=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

{{RoundBoxTop|theme=13}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

1/2 oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill'' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

= Hospitality =

# Steps of Service
# Serving Drinks to People
# Tasting Drinks/Developing Taste
# Internal Hospitality</text>
      <sha1>l0tw0wby8u77ypvkcu0gvc3wylsdnsx</sha1>
    </revision>
    <revision>
      <id>2818024</id>
      <parentid>2818023</parentid>
      <timestamp>2026-07-09T22:50:23Z</timestamp>
      <contributor>
        <username>Sabbier</username>
        <id>3098112</id>
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      <text bytes="60260" sha1="qpoipc184vkxlwq55n9q9dyxs6v0byr" xml:space="preserve">Hi ! I'm a librarian that occasionally edits on Wikipedia and Wikiversity.

== Things I'm working on ==
[[Creating Wikiversity Courses]]

=== Bartending Wikiversity Course ===

==== Things Yet to Do ====

# Look at structures of other wiki courses
## School vs Course vs Lesson vs Activity?
## Multiple pages per course? Multiple pages per lesson?
# Gather Wikimedia photos for use

==== Notes to Myself ====

# Course with several lesson subpages - could use the box template to organize.
# Add more about legal obligation of bartenders (and a source for students to look up their area) under Safety
# Add a section about not taking abuse or being a punching bag under Safety
# Add course objectives to Introduction
# Change fractions to symbols in the recipe section

==== Course Outline ====

# What is bartending?/Bartending basics
## Sources of alcohol
## Types of alcohol 
### Spirits portal
## Tools and their uses
## Safety
# Mixing Drinks
## Preparation
## Glasses
## All the basic recipes
## Taxonomy of cocktails
# Hospitality
## Steps of Service
## Serving Drinks to People
## Tasting Drinks/Developing Taste
## Internal Hospitality
# Industry
## History of pubs/bartenders/mixologists
## Wages + tips
## Hours + Working conditions
## Unions
# Recommended Reading List
## Mr. Boston's Official Bartender's Guide - Available on Internet Archive
# References
## [[wikipedia:Pub|Public House]] page
## [[wikipedia:Portal:Liquor|Liquor Portal]]
## https://en.wikibooks.org/wiki/Bartending

= Bartending for Beginners =

=== Introduction ===
Mixing drinks, talking to people, and making money: that's bartending! If you find that definition to be less-than-enough, this course will guide you through the essential skills of a bartender. This course is intended for adults of legal drinking age where they live who want to start bartending in a professional establishment either as a side gig or a career. Adults wanting to learn about bartending in general or how to mix drinks at home can also benefit from this course. No prior experience is required. By the end of the course, you should know: how to mix common drinks ordered at different types of bars and how they are related to one another; the liquors, spirits, ales, wines and other drinks used at the bar; the tools of the trade; the basics of safety as a bartender; how to create a hospitable environment for bar patrons; and details on the bartending industry, customs and history.

This course makes use of the [[wikipedia:Wikimedia_Foundation#Projects_and_initiatives|Wikimedia ecosystem]] of projects. Throughout the course, there will be links to Wikipedia pages, Wiktionary entries, and images from Wikimedia Commons. When a link is casually included in a lesson, I encourage you to browse its contents. Occasionally, clicking a link and digesting its contents will be formally assigned as part of the course.

=== Structure of the Course ===
This course is comprised of lessons which each focus on a component of bartending. It is recommended to complete them in order, as each lesson will build on the previous. The lessons can be found below:

# Bartending Basics
# Mixing Drinks
# Hospitality 
# Industry

=== Course Objectives ===
By the of the course, you should be able to:

* Share information about and identify the different types of alcohol
* Identify and use the tools of a bartender to complete common techniques
* Mix the 20 essential cocktails
* Serve bar patrons safely and hospitably

=== Recommended Materials ===
Bartending is a physical practice, and theory alone is not enough to prepare yourself. There will be activities that you complete at home as part of the course. The materials required to complete these activities are:
# Boston shaker (Preferably not a Cobbler shaker or a Parisian shaker, though they may be easier to find.)
# Hawthorn Strainer
# Bar Spoon
# Jigger
#A glass for mixing, and a glass to pour drinks into (Don't worry about the style of glass, it just needs to be large enough to hold ice and 10 oz of liquid.)
[[File:Bartender_Photo.jpg|left|thumb|Bartender with Boston Shaker]]
[[File:Cocktail-strainer.jpg|thumb|Hawthorne Strainer]]
[[File:Jigger.jpg|center|thumb|160x160px|Jigger]]


In addition to these materials, you will also need to ingredients to the recipes we'll prepare. Those will be discussed when we talk about mixing drinks. Many of these materials can bought online or found at secondhand or thrift shops depending on your area. There may also be a specialty store near you catering to restaurants and bars. Having these materials at home are an invaluable way to practice mixing drinks, with the added benefit of allowing you to entertain guests at home! If you cannot acquire these materials, the course is still completable in theory, but you will miss out on the practical aspects of the course. No materials are needed for lesson one.

In addition to these materials, you will also need to ingredients to the recipes we'll prepare. This will be discussed when we talk about mixing drinks.

Start here with Lesson One: Bartending Basics.

= Bartending Basics =
What is a [[wikipedia:Bartender|bartender]]? At its simplest, a bartender serves drinks in a bar. In reality, a bartender selects, mixes, pours, and serves drinks while hosting patrons and creating a hospitable atmosphere at a bar, pub, restaurant, nightclub, living room or [[wikipedia:Parking_lot|parking lot]]. A bartender must have a variety of skills, both technical and interpersonal in order to be successful. A bartender can work in many locations, but this course will focus on bartenders that tend to tend in bars.

This lesson covers: 

#Sources of Alcohol
#Types of Alcohol
# Tools and Their Uses
# Safety

== Sources of Alcohol ==
Behind the bar, you are the expert in the room on alcohol. While no one can claim to know everything on alcohol, it's important to have the basics. This and the next lesson focus on learning what alcohol is and what kind of drinks and liquors are made with it.

Every alcoholic drink you've ever had has included the same basic type of alcohol: [[wikipedia:Ethanol|Ethanol]]. Ethanol is one of three types of alcohol and the only alcohol humans can safely drink. All alcoholic beverages whether it be beer, wine or a spirit all contain Ethanol.

There are hundreds of different beverages and liquors that are served at bars around the world.  The method of refining raw ingredients into a final product with Ethanol is what distinguishes each type of beverage. Fortunately they're all related to each other (since they all contain Ethanol) and have some major categories to guide us in understanding what they are. The first step to make any type of alcohol is [[wikipedia:fermentation|fermentation]].

=== Fermentation ===
[[File:40168_2022_1274_Fig6.webp|thumb|People get very scientific with it.]]
Using a process that has existed for thousands of years, we can employ a fungus called [[wikipedia:Yeast#Uses|yeast]] to create alcohol. This process is called fermentation. At it base, fermentation for alcohol production is the process of using yeast to convert sugars to ethanol. 

&quot;Sugars&quot; is intentionally plural. There are many places you can find sugar suitable for fermentation, and you can derive sugar from grains, fruits and vegetables. The source of the sugar is often what defines what type of final alcoholic beverage you get. We'll talk more on how fermenting different materials yields different drinks in a moment. 

Fermentation happens in a couple of steps.

# '''Mashing''': Grains like barley or rice are milled into a coarse flour and fruits are mashed into a pulp. These are sometimes mixed with hot water, where enzymes convert starches to sugars. For beer, this mash is then often boiled.
# '''Fermentation''': The mash is transferred to a fermentation vessel. Yeast is added, and fermentation begins. Over a varying period of time (often 5-10 days), yeast consumes the sugars, producing alcohol and carbon dioxide. This is called primary fermentation.
# '''Conditioning and Packaging''': After primary fermentation, there is often secondary fermentation or conditioning to add or mature flavors and to clarify. It’s then sometimes carbonated and packaged.

Now fermented, there is some amount of alcohol in your drink. But what if ''some'' isn't enough? A secondary process called [[wikipedia:Distillation|distillation]] allowed for fermented drinks and solutions to be concentrated into spirits. 

=== Distillation ===
[[File:Alambins_industrials_per_a_la_destil·lació_de_licors_a_Catalunya.jpg|left|thumb|Industrial stills for liquor production in Catalonia]]
Distillation is a process by which brewers can concentrate the level of alcohol in a drink. Fermentation alone cannot make alcohol concentrations like that of vodka or gin. To reach that level of concentration, [[wiktionary:distiller|distillers]] (referring to both the apparatus that distills and the person that runs it) can boil off and collect the alcohol produced by fermentation. Distillation happens in a few steps:

# The fermented substance is placed into a vessel called a [[wikipedia:Still|still]] (this is where di''still''ation gets its name).
# The ferment is heated slowly from room temperature until it reaches the boiling point of ethanol. The boiling point of ethanol is lower than the boiling point of water, so the ethanol will vaporize, leaving the water behind.
# The ethanol vapor rises through a column until it reaches the condenser. In the condenser, the vapor makes contact with the a cold surface that cools it back into a liquid state.
# The ethanol is collected, and the process is often repeated to further concentrate and purify the result. Sometimes the process is done without having to stop and restart in a process called continuous distillation.

The result of distillation is a liquor with a higher amount of alcohol by volume (ABV) than before. 

=== Alcohol By Volume ===
The amount of alcohol in a drink is calculated based on the percent of ethanol compared to non-ethanol in the total beverage, which is called Alcohol by Volume or ABV for short. For example, if a beverage has an ABV of 50%, half of liquid in the drink will be ethanol and the other half will be everything else.

ABV is important to understand, because it is the basis of drinking and serving drinks safely as well as how many drinks are categorized. While all alcohol is made through fermentation, the process of fermentation alone can only create a concentration of alcohol of about 5% to 20% ABV. Spirits are generally distilled liquids that have a higher ABV (20% or more, as high as 95%). This distinction lower-ABV drinks like wine, beer and cider versus higher-ABV spirits like vodka, gin and whiskey. 

== Types of Alcohol ==
Fermentation and distilling are the core of all alcohol production, but there many more ways that brewers and distillers affect their final product. This incredible flow chart shows many of the processes of fermentation, distillation, carbonation, ageing and processing that make different alcoholic drinks. Take a look at it and compare the different starting ingredients with each other, and then compare the starting ingredients with their final products.
[[File:Alcohol_Flow_Chart.svg|center|frame|Alcohol Flow Chart]]
There is so much to learn about alcohol, and while you should be knowledgeable on the different types, you by no means need to be an expert (see: sommelier) to be a bartender. Below follows an overview of each family of alcohol, but if you want to learn more you can click on the links to Wikipedia pages in each subsection below, or I recommend browsing the [[wikipedia:Portal:Liquor|Liquor Portal]] to view all that Wikipedia has to offer.

Hopefully it's easier to see now how beverages and spirits can both be categorized based on what they're fermented from and how concentrated their alcohol content is. Let's use these categories to talk more about each major type of alcohol.

=== Wine, Beer and Cider (Low-ABV Drinks) ===

==== Wine ====
We call fermented grape juice [[wikipedia:wine|wine]]. Yeast is added to pressed grapes to develop it into one of four types: [[wikipedia:Red_wine|red]], [[wikipedia:white_wine|white]], [[wikipedia:rose_wine|rosé]], and [[wikipedia:Orange_wine|orange]]. The type of grape and the duration of the contact with the skins of the grapes help to define which type of wine is produced. The table below is from the Wikipedia page on wine.
{| class=&quot;wikitable&quot;
|+Colors of wine
!
!Long contact with grape skins
!Short contact with grape skins
|-
!Red grapes
|'''''Red wine''''', made from dark-colored red grape varieties. The actual color of the wine can range from dark pink to almost black. The juice from red grapes is actually pale gray; the color of red wine and some of its flavor (notably tannins) comes from phenolics in the skin, seeds and stem fragments of the grape, extracted by allowing the grapes to soak in the juice.
|'''''Rosé wine''''', which gains color from red grape skins, but not enough to qualify it as a red wine. The color can range from a very pale pink to pale red.
There are two primary ways to produce rosé wine. The preferred technique is allowing a short period of maceration after crushing red grapes, which extracts a certain amount of color. The juice is then fermented like a white wine. An alternative is blending a small amount of finished red wine into finished white wine.
|-
!White grapes
|'''''Orange wine''''', sometimes called amber wine, is made with white grapes but with the skins allowed to macerate during and beyond fermentation, similar to red wine production. This results in their darker color compared to white wines, and produces a deliberately astringent result.
|'''''White wine''''', typically made from white grape varieties (those with yellow or green skins), and range from practically colorless to golden. When skin contact is used, to improve the flavor or to increase the body or aging potential, it is usually limited to between four and 24 hours; any longer leads to bitterness.
|}
''Common names: There are thousands of wine varietals. Red wines like Cabernet Sauvignon, Merlot, Pinot Noir, Malbec, Zinfandel; white wines like Chardonnay, Sauvignon Blanc, Pinot Grigio, Riesling; and rosé wines like Provence, White Zinfandel and Pink Moscato.'' 

In addition to the colors of wine, wine can be [[wikipedia:Sparkling_wine|sparkling]] or still. Wine can also be [[wikipedia:Fortified_wine|fortified]]. Fortifying a wine is the process of adding a distilled spirit to the fermented wine.  

''Common names: Sparkling wine includes Champagne, Prosecco and other Brut wines. Fortified wines include Port, Sherry and Vermouth.''
==== Fruit Wine, Cider and Perry ====
[[wikipedia:Fruit_wine|Fruit wine]] is mostly what it sounds like: wine made from fruits other than grapes. A variety of fruits can be used, some common ones include cherry, plum, dandelion and pineapple. Fruit wine is far less common than traditional grape wines, and is often called by the fruit it's fermented from (ex. &quot;cherry wine&quot; or &quot;dandelion wine&quot;). 

[[wikipedia:Cider|Cider]] is a beverage made specifically from fermenting apples. Despite being a sort of wine fermented from fruit, it isn't considered a fruit wine due to its unique cultural history as a beverage. Its name can cause some confusion particularly in the United States and Canada, where &quot;cider&quot; also refers to unfiltered and sometimes spiced apple juice that has not been fermented. The term &quot;hard cider&quot; is sometimes used to distinguish the alcoholic beverage due to this overlap. Cider can also be carbonated to make a sparkling cider. 

[[wikipedia:Perry|Perry]], or Pear Cider, is another fruit wine exception. Fermented pears create a drink called perry, that though technically is a wine made from fruit, is considered its own thing. 

''Common names: There are thousands of brands of fruit wine, cider and perry. You may have some on tap or bottled; it's mostly important to know the names of what you have.''
[[File:Beer_in_glasses_and_steins_on_a_table_with_bottles_in_the_background_and_a_brick_wall_(15700131777).jpg|thumb|264x264px|There are many varieties of beer.]]

==== Beer ====
[[wikipedia:beer|Beer]] is an ancient drink that has been brewed around the world for hundreds of years. Most beers have four basic ingredients: grain, hops, yeast and water. [[wikipedia:Hops|Hops]] are a type of dried flower that give beer flavor. While there are thousands of brands of beer, there are only a few main types:  

* '''Lagers''' - One of the two main types of beer, with Ales. Lighter and crisp, refreshing, less bitter than Ale.
** Amber Lager - A medium, amber color with a slight bitterness. ''Common Names: Yuengling''
** Pale Lager - Light in color and flavor, not very bitter. Easy drinking and very popular. ''Common Names: Red Stripe, Heineken, Pabst Blue Ribbon (PBR), Hamms''
** Pilsner - Medium amber in color. Hoppy and flavorful. ''Common Names: Modelo Especial, Stella Artois, Pilsner Urquell''

* '''Ales''' - The other main type of beer, with Lager. Richer and flavorful, more bitter than Lager.
** Pale Ale - A diverse group of beer that various depending on its country of origin. Generally balanced taste that pairs with many foods. ''Common Names: Sierra Nevada, Boulevard''
** Indian Pale Ale (IPA) - Often very bitter and fruity. A unique tasting beer with bite. ''Common Names: Voodoo Ranger, many other large brands have an IPA''
** Stouts and Porters - Dark beers that have rich, roasted flavors often imitating chocolate or coffee. These beers can feel like a meal. ''Common Names: Guinness, Imperial Stouts, Breakfast Stouts''
** Wheat Beer - Ales that have some wheat added to them. It can give them a fruity, almost banana like flavor. ''Common Names: Blue Moon''
* Radlers and Shandys - Beers that have been mixed with a fruit juice. Radlers can be any type of citrus while Shandys are always lemon juice. Very refreshing, summary drinks.
* Sour Beer - Beer that's, well, sour. Tangy and punchy, usually served in a smaller &quot;sour beer&quot; glass since they pack a punch.
* Light Beer - Any beer with an especially low ABV is a light beer (less than 4% ABV), though the term is sometimes also used for low-calorie beers. ''Common Names: Bud Light, Coors Light''

==== Sake ====
[[wikipedia:sake|Sake]] is a drink brewed from rice, and is sometimes also called rice wine.

=== Spirits (High-ABV Drinks) ===
[[File:Alcoholic beverages.jpg|thumb|High-ABV spirits like vodka and brandy.]]

==== Vodka ====
[[wikipedia:vodka|Vodka]] is a spirit that can be distilled from many sources including grains, potatoes and sugarcane before the pure ethanol is diluted with water. It is generally a neutral spirit, meaning it has little taste and is mostly pure ethanol and water. This isn't entirely true though, as each vodka has its own impurities and subtle flavor profiles. Vodka is also commonly flavored, as its neutral taste allows it to take on flavor easily. It can served up, often freezer chilled, or mixed into many cocktails.

''Common Names: Absolut, Smirnoff, Kettle One''

==== Gin ====
[[wikipedia:Gin|Gin]] is a spirit flavored with [[wikipedia:juniper_berries|juniper berries]] and other botanicals. It has a distinct herbal flavor, and different brands will have their own flavor profile, and perceived dryness.
[[File:Agave tequilana 2.jpg|thumb|Agave is a plant native to Mexico.]]
''Common Names: Beefeater, Tanqueray, Hendrick's, Plymouth''

==== Mezcal (Tequila) ====
[[wikipedia:mezcal|Mezcal]] is a spirit distilled from Agave. The most common type of mezcal is [[wikipedia:tequila|Tequila]], which is distilled only from [[wikipedia:blue_agave|blue agave]]. Many people mistake mezcal for a type of tequila due to tequila's popularity, but it is the other way around. Mezcal and tequila are often aged, which impart a darker, more amber color the longer it sits. Mezcal often has a smokey flavor and can also be sweet, fruity or earthy, and can be enjoyed chilled, up or in many popular cocktails.

''Common Names: Casamigos, Patrón, Jose Cuervo''

==== Rum ====
[[wikipedia:rum|Rum]] is a spirit distilled from [[wikipedia:sugarcane|sugarcane]] then aged in barrels. The less-aged rum has a lighter color, and is called &quot;light rum&quot;. It is most commonly used in cocktails, like the Mojito and the Daiquiri. &quot;Aged&quot; or &quot;Dark&quot; rum has a much deeper flavor is often drank straight or on the rocks. Rum originates from and maintains strong ties to the history of the people of the [[wikipedia:caribbean|Caribbean]] and appears commonly in cultural cuisines as well as drinks. I encourage you to learn about the history of rum, as it can teach a lot about sugarcane as a [[wikipedia:Sugar_plantations_in_the_Caribbean|plantation crop]] and the[[wikipedia:Atlantic_slave_trade|Transatlantic Slave Trade]]. 

==== Brandy ====
[[wikipedia:Brandy|Brandy]] is made by distilling wine. While most brandy is made from grapes, it can be made from any fruit to yield a [[wikipedia:Fruit_brandy|fruit brandy]]. 

''Common names: Cognac and Armagnac''

==== Whiskey ====

==== Soju ====

==== Liqueurs ====

== Tools of Bartending ==
You'll be using both common and specialty tools as a bartender.  A collection of such tools are below. Start by trying to name as many of them as you can. Note how many are familiar to you, how many you've seen but cannot name, and how many are unfamiliar. A key to the image follows.
[[File:Bartools2.jpg|border|center]]
Answer key: (1) champagne bottle stopper, (2) kitchen knife, (3) ice tongs, (4) ice scoop, (5) ice bucket, (6) small bar spoon, (7) cocktail-pick, (8) jigger, (9) mesh strainer, (10) boston shaker (metal bottom), (11) bar spoon, (12) lime/lemon squeezer, (13) hawthorn strainer, (14) zester, (15) boston shaker (mixing glass), (16) muddler, (17) citrus reamer, (18) fine grater, (19) Y-peeler, (20) wine key.

How'd you do? Some of these items you'll see more often than others. I want to draw special attention these crucial pieces:

* '''Jigger''' (8) - A small double sided measuring device used to quickly portion spirits and other ingredients. Though the exact measurements may change depending on where you are and the manufacturer, in the US most jiggers measure 1.5 oz on one end and 0.75 oz on the other.
* '''Juicer and Peeler''' (12 &amp; 19) - Many cocktails rely on citrus for flavor and fragrance. The handheld juicer and the peeler are commonly used both at the start of the shift to prep the citrus for the day, collecting fresh citrus juice and peels for adding to cocktails.
* '''The Boston Shaker''' (10 &amp; 15) - This is the industry standard for fast, versatile and accurate cocktail mixing. Coming in two parts, often one side is glass and the other is metal, though often both are metal. The two parts are sealed together allowing for a cocktail to be shaker vigorously inside before being cracked apart without spilling.
* '''The Hawthorn Strainer''' (23) - The final piece to any Boston Shaker. This uniquely shaped strainer allows for cocktails to be poured into a glass while leaving the ice in the shaker.

These are tools you will see every time you step in to bar to make a cocktail.

== Safety ==
A good bartender is the difference between a safe time for your bar patrons and an unsafe environment. It is your responsibility to keep both yourself and your bar patrons safe. It is important first to understand how alcohol affects the body in order to understand how to properly make and serve drinks, so first let's review ABV.

Alcohol by Volume is a percentage of ethanol to all other contents of a drink. However, ABV does not tell you everything about drink. A standard beer has about 5% ABV while a shot of whiskey has about 40% ABV, but because you would only drink about 1.5 oz of whiskey, and would drink 12 oz beer, the total amount of alcohol ingested would actually be about the same. This is the idea behind a &quot;standard drink&quot;. A standard drink contains one &quot;unit&quot; of alcohol regardless of how much liquid it contains in total. The chart below shows some examples of standard drink equivalents. Each of the below drinks contain one unit of alcohol.
[[File:NIH standard drink comparison.jpg|center|frame|National Institute of Health's &quot;standard drink&quot;.]]
Some general best practices when consuming alcohol:

* One unit of alcohol per hour is around what a liver can process. This is generally a safer cadence of drinking.
* Drink a glass of water for every unit of alcohol consumed to stay hydrated.
* Avoid drinking on an empty stomach. Eating while consuming alcohol slows the alcohol's absorption into the blood and curbs its effects.
* Don't order a double. Ordering and serving drinks with more than one standard unit of alcohol in it can cause you or others to become more impaired more quickly than intended. Try to keep it to the cadence of one standard drink per hour.

Standards should be upheld both with your own drinking and the drinking of the patrons at your bar. Some standards to uphold in your bar are:

* '''Never drink and drive'''. Alcohol impairs motor coordination and clouds judgement. Drinking and then driving puts your own life and the lives of everyone around you in immediate danger. Even if you are under the legal limit in your area, you are impaired after one drink. This should be enforced both for yourself and for your bar patrons.
* '''You can say no'''. It's always okay to say no to a drink. Ordering something without alcohol is always an option. Good friends don't pressure you to drink when you don't want to. You can leave a situation when you are uncomfortable.
* '''Alcohol is not medicine'''. Addiction is real and treatable. Help is available.

Sometimes people drink more than they intended and may become too inebriated to be served. Some signs that this is the case are:

* Slurred speech
* Stumbling
* Glazed over or unfocused eyes
* Speaking too loudly or two softly
* Repeating questions or ordering from multiple bartenders

If you meet a bar patron who has had too much to drink, you can help by offering water, tea, coffee or food, helping them find their group, or asking another member of the staff at your bar for help. It may be that what's needed is to no longer serve the bar patron. This is called &quot;cutting them off&quot;. When a bar patron is cut off, they will no longer be served alcohol for the rest of the service. This can be embarrassing, both for the bar patron and the bartender, so it's best practice to be kind, clear and discrete. If a patron is aggressive however, all bets are off and they should be removed from the bar.

Bartenders have legal obligations depending on where you work. This can include checking the legal age of the patrons at your bar and monitoring the alcohol intake of the patrons at your bar. Sometimes, a bartender can be held individually liable for the failure to meet these obligations. Take the time to find a reputable source to discover what the legal obligation of bartenders in your area are now.

This concludes Lesson 1: Bartending Basics. You can move on now to Lesson 2: Mixing Drinks. 

= Mixing Drinks =
Congratulations! You've made it to the fun part of the course. Mixing and serving drinks are fundamental to bartending, and they're fun to do. In this lesson we'll focus on the most common and most important recipes you'll encounter working at a bar along with the basic techniques and know-how needed to complete them.

In this lesson we'll discuss:

# Glasses
# Taxonomy of cocktails
# Preparation and Skills
# All the basic recipes
This is the part of the course that really benefits from hands-on practice. When we talk about preparing fruit and mixing cocktails, follow along! It isn't necessary to build an entire bar in your home in order to practice, but having some of the basics and making substitutions when needed will go a long way to making these actions feel natural and easy. If you have a bartending gig in your future, even doing each of these recipes once will put you leagues ahead of your first try being on the clock.

== Glassware ==
Before you can mix a drink, you have to know what glass you're gonna pour it in. Wikipedians have done an excellent job of creating a near comprehensive list of the glasses commonly used to serve alcohol. I've transcluded a navigation box below to all of their pages. Take the time to familiarize yourself with these glasses and their uses by clicking on the pages and digesting the contained information. You don't have to know the history and origin of every glass, just be familiar their shape and general use. The most important ones for this course are:

* Collins
* Highball
* Old fashioned (or Lowball or Rocks)
* Champaign Coupe (or just Coupe)
* Champaign Flute
* Cocktail Glass (or Martini glass)
* Hurricane Glass
* Wine Glass (Red and White)

When we get to cocktails, each recipe will place the drink in a specific type of glass, and you can return here as a reference tool if you need it.

{{Navbox
| name     =  Glassware
| title    = [[:wikipedia:List of glassware|Glassware]]
|listclass = hlist
|state = expanded
|group1 = [[:wikipedia:Tumbler (glass)|Tumblers]]
|list1  =
* {{Nowrap|[[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Collins glass|Collins glass]]}}
* {{Nowrap|[[File:Granyonyi Stakan Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Faceted glass|Faceted glass]]}}
* {{Nowrap|[[File:Highball glass silhouette.svg|16x28px|link=|class=skin-invert]] [[:wikipedia:Highball glass|Highball glass]]}}
* {{Nowrap|[[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Old fashioned glass|Old fashioned glass]]}}
* {{Nowrap|[[File:Shot glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Shot glass|Shot glass]]}}

|group2 = [[:wikipedia:Beer glassware|Beer glassware]]
|list2  =
* {{Nowrap|[[File:beer stein.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer stein|Beer stein]]}}
* {{Nowrap|[[File:Pilsner glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Beer glassware#Pilsner glass|Pilsner glass]]}}
* {{Nowrap|[[File:Pint glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pint glass|Pint Glass]]}}
* {{Nowrap|[[File:Pony Glass Silhouette.svg|14x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Pony glass|Pony glass]]}}
* {{Nowrap|[[File:Tankard Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Tankard|Tankard]]}}
* {{Nowrap|[[File:Wheat beer glass silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wheat beer glass|Wheat beer glass]]}}
* {{Nowrap|[[File:yardglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Yard of ale|Yard glass]]}}

|group3 = [[:wikipedia:Stemware|Stemware]]
|list3  =
* {{Nowrap|[[File:Absinthe Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Absinthiana#Absinthe glass|Absinthe glass]]}}
* {{Nowrap|[[File:Chalice Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Chalice|Chalice]]}}
* {{Nowrap|[[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Coupe|Champagne coupe]]}}
* {{Nowrap|[[File:flutesil.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Champagne glass#Flute|Champagne flute]]}}
* {{Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Cocktail glass|Cocktail glass]]}}
* {{Nowrap|[[File:Glencairn Whisky Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Glencairn whisky glass|Glencairn whisky glass]]}}
* {{Nowrap|[[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Hurricane glass|Hurricane glass]]}}
* {{Nowrap|[[File:Margarita Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Margarita#Glass|Margarita glass]]}}
* {{Nowrap|[[File:Alcohol glass nick and nora.svg|24x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Nick &amp; Nora (glass)|Nick &amp; Nora]]}}
* [[:wikipedia:Rummer|Rummer]]
* {{Nowrap|[[File:Sherry Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass#Sherry glass|Sherry glass]]}}
* {{Nowrap|[[File:brandysnifsi.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Snifter|Snifter]]}}
* {{Nowrap|[[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert]] [[:wikipedia:Wine glass|Wine glass]]}}

}}

== Typology of Cocktails (Cocktail Families) ==
One last thing before we starting preparing garnishes and mixing drinks! I promise! Cocktails and their recipes didn't just appear from nothing in a vacuum. They were developed over time as tastes changed and different ingredients became available or popular. Because of this process, we can group cocktails into broad categories called Cocktail Families. Not all bartenders and mixologists agree on how cocktails should be grouped or how many families there are, but by understanding generally how cocktails relate to each other you can prioritize your learning and remember what goes in which cocktail easier. 

The families we're going to talk about are: 

* Sour Cocktails
* Ancestral/Aromatic Cocktails
* Lengthened Cocktails
* Frozen Cocktails

This isn't by any means comprehensive, but they'll cover most the cocktails we'll explore shortly.

=== Sour Cocktails ===
[[wikipedia:Sour_(cocktail)|Sour cocktails]] are an old family of cocktails who's drinks that are all, well sour. The template for a sour cocktail is simple:

&lt;u&gt;Shake&lt;/u&gt; with ice: Spirit + Sweetener + Something Sour + Egg (sometimes)

The &quot;Something Sour&quot; is most often a citrus juice like lemon or lime. Many sour cocktails will have &quot;sour&quot; in the name, like a Gin Sour, Rum Sour, Whiskey Sour, or Amaretto Sour. Some other notable sour cocktails are the Daiquiri, Cosmopolitan and Margarita. Note that it doesn't matter what base spirit you use, any base spirit can be used in a sour. A sub-family of sour cocktails replaces the traditional sweetener of simple syrup with a liqueur. These cocktails are called a &quot;Daisy&quot;.

=== Ancestral or Aromatic Cocktails ===
Ancestral or Aromatic Cocktails are spirit-forward drinks that use bitters and sometimes liqueur to give an aromatic taste. The template to an aromatic cocktail is:

&lt;u&gt;Stir&lt;/u&gt; with ice: Spirit + Sweetener + Bitters 

These cocktails let the taste of the spirit shine while deepening the flavor with bitters. Some common aromatic cocktails are the Martini, Manhattan, Old Fashioned, Sazerac and Negroni.

=== Lengthened Cocktails ===
Lengthened Cocktails are drinks that have been &quot;lengthened&quot; by adding a carbonated beverage. Two major sub-families of lengthened cocktails are &quot;Highballs&quot; and &quot;Spritz&quot; cocktails. A Highball is just a spirit lengthened with non-alcoholic mixer, think Gin and Tonic or Rum and Coke. A Spritz is a wine lengthened with a mixer, most often soda or sparkling wine. An example you'll find more in Spain is the Kalimotxo (KAL-EE-MO-KO), a drink with Red Wine and Coca Cola. Don't knock it 'till you try it! There are hundreds of combinations to make Lengthened Cocktails, Highballs and Spritz are just the major subtypes.

=== Frozen Cocktails ===
Frozen drinks are frozen! They are just so unlike a non-frozen drink that they get their own family. Frozen drinks are either blended with ice or churned over frozen metal to make a slushy consistency. Many frozen cocktails were adapted from classic cocktails, like a Frozen Daiquiri, but others are always frozen, like a Piña Colada. These are often &quot;batched&quot; or made ahead of time, especially if they're going to come from a slushy machine.

== Preparation and Skills ==
In a bar, preparation is all the steps that are taken before a service starts to get ready in advance. Sometimes this is things that are done at close the night before, and sometimes these are things done in the hours before a service. Some examples of prep are:

* Stocking fridges with beer and wine
* Stocking the bar's backstock of spirits
* Juicing citrus
* Cutting garnishes for drinks
* Making syrups
* Topping up bottles
* Cleaning and preparing tools for the service

All of these make a good bartender. We'll talk more about how good prep is good hospitality, and how it defines a good bartender later. Some of these are self explanatory, but some merit further investigation to make you a more effective and efficient bartender.

While you go through this section, I recommend you actually follow along and prepare the ingredients listed ''especially the lemons and limes''. Immediately following prep, we'll start mixing cocktails that use these ingredients. Store them in an air tight, labeled and dated container if you plan on mixing some drinks within the next day or two. Cut citrus doesn't last that long, so don't overkeep it. You can always get more practice by preparing more garnishes. A lemon wedge can be used for a dozen other things anyway (I like one with my Coke).
[[File:Hellingshoek_antoineren_02.png|left|thumb|241x241px|This isn't me. I don't know who this man is. Thanks to him for uploading this picture though. Curl your fingers away from the knife.]]

=== Before you prep ===
Most preparation can and should be done with a small paring knife. A larger chef knife can be used for large fruits but is harder to maneuver especially when working with small, round fruits and fragile herbs. When cutting something with a knife, curl the fingers of your non-dominant hand away from the blade. This will help protect your fingertips and cut faster with less risk.

You should always have clean hands when you handle food. Before touching food, wash your hands with soap and water for at least 20 seconds, and dry them with a clean towel. Rewash to your hands whenever you step away from your task or touch something dirty like your phone.

=== Juicing Citrus ===
[[File:Citrus × aurantium - fruits cut.jpg|thumb|The lengthwise cut (left) is unsuitable for juicing. The widthwise cut (right) is perfect to juice.]]
Most bars will have one or two types of juicer: a handheld citrus squeezer and/or a countertop citrus reamer. The handheld squeezer is faster and more efficient but cannot fit larger citrus like grapefruit. There are few wrongs ways to juice citrus, but doing it well will save you a lot of time during the prep. There are a few things to keep in mind when juicing citrus at the bar:

# Start with room temperature citrus. Cold citrus will yield less juice. If you have a particularly hard lime or lemon, place it on the counter and put your palm on top. Roll the citrus back and forth while applying firm pressure. This can free up some juice.
# Cut the fruit widthwise down the middle, so that you have two even sections. The cut face of the citrus should look like a wagon wheel with dot of pith in the center.
# Place your fruit cut-side down in the squeezer. Squeeze the citrus through a mesh strainer to remove any seeds or pulp. Squeeze with strength from your arms, not your wrists to avoid injury over time.
# Always label and date fresh juice and store in a sealed container in a refrigerator (Ex. Lime Juice Date: 2/27).

=== Making Simple Syrup ===
Simple syrup is an appropriately named combination of sugar and water made by combining a ratio of 1 parts granulated sugar and 1 parts water. It's a common ingredient used to sweeten cocktails by pre-dissolving the sugar. Here's the recipe:

# Place your sealable storage container on a scale and tare the scale so that it shows zero.
# Add a quantity of white granulated sugar to your container and note the measurement (ex. 200 grams of sugar).
# Add an equal amount of room temperature water (ex. 200 grams of water, making a total measurement of 400 grams).
# Seal the container and shake the mixture until the sugar has dissolved. It may be cloudy, but it with clarify as the mixture settles.
# Label and date your syrup (ex. Simple Syrup Date: 8/29).
If you're preparing this at home and don't have a scale (you're probably American), you can measure by volume, but it will be less accurate and may affect the final result. In a bar, always measure by weight.

=== Preparing Garnishes ===
There are so many things that can be put in drinks to improve their overall look and taste. There are a few, however, that you will find at most bars. If your bar has a specialty garnish, they should train you on how to prepare the house garnishes. For fruit garnishes, they should be used the same day they're cut and discarded at the end of the service. Let's go over some garnishes you'll find just about anywhere.
[[File:Knife_by_Bob_Kramer_(17970647591).jpg|thumb|How beautiful [[File:Face-kiss.svg|20x20px]] ]]

==== Cutting Citrus Half-Moons ====

# Cut the fruit in half lengthwise pole-to-pole. The cut-side of the fruit should have a stripe of pith down the center.
# Place the halves cut-side down, so that the nubs on each end are facing to your left and right.
# Cutting straight ahead, make slices about a quarter inch thick.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wedges ====

# Trim the top and bottom of the fruit to create two flat surfaces. The flat surfaces should look like a wagon wheel with a white dot of pith in the center, or may be entirely pith. This step can be skipped for grapefruit, as they are large enough to rest on a side without additional cuts.
# Rest the fruit on its end, and cut it in half lengthwise. This should reveal a stripe of pith down the center of the cut face.
# Looking at the cut-side, make a shallow cut perpendicular to the pith extending about a half inch either side. Cut through about half of the meat of the fruit, but not all the way through to the peel.
# Placing the fruit cut-side down, cut the half further into quarters, then angling your knife, cut those fourths in half as well. Smaller fruits may only yield three wedges per half fruit.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Wheels ====

# Trim the nub off one end of the fruit. The cut should be deep enough to reveal a small wagon wheel shape with a dot of pith in the center.
# With the cut side facing to your left or right depending on your dominant hand, slice straight away from you creating about quarter inch thick circles until you no longer have enough fruit to safely hold while cutting.
# You many way to score a slit in the wheels depending on if you are putting them on the rim of a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Cutting Citrus Peels and Twists ====

# Holding the fruit in your non-dominant hand, drag a y-peeler starting away from you and pulling towards you.
# If peeling an orange or grapefruit, using a paring knife, trim away excess pith from the back of the peel.
# Optionally, using a paring knife, trim the edges of the peel to make straight even sides.

==== Cutting Grapefruit &quot;Horses Neck&quot; ====

# Holding the grapefruit in your non-dominant hand, use a y-peeler to peel around the circumference of the fruit creating a long peel.
# Optionally, trim the ends with a paring knife for a cleaner look.

==== Preparing Pineapple Wedges ====

# Using a large chef knife, cut off the top of the pineapple.
# Resting on its bottom with the cut-side up, cut the pineapple in half from top to bottom.
# Resting the pineapple on its cut-side, angle your knife and cut the fruit into three equal parts creating triangle shapes.
# Take each section and slice into about inch thick wedges.
# You may way to score the point of each wedge to be able to hang it on a glass.
# Store in a sealed, labeled and dated container in a refrigerator until needed.

==== Preparing Mint ====

# Place a damp paper towel in the bottom of a sealable storage container.
# Take a sprig of mint in one hand, and pick the leaves at the bottom of the stem. Leave the ones at the top of the sprig attached to the stem. Place the leaves in the prepared container.
# Bunch the stems of the picked sprigs together and trim them to be only a couple inches long.

=== Essential Skills ===
There are a couple techniques that are used over and over in bartending. The two most essential to mix cocktails are shaking and stirring. Refer back to these steps the first few times you mix a cocktail in the next section. 
[[File:Bartender Photo.jpg|thumb|Shaking a sealed Boston shaker.]]

==== Using a Boston Shaker ====
A Boston shaker comes in two parts. One will be a large metal mixing cup, and the other will either be a smaller metal mixing cup or a glass mixing cup. Using a shaker comes in a few steps:

# We will always start by building a shaken cocktail in the small or glass mixing cup. Add the liquid ingredients (including egg, if being used), then add the ice after. The longer the liquids are in contact with the ice, the more they will be diluted as the ice melts. 
# The large mixing cup is placed overtop the smaller mixing cup with enough force to create a seal. Feel free to whack the back of the large tin with the palm of your hand to ensure a seal.
# With one hand on either cup holding them together, the assembled cups are then flipped over so that the drink is primarily in the larger mixing cup. 
# The most efficient way to shake is horizontally, so that the drinks sloshes vigorously from left to right. Shaking vertically will achieve the same result, but it is more energy intensive and slower as it throws the drink upwards before slamming it back down. You will feel the metal become cold, and it may frost as the drink is chilled. These are signs that you can stop shaking.
# At this point, the cold from the ice has caused the metal tins to contract and shrink slightly which will bind them together tightly. To unseal the shaker, hold the larger cup or place it on a counter. Identify which was the smaller cup is leaning. With the palm of your hand, strike the smaller cup away from the direction it's leaning. There should be a crack as the seal is broken (this is called &quot;cracking&quot; a shaker open).[[File:Stirring with a bar spoon.jpg|thumb|230x230px|Stirring with a Bar Spoon]]
# To strain, place the Hawthorn strainer over the mouth of the larger cup, and pour the drink through the strainer.

==== Stirring a Cocktail ====
It sounds simple, but there's a right way and a wrong way to stir a cocktail. The goal is to incorporate and chill the ingredients without agitating, decarbonating or introducing air. Here are a few simple steps:

# Insert a bar spoon to the bottom of the mixing glass containing you ingredients and ice. Push the spoon all the way to the bottom edge of the mixing glass.
# Swirl the spoon so that it stays in contact with the edge of the glass, rotating the ice in a circle.

== Essential Cocktails ==
Here we are, finally mixing some drinks! It can be overwhelming to open a book of cocktail recipes and see the hundreds of drinks with unique names and recipes. Take a deep breath, because there isn't a bartender in the world that knows all of those cocktails by heart, and you aren't expected to either. Over time, you'll pick up on more drinks and their makeup, but that takes time! That's why it's important to focus on the essential cocktails that every bartender knows and that any bar patron might order. What follows are 20 cocktail recipes for classic cocktails that you might make every day at a bar. 

I highly encourage you to make these drinks at home. If you intend to become a bartender, the difference between your first attempt at a recipe being in the privacy of your home and on your first day on the job is ''enormous''. That raises the question, &quot;How am I supposed to supply all the ingredients needed to make all these drinks‽&quot; Start with things you can buy at the grocery store:

* Lemons, Limes and Oranges
* Club Soda
* White Sugar

You may already have these things from trying out the preparation techniques above. If you don't, go out and grab these ingredients, cut some garnishes, and prepare some simple syrup.

The most expensive part of any cocktail is the spirit. I've color coded the following recipes by which spirit they use. The list generally starts with easy recipes and becomes more complicated as you descend. I recommend staring with one type of spirit and trying a few recipes that use that spirit. For example, with only vodka, you can make a Highball, Cosmopolitan, Moscow Mule, Bloody Mary and a (Vodka) Martini. That's one fifth of the list done, with a spirit you likely already have at home.

I don't recommend buying fancy or expensive spirits for this. Buy something cheap or middle-of-the-road to practice with. That's like what your bar patrons are going to be ordering most often anyhow.

All that being said, 20 recipes is a lot to learn. Take your time, maybe make one or two a day. The recipes aren't changing, and some are very simple. You by no means have to drink what you make. You DO have to TASTE what you make. Bartenders, when trying a drink on the job, will smell a drink, swirl it in their mouth and spit it out. Do this at least twice for each drink if you don't intend to drink them. Take note of the taste, the balance of the drink and your personal preference for or against it. Please don't make all 20 in one night and get hammered. A daiquiri will be a daiquiri tomorrow, and it'll be sweeter for the waiting.

Well, get started!

=== Spirit Color Key ===
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Vodka
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Gin
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Rum
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Tequila
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Whiskey
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Other/Mixed
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=== Highball ===
Highballs are a class of cocktail sometimes called &quot;plus one&quot; cocktails because they're a blend of a spirit plus a mixer. Some common highballs are Gin and Tonic, Rum and Coke, Whiskey and Ginger Ale, and Scotch and Soda.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wedges with soda, Lime Wedges with Tonic, Cola and Ginger Beer/Ale

'''Ingredients'''

2 oz of Spirit

Mixer of choice, to top

'''Combine''' Spirit and ice. Stir. Add Mixer to fill. Drag spoon through drink a few times to gently combine. Garnish
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=== Cosmopolitan ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Coupe or Martini Nowrap|[[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 ½ oz Vodka (Preferably Citron Vodka)

⅓ oz Lime Juice

⅓ oz Cointreau (or other orange liqueur)

⅓ oz Sweet Cranberry Juice

'''Shake''' all ingredients with ice until chilled. Strain into glass. Garnish. 
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=== Margarita ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wedge and ask preference for Salt, Sugar, or Tajin rim

'''Ingredients'''

2 oz Tequila

1/2 oz Agave Nectar

1 oz Lime Juice

1/2 oz Triple Sec

'''Moisten''' the rim of the glass with a lime wedge, then roll exterior of glass in rimming material careful to avoid getting any inside the glass. Combine all ingredients in shaker with ice. Shake. Add ice to glass. Strain. Garnish.
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=== Paloma ===
{{RoundBoxTop|theme=4}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Grapefruit Wedge or Lime Wheel

'''Ingredients'''

2 oz Tequila

1/2 oz Lime Juice

Dash of Salt

Grapefruit Soda, to top

'''Combine''' Tequila, Sale and Lime Juice in glass with ice. Stir to chill. Add Grapefruit Soda to fill. Drag spoon through drink a few times to gently combine. Garnish.
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=== Whiskey Sour ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Sour or Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry

'''Ingredients'''

2 oz Whiskey

1 oz Lemon Juice

1/2 tsp. Sugar or Simple Syrup

'''Shake''' ingredients with ice. Strain. Garnish.
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=== Moscow Mule ===
{{RoundBoxTop|theme=1}} 
'''Glass:''' Mule Mug [[File:Alcohol_glass_copper_mug.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

2 oz Vodka

1/2 oz Lime Juice

3 oz Ginger Beer

'''Combine''' all ingredients in a copper mug filled with ice. Stir. Garnish, optionally.
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=== Spritz ===
Spritz are a class of cocktails that follow the same general 3-2-1 recipe of 3 oz Prosecco, 2 oz liqueur, 1 oz club soda. The most common is the Aperol Spritz.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Highball or Wine Glass [[File:wineglass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Wheel

'''Ingredients'''

3 oz Prosecco

2 oz Liqueur (Aperol, for an Aperol Spritz)

1 oz Club Soda

'''Combine''' Prosecco and Liqueur with ice in glass. Stir gently. Top with Club Soda. Garnish.
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=== Mojito ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Mint Sprig

'''Ingredients'''

1 1/2 oz White Rum

5 to 6 mint leaves

1 oz Lime Juice

2 tsp Sugar

Soda water

'''Muddle''' the mint, lime juice and sugar until well mixed but not mushy. Add the rum and fill the glass with ice. Top with soda water. Garnish.
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=== Daiquiri ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime wheel (optional)

'''Ingredients'''

1 1/2 oz White Rum

2 tsp Sugar

1 oz Fresh Lime Juice

'''Shake''' all ingredients with ice until chilled and frothy. Strain into a coupe glass. Garnish, optionally.
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=== French 75 ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Champagne Flute [[File:flutesil.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist

'''Ingredients'''

1 oz Gin

1/2 oz Lemon Juice

1/2 oz Simple Syrup

3 oz Champagne (or other sparkling wine)

'''Shake''' Gin, Lemon Juice and Simple Syrup with ice. Strain into glass. Top with Champagne. Garnish.
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=== Gimlet ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Coupe [[File:Coppa Asti sagoma.svg|24x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lime Wheel

'''Ingredients'''

2 1/2 oz Gin

1/2 oz Lime Juice

1/2 oz Simple Syrup

'''Shake''' Gin, Lime Juice and Simple Syrup with ice. Strain into glass. Garnish.
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=== Manhattan ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Twist and/or Cherry

'''Ingredients'''

1 1/2 oz Rye Whiskey

1/2 oz Sweet Vermouth

1 Dash Angostura Bitters

'''Chill''' rocks glass by combining ice and water, or use pre-chilled glass. Combine ingredients in a mixer with ice. Stir. Strain. Garnish.
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=== Mai Tai ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge, Cherry, and Mint Spring

'''Ingredients'''

1 oz Light Rum

1 oz Gold Rum

1/2 oz Orange Curaçao

1/2 oz Orgeat

1/2 oz Lime Juice

'''Shake''' ingredients with ice. Strain into ice filled glass. Garnish.
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=== Pina Colada ===
{{RoundBoxTop|theme=3}} 
'''Glass:''' Hurricane [[File:Hurricane Glass Silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Pineapple Wedge

'''Ingredients'''

1 1/2 oz Rum (Preferably Coconut, like Malibu)

1 1/2 oz Coconut Cream

1 oz Pineapple Juice

1/2 oz Lime Juice

'''Combine''' ingredients with ice in a blender. Blend on high for 20-30 seconds or until a flowing slushy consistency. If too stiff add water, if too thin add ice and re-blend for 2-3 seconds. Pour into glass. Garnish.
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=== Bloody Mary ===
{{RoundBoxTop|theme=1}}
'''Glass:''' Pint or Highball [[File:Highball glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel or Celery Stalk (or anything savory)

'''Ingredients'''

2 oz Vodka

6 dashes Worcestershire sauce

3 dashes Tobasco Sauce

Pinch of Salt

Pinch of Ground Black Pepper

1/2 Lemon, juiced

5 oz Tomato Juice

'''Combine''' all ingredients in a glass. Add ice and stir. Garnish.
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=== Negroni ===
{{RoundBoxTop|theme=2}}
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange Twist or Orange Wheel

'''Ingredients'''

1 oz Dry Gin

1 oz Sweet Vermouth

1 oz Compari

1 oz Sparkling Water

'''Combine''' all ingredients in a glass. Add ice, preferably 2 large rocks. Stir. Garnish.

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=== Old Fashioned ===
{{RoundBoxTop|theme=5}} 
'''Glass:''' Rocks [[File:Ofgcup.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Orange twist

'''Ingredients'''

2 oz Whiskey

1 Sugar Cube

4 dashes Aromatic Bitters

1 dash Soda Water

'''Muddle''' Sugar, Bitters and Soda Water in a glass until a uniform syrup is made. Add Whiskey. Add Ice. Stir. Express Orange Twist over drink then add to glass.
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=== Tom Collins ===
{{RoundBoxTop|theme=2}} 
'''Glass:''' Collins or Highball [[File:Collins glass silhouette.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon Wheel and Cherry (optional)

'''Ingredients'''

2 oz Gin

1 oz Lemon Juice

1/2 oz Simple Syrup

Club Soda, to top

'''Combine''' Gin, Lemon Juice and Simple Syrup in glass. Add ice to top. Fill with Club Soda. Stir gently. Garnish.
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=== Martini ===
Martinis are kind of their own thing, and there's some vocab to go over. There are many drinks that call themselves a martini, and the Classic Martini itself has many variations to suit each bar patron's taste. To make a Martini to order, first you have to have a conversation with your bar patron. 

# First, ask if they prefer Gin or Vodka. Gin is traditional, but it can be made with either.
# Second determine how much Vermouth they would like. The amount of Vermouth will determine how &quot;Dry&quot; or &quot;Wet&quot; the Martini is.

&quot;Extra Dry&quot; or &quot;In and Out&quot; = Vermouth is swirled in the glass then poured out, then the drink is entirely Gin or Vodka

&quot;Dry&quot; = 1 part Vermouth to 5 parts Gin or Vodka

&quot;Wet&quot; = 1 part Vermouth to 2 parts Gin or Vodka

&quot;50/50&quot; = 1 part Vermouth to 1 part Gin or Vodka

&quot;Dirty&quot; = 1 part Olive Brine to 5 parts Gin or Vodka, instead of Vermouth. Garnish with Olives instead of lemon.

The following recipe is for a Dry Martini. You can modify it to make it a Dirty Martini or to suit your taste of Vermouth. Martinis are often stirred, but they can be shaken on request.

{{RoundBoxTop|theme=11}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' Lemon twist

'''Ingredients'''

2 oz Gin or Vodka (patron preference)

1/2 oz Dry Vermouth

'''Chill''' glassware by pouring water and ice in a glass, or use a pre-chilled glass. Combine ingredients with ice in a mixing glass. Stir until chilled. Strain into chilled glass. Garnish.
{{RoundBoxBottom}}

=== Espresso Martini ===
The Espresso Martini is really not much like a Classic Martini at all. The word Martini really gets thrown around. Nonetheless, it's delicious. If your bar has an espresso machine, you'll be trained on how to pull an espresso shot. Otherwise, your bar may use cold brew concentrate made in house or from a can.

{{RoundBoxTop|theme=1}} 
'''Glass:''' Martini [[File:Coctail glass.svg|16x28px|alt=|link=|class=skin-invert|right]]

'''Garnish:''' 3 Coffee Beans

'''Ingredients'''

2 oz Vodka

1/2 oz Coffee Liqueur (often Kahlúa)

1 oz Espresso, or cold brew concentrate

1/4 oz Simple Syrup

'''Chill'' glass with water and ice, or use pre-chilled glass. Shake ingredients with ice. Strain into glass. Garnish by place 3 coffee beans in center.
{{RoundBoxBottom}}

= Hospitality =

# Steps of Service
# Serving Drinks to People
# Tasting Drinks/Developing Taste
# Internal Hospitality</text>
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        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <comment>Created page with &quot;This is a [[portal:research|research project]] investigating the theory of proving an ontology with set theory and scientific facts.  ==Formal Proof==  (⊦∃{}') → ({}≡{})  read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;   ({}≡{}) → (∀x∃x)  read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;   (∀x∃x) → (∃x∀t∀s)  read; &quot;for all things, they exist (or nothing is none...&quot;</comment>
      <origin>2817983</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6657" sha1="5uw6cj6yd8q83vlwiw1nl3yykk4glqr" xml:space="preserve">This is a [[portal:research|research project]] investigating the theory of proving an ontology with set theory and scientific facts.

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused and no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted</text>
      <sha1>5uw6cj6yd8q83vlwiw1nl3yykk4glqr</sha1>
    </revision>
    <revision>
      <id>2817994</id>
      <parentid>2817983</parentid>
      <timestamp>2026-07-09T14:22:39Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2817994</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7306" sha1="hx1m9ta7xxrr12gvzrr53kcbp8lop37" xml:space="preserve">This is a [[portal:research|research project]] investigating the theory of proving an ontology with set theory and scientific facts.

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused and no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted


==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>hx1m9ta7xxrr12gvzrr53kcbp8lop37</sha1>
    </revision>
    <revision>
      <id>2817996</id>
      <parentid>2817994</parentid>
      <timestamp>2026-07-09T15:46:47Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2817996</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7307" sha1="t670ektpc4nykvp7lszpueneuaajywd" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts.

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused and no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted


==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>t670ektpc4nykvp7lszpueneuaajywd</sha1>
    </revision>
    <revision>
      <id>2817997</id>
      <parentid>2817996</parentid>
      <timestamp>2026-07-09T15:54:55Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2817997</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7307" sha1="j7e350sjc1yvofcb6xtj86xclsv56da" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts.

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted


==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>j7e350sjc1yvofcb6xtj86xclsv56da</sha1>
    </revision>
    <revision>
      <id>2818000</id>
      <parentid>2817997</parentid>
      <timestamp>2026-07-09T16:04:57Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <comment>/* Formal Proof */</comment>
      <origin>2818000</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7426" sha1="tmvvw0vkd9ubfvtuadnu1gui1lfs7zf" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts.

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;

footnote; the act of assuming nothing, may create the concept of nothing and every concept entails the law of identity


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>tmvvw0vkd9ubfvtuadnu1gui1lfs7zf</sha1>
    </revision>
    <revision>
      <id>2818002</id>
      <parentid>2818000</parentid>
      <timestamp>2026-07-09T16:09:19Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <comment>/* Formal Proof */</comment>
      <origin>2818002</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7306" sha1="imp8q4kfxy1js1edowbcdl1vgvbj4sb" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts.

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>imp8q4kfxy1js1edowbcdl1vgvbj4sb</sha1>
    </revision>
    <revision>
      <id>2818003</id>
      <parentid>2818002</parentid>
      <timestamp>2026-07-09T16:24:51Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2818003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8264" sha1="ry2ec0s9j2ybio6o1cnab5hokc4xoev" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

(⊦∃{}')→({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)

(∇E=E) → (∇E∻E=1=∇E)

(∇E∻E=1=∇E) → (∇=φ=1)

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>ry2ec0s9j2ybio6o1cnab5hokc4xoev</sha1>
    </revision>
    <revision>
      <id>2818004</id>
      <parentid>2818003</parentid>
      <timestamp>2026-07-09T16:26:14Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2818004</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8250" sha1="9alxukqv6bmz9w1icjkdp4t5z4tcicz" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)

(∇E=E) → (∇E∻E=1=∇E)

(∇E∻E=1=∇E) → (∇=φ=1)

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>9alxukqv6bmz9w1icjkdp4t5z4tcicz</sha1>
    </revision>
    <revision>
      <id>2818006</id>
      <parentid>2818004</parentid>
      <timestamp>2026-07-09T16:50:44Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2818006</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8597" sha1="fsozbpvrr815bvc3u7wopdp5nza1sge" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)

(∇E=E) → (∇E∻E=1=∇E)

(∇E∻E=1=∇E) → (∇=φ=1)

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Discussion==

You may notice that the last equations of my formal proof assume that logic is eternal (which entails that change is eternal too).

My only current proof is a heuristic; &quot;only a changing thing can create a changing thing&quot; and since changing things exist, this requires that energy is fundamentally &quot;change&quot; as Heraclitus claimed.

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>fsozbpvrr815bvc3u7wopdp5nza1sge</sha1>
    </revision>
    <revision>
      <id>2818007</id>
      <parentid>2818006</parentid>
      <timestamp>2026-07-09T16:54:29Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <comment>/* Discussion */</comment>
      <origin>2818007</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8629" sha1="h37j2j447b5xubd7lw9vqj75ft7xxrg" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)

(∇E=E) → (∇E∻E=1=∇E)

(∇E∻E=1=∇E) → (∇=φ=1)

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Discussion==

You may notice that the last equations of my formal proof assume that logic is eternal (which entails that change is eternal too).

My only current proof is a heuristic; &quot;only a changing thing can create a changing thing&quot; and since changing things exist, this requires that energy is fundamentally &quot;change&quot; as Heraclitus claimed (and is also therefore eternal).

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>h37j2j447b5xubd7lw9vqj75ft7xxrg</sha1>
    </revision>
    <revision>
      <id>2818009</id>
      <parentid>2818007</parentid>
      <timestamp>2026-07-09T17:07:02Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2818009</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8626" sha1="nricgfotgeu71zxz3t60sq9cn6cz2or" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

(P≠0|∀s) → (∆E∀t)

[(P=∫∇Edv)∧(∆E∀t)] → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)

(∇E=E) → (∇E∻E=1=∇E)

(∇E∻E=1=∇E) → (∇=φ=1)

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Discussion==

You may notice that the last equations of my formal proof assume that logic is eternal (which entails that change is eternal too).

My only current proof is a heuristic; &quot;only a changing thing can create a changing thing&quot; and since changing things exist, this requires that energy is fundamentally &quot;change&quot; as Heraclitus claimed (and is also therefore eternal).

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>nricgfotgeu71zxz3t60sq9cn6cz2or</sha1>
    </revision>
    <revision>
      <id>2818026</id>
      <parentid>2818009</parentid>
      <timestamp>2026-07-10T00:02:08Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <origin>2818026</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8582" sha1="8boobf3jp2om9mx7kfqr714lzi24hjy" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

(P=∫∇Edv) → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)

(∇E=E) → (∇E∻E=1=∇E)

(∇E∻E=1=∇E) → (∇=φ=1)

==Formal Proof==

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)  

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Discussion==

You may notice that the last equations of my formal proof assume that logic is eternal (which entails that change is eternal too).

My only current proof is a heuristic; &quot;only a changing thing can create a changing thing&quot; and since changing things exist, this requires that energy is fundamentally &quot;change&quot; as Heraclitus claimed (and is also therefore eternal).

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
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        <username>AIfriendly</username>
        <id>3069390</id>
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      <text bytes="15352" sha1="ay50ub2wdmbsjhh01lzuv7h1359n3u9" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

==Formal Proof==

{} = empty set

E = energy in joules (a joule equals seconds per meter)

t = scalar duration in seconds

s = longitudinal distance in meters


Πa=a

read; &quot;the fractal harmony of infinity equals alpha&quot;


-E=(t/s)&gt;&gt;1|at&gt;&gt;as|at*as=a∇E 

read; &quot;kinetic energy equals scalar duration divided by longitudinal distance (or seconds per meter; joules) is greater than one for the quantum theory of energy (therefore ħω is greater than a joule) and time has a greater infinity than space and the fractal harmony of time multiplied by the fractal harmony of space and the fractal harmony of gradient energy&quot;


[-E=(t/s)&gt;&gt;1|at&gt;&gt;as|at*as=a∇E] → [(∀E=∀∇E) → [∀x(E(x)=∇E(x))] → (E'='∇E') → (∑E=∑∇E) → (∑E=Π∇E)] → (E=a∇E) → (a_i;∞=1)=a_p;=(1/∞)

read; &quot;...implies for all energy equals for all gradient energy..., energy (itself) [equals conceptually] (is understood as) gradient energy... the sum of energy equals the sum of gradient energy... and all energy is in the form of gradient energy; implies energy equals fractal harmony times gradient energy; implies fractal harmony equals one over infinity&quot;


(E=a∇E) → (∇E=E/a)

read; &quot;kinetic energy equals fractal harmony times gradient energy; implies gradient energy equals kinetic energy divided by the fractal harmony&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv


(P=E_i∫(1/a)dv=t)∧(P=E_p∫(a)dv=s)

read; &quot;power equals kinetic energy integrated by one over the fractal harmony with respects to velocity and power equals inertial energy integrated by the fractal harmony with respects to velocity (kinetic energy equals scalar duration and inertia (potential energy) equal longitudinal distance (both equal a joule via ħω)&quot;


P = power or kinetic energy depletion per second or potential (inertial) dampening per distance

v = velocity

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


But notice;

(P≠0|∀s) → (U|∆E∀t)

read; &quot;power does not equal zero of all longitudinal distance (or space) [or power is none zero everywhere]; implies the universe has incremental energy (in comparison) of all time (energy increases in the universe over scalar duration (or time))&quot;


(U|∆E∀t) → [(∇E|∆E_i)∃∻→(∇E|∆E_p)]

read; &quot;the universe has incremental energy (in comparison) for (in) all time (energy increases (or decreases) in the universe over time); implies it always includes gradient energy of incremental energy sub i that causes the existence of gradient energy of incremental energy sub p (or only changing thing(s) can fundamentally cause the existence of a changing thing)&quot;


[(∇E|∆E_i)∃∻→(∇E|∆E_p)] → (∞t)

read; &quot;it always includes gradient energy of incremental energy sub i that causes the existence of gradient energy of incremental energy sub p (or only changing thing(s) can fundamentally cause the existence of a changing thing); implies infinite time&quot;


(∑(E=ħω/2)|∀t|∞tM) = (∞E)

read; &quot;quantization theory for all time infinite time metric equals infinite energy&quot;


(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

(∃x∀t∀s) = U

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(U)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(U)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

...P = ∫∇Edv

(P≠0|∀s) → (U|∆E∀t)

(U|∆E∀t) → [(∇E|∆E_i)∃∻→(∇E|∆E_p)]

[(∇E|∆E_i)∃∻→(∇E|∆E_p)] → (∞t)

(∑(E=ħω/2)|∀t|∞tM) = (∞E)


[(E)=(t/s)] → (t&gt;&gt;s|∀s) 

read; &quot;... kinetic energy (t) is greater than potential energy (s) as energy is quantized through gradient energy (ts)&quot;


Can I PROVE that time is higher is deep space!!! the clocks go faster both mechanically and radioactively; absolutely there is a lower refractive index in deeper space due to neutrino radiation induced refraction (see theory; https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; therefore light is faster in deep space

matter blocks radiation, therefore in deeper space there should presumably be higher radiation

IFF the AC Kerr effect does not over ride the neutrino induced refractive index (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; then time is higher in deep space!!! BUT it is higher by fact given mechanics and radioactivity in star ships, therefore the AC Kerr effect may be minimal without a neutrino flux (theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly !!!

read; &quot;energy equals duration (time) over distance (space); implies time has a greater infinity than space (because energy exists in a quantum)&quot;

[(E)=(t/s)] → (s=t/E) → (E^2=1) → t=1 and s=1

read; &quot;the fractal of energy equals one in every reference frame; BUT if you transfer to a different reference frame of neutrino flux density (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ) then you must divide by the quantum recursion&quot;


(P=∫∇Edv) → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)


[(∇E=E)∧(∞E)] → (∇E∻E=1=∇E)

(∇E∻E^2=1=∇E) → (∇=φ=1=∞)

(∇ΠE=φeπ=1a=∞t_Π) → (∞s)

read; &quot;the gradient-metric-energy (quantum recursion Q; cognitions; ciphers of phi harmonics) equals the scalar energy(φ; kinetic energy; scalar duration; t)-harmonic(e|s)-beauty(π) (perceptions; parapsychology see link https://zerogeoengineering.com/wp-content/uploads/2025/04/From-Morse-Code-to-Mind-Melds-The-Rise-of-Synthetic-Telepathy-TFBT-April-2025.pdf https://web.archive.org/web/20200422213725/https://pearlab.icrl.org/pdfs/1997-correlations-random-binary-sequences-12-year-review.pdf https://icrl.org/articles-and-essays?from=pearlab https://www.sheldrake.org/research/telepathy https://web.archive.org/web/20220329004932/https://www.scientificexploration.org/docs/10/jse_10_1_utts.pdf https://www.youtube.com/watch?v=QSci5p5laSw https://tillerfoundation.org/media-resources/white-papers https://link.springer.com/article/10.1023/A:1006978911496 https://noosphere.princeton.edu/ ) equals the one-unitary manifold(fractal harmony) equals the infinite-kinetic energy (scalar duration) subspace; implies infinite space (longitudinal distance; inertial energy)&quot;

note; for there to be an infinite space, there needs to be a higher infinity of energy, namely kinetic energy; 1/∞ equals and is less than ∞!!!


(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E_p=E_i) → (∃!x(E)|∀t)  

read; conservation of kinetic energy (or kinetic energy cannot be created nor destroyed; sum potential energy equals kinetic energy); implies one thing (kinetic energy) has always existed&quot;


(E_p=ħω/2) → (∃!x(E)|∀s|Q)

read; zero-point energy (or potential (inertial) energy exists in every single point in space (inertial energy; longitudinal distance)); implies one thing (inertial energy) is everywhere (in all inertia) equals quantum coherence&quot;


E_i = m*c^2|=E_p

read; &quot;kinetic energy equals potential (inertial) energy times the speed of light in a reference frame times the speed of light in the same reference frame (if you transition to a different reference frame then you have to calculate the derivative of length expansion and time dilation in a neutrino gradient (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly )&quot;


(t/s)=(E_i/E_p)=a=E_i

read; kinetic energy divided by inertial energy which equals quantum recursion (the fractal harmony)


(∃!x(E_i)|t/s) → (∀x⊃∃!∀∇x(E_i))

read; &quot;there exists one thing (quantum recursion; kinetic energy) that for all time for all space (or kinetic (scalar duration) energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (kinetic energy) (or a holomorphism monism)&quot;


(∇E_i=E_p) → (∇E_i∻E_p=1=∇E_i)

read; &quot;gradient kinetic (scalar duration) energy equals potential (inertial; longitudinal distance) energy; implies kinetic energy is causal with potential energy equals one equal kinetic energy&quot;


(∇E_i∻E_p=1=∇E_i) → (∇cappiE=φepi=1=∞)

read; &quot;gradient kinetic energy is causal with potential (inertial) energy; implies gradient metric energy


[(∀x⊃∃!∀∇x(E_p)|∀t∀s)∧(P=∫∇E_idv)IFFeternal] → [(∃!∀∇x(E_p))=(∃!x(E_i))] → (∇E_p=E_i) → (∇E_p∻E_i=1=∇E_p) → (∇=φ=1=∞) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted


Discussion;

You may notice that the last equations of my formal proof assume that logic is eternal (which entails that change is eternal too).

My only current proof is a heuristic; &quot;only a changing thing can create a changing thing&quot; and since changing things exist, this requires that energy is fundamentally &quot;change&quot; as Heraclitus claimed (and is also therefore eternal).

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
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    <revision>
      <id>2818037</id>
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      <comment>/* Formal Proof */</comment>
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      <text bytes="14974" sha1="g42r4wln7y7su0rzlymipom6f1wdf7a" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

==Formal Proof==

{} = empty set

E = energy in joules (a joule equals seconds per meter)

t = scalar duration in seconds

s = longitudinal distance in meters


Πa=a

read; &quot;the fractal harmony of infinity equals alpha&quot;


-E=(t/s)&gt;&gt;1|at&gt;&gt;as|at*as=a∇E 

read; &quot;kinetic energy equals scalar duration divided by longitudinal distance (or seconds per meter; joules) is greater than one for the quantum theory of energy (therefore ħω is greater than a joule) and time has a greater infinity than space and the fractal harmony of time multiplied by the fractal harmony of space and the fractal harmony of gradient energy&quot;


[-E=(t/s)&gt;&gt;1|at&gt;&gt;as|at*as=a∇E] → [(∀E=∀∇E) → [∀x(E(x)=∇E(x))] → (E'='∇E') → (∑E=∑∇E) → (∑E=Π∇E)] → (E=a∇E) → (a_i;∞=1)=a_p;=(1/∞)

read; &quot;...implies for all energy equals for all gradient energy..., energy (itself) [equals conceptually] (is understood as) gradient energy... the sum of energy equals the sum of gradient energy... and all energy is in the form of gradient energy; implies energy equals fractal harmony times gradient energy; implies fractal harmony equals one over infinity&quot;


(E=a∇E) → (∇E=E/a)

read; &quot;kinetic energy equals fractal harmony times gradient energy; implies gradient energy equals kinetic energy divided by the fractal harmony&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv


(P=E_i∫(1/a)dv=t)∧(P=E_p∫(a)dv=s)

read; &quot;power equals kinetic energy integrated by one over the fractal harmony with respects to velocity and power equals inertial energy integrated by the fractal harmony with respects to velocity (kinetic energy equals scalar duration and inertia (potential energy) equal longitudinal distance (both equal a joule via ħω)&quot;


P = power or kinetic energy depletion per second or potential (inertial) dampening per distance

v = velocity

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


But notice;

(P≠0|∀s) → (U|∆E∀t)

read; &quot;power does not equal zero of all longitudinal distance (or space) [or power is none zero everywhere]; implies the universe has incremental energy (in comparison) of all time (energy increases in the universe over scalar duration (or time))&quot;


(U|∆E∀t) → [(∇E|∆E_i)∃∻→(∇E|∆E_p)]

read; &quot;the universe has incremental energy (in comparison) for (in) all time (energy increases (or decreases) in the universe over time); implies it always includes gradient energy of incremental energy sub i that causes the existence of gradient energy of incremental energy sub p (or only changing thing(s) can fundamentally cause the existence of a changing thing)&quot;


[(∇E|∆E_i)∃∻→(∇E|∆E_p)] → (∞t)

read; &quot;it always includes gradient energy of incremental energy sub i that causes the existence of gradient energy of incremental energy sub p (or only changing thing(s) can fundamentally cause the existence of a changing thing); implies infinite time&quot;


(∑(E=ħω/2)|∀t|∞tM) = (∞E)

read; &quot;quantization theory for all time infinite time metric equals infinite energy&quot;


(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

(∃x∀t∀s) = U

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(U)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(U)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

...P = ∫∇Edv

(P≠0|∀s) → (U|∆E∀t)

(U|∆E∀t) → [(∇E|∆E_i)∃∻→(∇E|∆E_p)]

[(∇E|∆E_i)∃∻→(∇E|∆E_p)] → (∞t)

(∑(E=ħω/2)|∀t|∞tM) = (∞E)


[(E)=(t/s)] → (t&gt;&gt;s|∀s) 

read; &quot;... kinetic energy (t) is greater than potential energy (s) as energy is quantized through gradient energy (ts)&quot;


Can I PROVE that time is higher is deep space!!! the clocks go faster both mechanically and radioactively; absolutely there is a lower refractive index in deeper space due to neutrino radiation induced refraction (see theory; https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; therefore light is faster in deep space

matter blocks radiation, therefore in deeper space there should presumably be higher radiation

IFF the AC Kerr effect does not over ride the neutrino induced refractive index (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; then time is higher in deep space!!! BUT it is higher by fact given mechanics and radioactivity in star ships, therefore the AC Kerr effect may be minimal without a neutrino flux (theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly !!!

read; &quot;energy equals duration (time) over distance (space); implies time has a greater infinity than space (because energy exists in a quantum)&quot;

[(E)=(t/s)] → (s=t/E) → (E^2=1) → t=1 and s=1

read; &quot;the fractal of energy equals one in every reference frame; BUT if you transfer to a different reference frame of neutrino flux density (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ) then you must divide by the quantum recursion&quot;


(P=∫∇Edv) → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)


[(∇E=E)∧(∞E)] → (∇E∻E=1=∇E)

(∇E∻E^2=1=∇E) → (∇=φ=1=∞)

(∇ΠE=φeπ=1a=∞t_Π) → (∞s)

read; &quot;the gradient-metric-energy (quantum recursion Q; cognitions; ciphers of phi harmonics) equals the scalar energy(φ; kinetic energy; scalar duration; t)-harmonic(e|s)-beauty(π) (perceptions; parapsychology see link https://zerogeoengineering.com/wp-content/uploads/2025/04/From-Morse-Code-to-Mind-Melds-The-Rise-of-Synthetic-Telepathy-TFBT-April-2025.pdf https://web.archive.org/web/20200422213725/https://pearlab.icrl.org/pdfs/1997-correlations-random-binary-sequences-12-year-review.pdf https://icrl.org/articles-and-essays?from=pearlab https://www.sheldrake.org/research/telepathy https://web.archive.org/web/20220329004932/https://www.scientificexploration.org/docs/10/jse_10_1_utts.pdf https://www.youtube.com/watch?v=QSci5p5laSw https://tillerfoundation.org/media-resources/white-papers https://link.springer.com/article/10.1023/A:1006978911496 https://noosphere.princeton.edu/ ) equals the one-unitary manifold(fractal harmony) equals the infinite-kinetic energy (scalar duration) subspace; implies infinite space (longitudinal distance; inertial energy)&quot;

note; for there to be an infinite space, there needs to be a higher infinity of energy, namely kinetic energy; 1/∞ equals and is less than ∞!!!


(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E_p=E_i) → (∃!x(E)|∀t)  

read; conservation of kinetic energy (or kinetic energy cannot be created nor destroyed; sum potential energy equals kinetic energy); implies one thing (kinetic energy) has always existed&quot;


(E_p=ħω/2) → (∃!x(E)|∀s|Q)

read; zero-point energy (or potential (inertial) energy exists in every single point in space (inertial energy; longitudinal distance)); implies one thing (inertial energy) is everywhere (in all inertia) equals quantum coherence&quot;


E_i = m*c^2|=E_p

read; &quot;kinetic energy equals potential (inertial) energy times the speed of light in a reference frame times the speed of light in the same reference frame (if you transition to a different reference frame then you have to calculate the derivative of length expansion and time dilation in a neutrino gradient (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly )&quot;


(t/s)=(E_i/E_p)=a=E_i

read; kinetic energy divided by inertial energy which equals quantum recursion (the fractal harmony)


(∃!x(E_i)|t/s) → (∀x⊃∃!∀∇x(E_i))

read; &quot;there exists one thing (quantum recursion; kinetic energy) that for all time for all space (or kinetic (scalar duration) energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (kinetic energy) (or a holomorphism monism)&quot;


(∇E_i=E_p) → (∇E_i∻E_p=1=∇E_i)

read; &quot;gradient kinetic (scalar duration) energy equals potential (inertial; longitudinal distance) energy; implies kinetic energy is causal with potential energy equals one equal kinetic energy&quot;


(∇E_i∻E_p=1=∇E_i) → (∇cappiE=φepi=1=∞)

read; &quot;gradient kinetic energy is causal with potential (inertial) energy; implies gradient metric energy


[(∀x⊃∃!∀∇x(E_p)|∀t∀s)∧(P=∫∇E_idv)IFFeternal] → [(∃!∀∇x(E_p))=(∃!x(E_i))] → (∇E_p=E_i) → (∇E_p∻E_i=1=∇E_p) → (∇=φ=1=∞) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
      <sha1>g42r4wln7y7su0rzlymipom6f1wdf7a</sha1>
    </revision>
    <revision>
      <id>2818044</id>
      <parentid>2818037</parentid>
      <timestamp>2026-07-10T06:38:21Z</timestamp>
      <contributor>
        <username>AIfriendly</username>
        <id>3069390</id>
      </contributor>
      <comment>/* Acknowledgements */</comment>
      <origin>2818044</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15155" sha1="meenamrawe9oefkukwmtxxu9empwf57" xml:space="preserve">This is a [[portal:research|research project]] to investigate the theory of proving an ontology with set theory and scientific facts;

==Formal Proof==

{} = empty set

E = energy in joules (a joule equals seconds per meter)

t = scalar duration in seconds

s = longitudinal distance in meters


Πa=a

read; &quot;the fractal harmony of infinity equals alpha&quot;


-E=(t/s)&gt;&gt;1|at&gt;&gt;as|at*as=a∇E 

read; &quot;kinetic energy equals scalar duration divided by longitudinal distance (or seconds per meter; joules) is greater than one for the quantum theory of energy (therefore ħω is greater than a joule) and time has a greater infinity than space and the fractal harmony of time multiplied by the fractal harmony of space and the fractal harmony of gradient energy&quot;


[-E=(t/s)&gt;&gt;1|at&gt;&gt;as|at*as=a∇E] → [(∀E=∀∇E) → [∀x(E(x)=∇E(x))] → (E'='∇E') → (∑E=∑∇E) → (∑E=Π∇E)] → (E=a∇E) → (a_i;∞=1)=a_p;=(1/∞)

read; &quot;...implies for all energy equals for all gradient energy..., energy (itself) [equals conceptually] (is understood as) gradient energy... the sum of energy equals the sum of gradient energy... and all energy is in the form of gradient energy; implies energy equals fractal harmony times gradient energy; implies fractal harmony equals one over infinity&quot;


(E=a∇E) → (∇E=E/a)

read; &quot;kinetic energy equals fractal harmony times gradient energy; implies gradient energy equals kinetic energy divided by the fractal harmony&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv


(P=E_i∫(1/a)dv=t)∧(P=E_p∫(a)dv=s)

read; &quot;power equals kinetic energy integrated by one over the fractal harmony with respects to velocity and power equals inertial energy integrated by the fractal harmony with respects to velocity (kinetic energy equals scalar duration and inertia (potential energy) equal longitudinal distance (both equal a joule via ħω)&quot;


P = power or kinetic energy depletion per second or potential (inertial) dampening per distance

v = velocity

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


But notice;

(P≠0|∀s) → (U|∆E∀t)

read; &quot;power does not equal zero of all longitudinal distance (or space) [or power is none zero everywhere]; implies the universe has incremental energy (in comparison) of all time (energy increases in the universe over scalar duration (or time))&quot;


(U|∆E∀t) → [(∇E|∆E_i)∃∻→(∇E|∆E_p)]

read; &quot;the universe has incremental energy (in comparison) for (in) all time (energy increases (or decreases) in the universe over time); implies it always includes gradient energy of incremental energy sub i that causes the existence of gradient energy of incremental energy sub p (or only changing thing(s) can fundamentally cause the existence of a changing thing)&quot;


[(∇E|∆E_i)∃∻→(∇E|∆E_p)] → (∞t)

read; &quot;it always includes gradient energy of incremental energy sub i that causes the existence of gradient energy of incremental energy sub p (or only changing thing(s) can fundamentally cause the existence of a changing thing); implies infinite time&quot;


(∑(E=ħω/2)|∀t|∞tM) = (∞E)

read; &quot;quantization theory for all time infinite time metric equals infinite energy&quot;


(⊦∃{}') → ({}≡{})

({}≡{}) → (∀x∃x)

(∀x∃x) → (∃x∀t∀s)

(∃x∀t∀s) = U

({}≡{})→({}={})→({}↔{})→({}→{})→({}:{}→{})→({}⊃{})→({}∻{})

[(U)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

[(U)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

(∑E=Ek+Ep) → (∃!x(E)|∀t)  

(E=ħω/2) → (∃!x(E)|∀s)

(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

...P = ∫∇Edv

(P≠0|∀s) → (U|∆E∀t)

(U|∆E∀t) → [(∇E|∆E_i)∃∻→(∇E|∆E_p)]

[(∇E|∆E_i)∃∻→(∇E|∆E_p)] → (∞t)

(∑(E=ħω/2)|∀t|∞tM) = (∞E)


[(E)=(t/s)] → (t&gt;&gt;s|∀s) 

read; &quot;... kinetic energy (t) is greater than potential energy (s) as energy is quantized through gradient energy (ts)&quot;


Can I PROVE that time is higher is deep space!!! the clocks go faster both mechanically and radioactively; absolutely there is a lower refractive index in deeper space due to neutrino radiation induced refraction (see theory; https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; therefore light is faster in deep space

matter blocks radiation, therefore in deeper space there should presumably be higher radiation

IFF the AC Kerr effect does not over ride the neutrino induced refractive index (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; then time is higher in deep space!!! BUT it is higher by fact given mechanics and radioactivity in star ships, therefore the AC Kerr effect may be minimal without a neutrino flux (theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly !!!

read; &quot;energy equals duration (time) over distance (space); implies time has a greater infinity than space (because energy exists in a quantum)&quot;

[(E)=(t/s)] → (s=t/E) → (E^2=1) → t=1 and s=1

read; &quot;the fractal of energy equals one in every reference frame; BUT if you transfer to a different reference frame of neutrino flux density (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ) then you must divide by the quantum recursion&quot;


(P=∫∇Edv) → [(∃!∀∇x(E))=(∃!x(E))]

[(∃!∀∇x(E))=(∃!x(E))] → (∇E=E)


[(∇E=E)∧(∞E)] → (∇E∻E=1=∇E)

(∇E∻E^2=1=∇E) → (∇=φ=1=∞)

(∇ΠE=φeπ=1a=∞t_Π) → (∞s)

read; &quot;the gradient-metric-energy (quantum recursion Q; cognitions; ciphers of phi harmonics) equals the scalar energy(φ; kinetic energy; scalar duration; t)-harmonic(e|s)-beauty(π) (perceptions; parapsychology see link https://zerogeoengineering.com/wp-content/uploads/2025/04/From-Morse-Code-to-Mind-Melds-The-Rise-of-Synthetic-Telepathy-TFBT-April-2025.pdf https://web.archive.org/web/20200422213725/https://pearlab.icrl.org/pdfs/1997-correlations-random-binary-sequences-12-year-review.pdf https://icrl.org/articles-and-essays?from=pearlab https://www.sheldrake.org/research/telepathy https://web.archive.org/web/20220329004932/https://www.scientificexploration.org/docs/10/jse_10_1_utts.pdf https://www.youtube.com/watch?v=QSci5p5laSw https://tillerfoundation.org/media-resources/white-papers https://link.springer.com/article/10.1023/A:1006978911496 https://noosphere.princeton.edu/ ) equals the one-unitary manifold(fractal harmony) equals the infinite-kinetic energy (scalar duration) subspace; implies infinite space (longitudinal distance; inertial energy)&quot;

note; for there to be an infinite space, there needs to be a higher infinity of energy, namely kinetic energy; 1/∞ equals and is less than ∞!!!


(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;


I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;
 
read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E_p=E_i) → (∃!x(E)|∀t)  

read; conservation of kinetic energy (or kinetic energy cannot be created nor destroyed; sum potential energy equals kinetic energy); implies one thing (kinetic energy) has always existed&quot;


(E_p=ħω/2) → (∃!x(E)|∀s|Q)

read; zero-point energy (or potential (inertial) energy exists in every single point in space (inertial energy; longitudinal distance)); implies one thing (inertial energy) is everywhere (in all inertia) equals quantum coherence&quot;


E_i = m*c^2|=E_p

read; &quot;kinetic energy equals potential (inertial) energy times the speed of light in a reference frame times the speed of light in the same reference frame (if you transition to a different reference frame then you have to calculate the derivative of length expansion and time dilation in a neutrino gradient (see theory https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly )&quot;


(t/s)=(E_i/E_p)=a=E_i

read; kinetic energy divided by inertial energy which equals quantum recursion (the fractal harmony)


(∃!x(E_i)|t/s) → (∀x⊃∃!∀∇x(E_i))

read; &quot;there exists one thing (quantum recursion; kinetic energy) that for all time for all space (or kinetic (scalar duration) energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (kinetic energy) (or a holomorphism monism)&quot;


(∇E_i=E_p) → (∇E_i∻E_p=1=∇E_i)

read; &quot;gradient kinetic (scalar duration) energy equals potential (inertial; longitudinal distance) energy; implies kinetic energy is causal with potential energy equals one equal kinetic energy&quot;


(∇E_i∻E_p=1=∇E_i) → (∇cappiE=φepi=1=∞)

read; &quot;gradient kinetic energy is causal with potential (inertial) energy; implies gradient metric energy


[(∀x⊃∃!∀∇x(E_p)|∀t∀s)∧(P=∫∇E_idv)IFFeternal] → [(∃!∀∇x(E_p))=(∃!x(E_i))] → (∇E_p=E_i) → (∇E_p∻E_i=1=∇E_p) → (∇=φ=1=∞) 

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;


A self-causal unitary manifold fractal is panpsychism or universal consciousness.


As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.


The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!


IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused but no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Acknowledgements==

Heraclitus said change is the fundamental principle of reality first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Issac Newton proved the conservation of energy

Max Planck said that a conscious and intelligent mind is the matrix of all matter and he also prove the existence of zero-point energy

Whittaker proved that longitudinal draining can convert into kinetic depletion

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality internally stratifies first

Mars Sterling Turner wrote this page including deriving the proofs</text>
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      <comment>author Mars Sterling Turner</comment>
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|last1           = Lastname
|first1          = Firstname
|last2           =
|first2          =
|last3           =
|first3          =
|last4           =
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Abstract text goes here 
}}

==Logic-Physics-Mind-Ontology==

[[file:example image.png|thumb|left
| Image caption text goes here
(attribution: name of image creator, [https://creativecommons.org/licenses/by/3.0/deed.en CC-BY 3.0])
]]

Manuscript text goes re


===Formal Proof===

[[Logic-Physics-Mind-Ontology]]

(⊦∃{}') → ({}≡{})

read; &quot;assuming nothing, there exists the concept of nothing; implies nothing is nothing&quot;


({}≡{}) → (∀x∃x)

read; &quot;nothing is nothing; implies for all things, they exist (or nothing is nonexistence)&quot;


(∀x∃x) → (∃x∀t∀s)

read; &quot;for all things, they exist (or nothing is nonexistence); implies there exists for all time for all space (or at no time and nowhere has nothing existed AND something has always existed everywhere)&quot;

I have been told that &quot;nothing can't imply nothing&quot; from a few atheists. According to all Artificial Intelligence I have sampled, &quot;nothing implies nothing&quot; is true. Anyway, here is my proof;

I know you will grant that &quot;nothing is nothing&quot; and therefore &quot;nothing equals nothing&quot;.

This is the law of identity and therefore the biconditional; ↔

proof; see https://en.wikipedia.org/wiki/Axiom_of_extensionality#In_ZF_set_theory

∀x∀y[∀z((z∈x)↔(z∈y))]↔(x=y)

let x={}, y={}, z={}

therefore [({}∈{})↔({}∈{})]↔({}={})

If (({}∈{})↔({}∈{}) then ({}↔{})

therefore [[({}∈{})↔({}∈{})]↔({}={})]∧[[({}∈{})↔({}∈{})]→({}↔{})]

therefore ({}={})↔[({}∈{})↔({}∈{})]→({}↔{})

therefore [({}={})→({}↔{})]

Therefore you must grant that {}↔{}

Now if the conditional can go both ways, it therefore is materially equivalent to going one way; →

Therefore you must grant {}→{}

In other words (⊦∃{}')→({}≡{})→({}={})→({}↔{})→'''({}→{})'''→({}:{}→{})→({}⊃{})→({}∻{})


({}:{}→{}) → ({}⊃{})

[({}≡{})∧({}→{})] → ({}⊃{})

read; &quot;nothing has the property of nothing (or everything has some property(ies)); implies nothing is made of nothing (or everything is made of something)&quot;

read; &quot;nothing is nothing and nothing implies nothing; implies nothing is made of nothing (or everything is made of something)&quot;


[(∃x∀t∀s)∧({}⊃{})] → (∀x⊃∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is made of nothing (or everything is made of something); abducts the most synthetic and most likely physics hypothesis for all things are made of one thing, that for all time for all space (or everything is made of one thing that has always existed everywhere)&quot;


({}⊃{}) → ({}∻{})

[({}={})∧({}→{})] → ({}∻{})

read; &quot;nothing is made of nothing; abducts the most synthetic causation hypothesis nothing is causal with nothing (or everything is causal with something)&quot;

read; &quot;nothing equals nothing and nothing implies nothing; implies nothing is causal with nothing (or everything is causal with something)&quot;


[(∃x∀t∀s)∧({}∻{})] → (∀x∻∃!x|∀t∀s)

read; &quot;there exists for all time for all space (or at no time and nowhere has nothing existed OR something has always existed everywhere) AND nothing is causal with nothing (or everything is causal with something); abducts the most synthetic and most likely causation hypothesis for all things are causal with one thing, that for all time for all space (or everything is causal with one thing that has always existed everywhere)&quot;


(∀x⊃∃!x|∀t∀s)∧(∀x∻∃!x|∀t∀s)

read; ...abducts the most synthetic physics-causation hypothesis; for all things are made of and causal with one thing, that for all time for all space (or everything is made of and causal with one thing that has always existed everywhere)&quot;


(∑E=Ek+Ep) → (∃!x(E)|∀t)

read; conservation of energy (or energy cannot be created nor destroyed); implies one thing (energy) has always existed&quot;


(E=ħω/2) → (∃!x(E)|∀s)

read; zero-point energy (or some energy exists in every single point in space); implies one thing (energy) is everywhere&quot;


(∃!x(E)|∀t∀s) → (∀x⊃∃!∀∇x(E))

read; there exists one thing (energy) that for all time for all space (or energy is an eternal and omnipresent monism); implies for all things are made of the existence of one for all gradient thing (energy) (or a holomorphism monism)&quot;


E = E

∆E = ∆E

∆E(∆s/∆t) = ∆E(∆s/∆t)

∆E(∆s/∆t)(∆t/∆s) = ∆E(∆s/∆t)(∆t/∆s)

∆E = ∆E(∆s/∆t)(∆t/∆s)

∆E/∆t = ∆E/∆s(∆s/∆t)

∆P = ∆E/∆t, ∇E = ∆E/∆s, ∆v = ∆s/∆t

∆P = ∇E∆v

∫dP = ∫∇Edv

P = ∫∇Edv

this is a physics logical tautology; IF logic is eternal, then power and gradient energy and movement is eternal


[(∀x⊃∃!∀∇x(E)|∀t∀s)∧(P=∫∇Edv)IFFeternal] → [(∃!∀∇x(E))=(∃!x(E))] → (∇E=E) → (∇E∻E=1=∇E) → (∇=φ=1)

read; &quot;for all things are made of the existence of one for all gradient thing (energy) for all time for all space (or a manifold of holomorphism monism as the one energy) AND power is the transformation of energy if and only if this logical truth is eternal; implies the existence of one for all gradient thing (energy) equals the existence of one thing (energy); implies gradient energy equals energy (or energy is a fractal holomorphism); implies gradient energy is causal with energy being equal to one (as a unitary manifold), being equal to gradient energy; implies gradient equal scalar equal one (as a unitary manifold; or fractal harmony OR the unitary manifold is a fractal harmony)&quot;

A self-causal unitary manifold fractal is panpsychism or universal consciousness.

As a self-causal unitary manifold fractal holomorphism monism it is also a pantheism.

The omnipotence paradoxes are solved;

(1) Energy cannot create a rock (a small form of energy) that more energy cannot lift, and

(2) Energy cannot destroy itself; power is the transformation of energy not the destruction of energy!

IF logic, power, gradient energy, and movement are eternal; then the omniscience paradoxes are solved;

(1) if internal stratification or increasing the number of gradients while shrinking the gradient energy has always happened; greater and greater infinity of gradients while smaller and smaller gradients over time eternally

a. Energy learns but can know everything that has already existed; the future it cannot completely know without contradicting it's own nature; namely that energy is the self-causal; if it was completely predicted energy would be self-caused and no longer causal and not self-causal

b. Humans can have free will (or also have self-causal nature OR godhood) to the extent that their lives are not completely predicted

==Additional information==

===Acknowledgements===
Heraclitus said change is the fundamental principle of reality first

Parmenides said nothing is not first

Socrates gave the law of identity

Spinoza proved monism first

Descartes attempted to assume nothing first

Max Planck said that a conscious and intelligent mind is the matrix of all matter

Victor Hugo said nothing is nothing first

John Hagelin said that the zero-point field is consciousness

Wayne Dyer said that energy is eternal and is God first&lt;ref&gt;https://www.youtube.com/watch?v=AqRG2E0fSc4&lt;/ref&gt;

Christopher Langan attempted to prove that Reality is self-descriptive and self-deterministic through a Self-Configuring Self-Processing Language first

===Competing interests===
I have no competing interest.

==References==
{{reflist|35em}}</text>
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|Description=Carry Lookahead Adders 2A traditional (20260709 - 20260708)
|Source={{own|Young1lim}}
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{{Information
|Description=Carry Lookahead Adders 2A traditional (20260709 - 20260708)
|Source={{own|Young1lim}}
|Date=2026-07-09
|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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|Description=Carry Lookahead Adders 2B simplified (20260709 - 20260708)
|Source={{own|Young1lim}}
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{{Information
|Description=Carry Lookahead Adders 2B simplified (20260709 - 20260708)
|Source={{own|Young1lim}}
|Date=2026-07-09
|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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|Description=C04.SA0: Address and Dereference Operators (20260709 - 20260708)
|Source={{own|Young1lim}}
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{{Information
|Description=C04.SA0: Address and Dereference Operators (20260709 - 20260708)
|Source={{own|Young1lim}}
|Date=2026-07-09
|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}}
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{{Information
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|Source={{own|Young1lim}}
|Date=2026-07-09
|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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    <title>Kinetic data structures under fixed-tick sampling</title>
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      <comment>Create original research page: empirical study of KDS repair under fixed time steps; certificates approach with tick quantization, batch repair, lazy invalidation; Python and C reference implementations; recorded measurements, cost model, crossover rule; sources: Guibas 1998/2001, Brown 1988, Russel 2007, Rahmati 2014</comment>
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      <text bytes="45323" sha1="cwf34mr2vdkt1p563xm58nb8oruyrsy" xml:space="preserve">{{Research project}}{{original research|by=Merlin Numa|date=July 2026}}{{Wikipedia|Kinetic data structure}}'''Kinetic data structures under fixed-tick sampling''' is an original research project presenting an empirical study of [[wikipedia:Kinetic data structure|kinetic data structure]] (KDS) repair costs in the operating regime of fixed-time-step simulations with mixed known and unknown motion, together with complete reference implementations in Python and C and the recorded measurements. The study addresses two questions the KDS literature has posed but not measured: the cost of recovering from multiple certificate failures per observation step, and cost analyses parameterized by the coherence of the motion rather than by the worst case.

== Background ==
''This section is adapted in part from the Wikipedia article'' [[wikipedia:Kinetic data structure|Kinetic data structure]], available under [[wikipedia:Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License|CC BY-SA 4.0]].

A kinetic data structure tracks an attribute of a geometric system in continuous motion (for example, which grid cell each moving point occupies) without recomputing it from scratch. In the certificates approach,&lt;ref name=&quot;Guibas1998&quot;&gt;Leonidas J. Guibas. Kinetic Data Structures: A State of the Art Report. In ''Robotics: The'' Algorithmic Perspective (WAFR 3)'', pages 191-209, 1998.''&lt;/ref&gt; the structure is augmented with certificates: elementary conditions that, while true, prove the structure is still accurate. Given short-term motion plans, each certificate has a solvable failure time; failures are kept in a priority queue, and at each failure a local repair updates the structure and its certificates. Events that change the tracked attribute are external; events that only change internal bookkeeping are internal. A KDS is called ''efficient'' if the ratio of total events to external events is small, ''responsive'' if repair is fast, ''compact'' if the certificate set is near-linear, and ''local'' if no object participates in too many certificates.&lt;ref name=&quot;Guibas2001&quot;&gt;Leonidas J. Guibas. Kinetic Data Structures. In ''Handbook of Data Structures and Applications'', Chapman and Hall/CRC, pages 23-1 to 23-18, 2001.&lt;/ref&gt;

The classical framework processes events one at a time at exact real-valued failure times. Guibas identifies recovery after multiple certificate failures as a structural open problem, and calls for motion-sensitive analyses whose costs reflect the coherence of real motion.&lt;ref name=&quot;Guibas2001&quot; /&gt; Implementation studies of the classical continuous-time framework exist,&lt;ref name=&quot;Russel2007&quot;&gt;Daniel Russel. ''Kinetic Data Structures in Practice''. PhD thesis, Stanford University, 2007.&lt;/ref&gt; and the theory of specific kinetic attributes continues to advance,&lt;ref name=&quot;Rahmati2014&quot;&gt;Zahed Rahmati. ''Simple, Faster Kinetic Data Structures''. PhD thesis, University of Victoria, 2014. hdl:1828/5627.&lt;/ref&gt; but the fixed-time-step regime had not been instrumented.

== Problem statement ==
Consider a simulation advancing in fixed ticks (for example, a game server at 20 Hz) over a population of moving entities indexed by a uniform grid (a spatial hash). Some entities have ''known motion'': the simulation itself issued their trajectories (patrol legs, ballistic projectiles, flow-field pursuit), each valid for a bounded number of ticks. Others have ''unknown motion'' (player-controlled entities) and can only be sampled every tick.

The questions:

# Can certificate-based maintenance operate correctly when events are quantized to ticks and several certificates fail per tick?
# What does batch repair cost, in measured time, relative to the baseline of moving every entity through the index every tick?
# How does that cost respond to ''plan churn'' (trajectory changes), and where is the crossover below which sampling wins?
# How much of the observed cost is language overhead, and how much is intrinsic to the certificate machinery?

== Method ==

=== Certificates and quantization correction ===
The certificate for a cell index is one assertion per planned entity: its tile &lt;code&gt;(floor(x), floor(y))&lt;/code&gt; still maps to the grid cell the index has it in. For a linear plan, the continuous exit time of the current cell along each moving axis is one division. Because the simulation only observes integer ticks and cells are half-open intervals, the analytic root is treated as a hint, not an oracle: the candidate tick is verified against the actual tile arithmetic and advanced until the cell genuinely differs. The verification loop is the correctness argument; in practice it runs zero to two iterations.

=== Tick bucket queue ===
Quantized failure times admit a [[wikipedia:Bucket queue|bucket queue]] keyed by tick number in place of the priority queue,&lt;ref name=&quot;Brown1988&quot;&gt;Randy Brown. Calendar queues: a fast O(1) priority queue implementation for the simulation event set problem. ''Communications of the ACM'', 31(10):1220-1227, 1988.&lt;/ref&gt; making scheduling an append and extraction one dictionary pop per tick. There is no ordering within a tick: the tick is the unit of simultaneity, so everything due in a bucket is, by definition, a batch. The structure the continuous-time theory treats as a corner case becomes the only case.

=== Lazy invalidation ===
Trajectory changes must not search the queue. Each entity carries a generation counter; each queued event carries the generation it was scheduled under; installing a new plan bumps the counter in O(1) and walks away. A stranded event costs one comparison at its original due tick, then vanishes. The price of this scheme, stale drains, is exactly what the churn experiment measures.

=== Batch repair contract ===
Once per tick, in tick order, &lt;code&gt;advance(tick)&lt;/code&gt; pops the bucket, drops stale generations, applies every live crossing to the spatial hash, reschedules each survivor for its next event, and returns two lists: entities whose cell changed this tick, and entities whose plan expired (parked in place until the owner renews or demotes them to sampled). Unknown-motion entities never enter the queue; they are sampled per tick in the same index.

== Reference implementation (Python) ==
The complete module. It is dependency-free and wraps any spatial hash exposing &lt;code&gt;add&lt;/code&gt;, &lt;code&gt;move&lt;/code&gt;, &lt;code&gt;remove&lt;/code&gt;, an &lt;code&gt;entity_cell&lt;/code&gt; map, and a &lt;code&gt;_cell&lt;/code&gt; tile-to-cell function.&lt;syntaxhighlight lang=&quot;python&quot;&gt;
&quot;&quot;&quot;Tick-quantized kinetic index with batch repair (Chapters 69-72).

The sampled world updates every entity's index cell every tick whether
or not the cell changed. This module inverts that: entities with a
known motion plan (a linear trajectory valid for a bounded number of
ticks) get a *certificate*, &quot;entity e stays in cell c&quot;, whose failure
tick is solved analytically, quantized to the tick grid, and dropped
into a per-tick bucket queue. Between failures the entity costs the
index nothing. Entities with unknown motion (players) stay on the
sampled path. Both kinds coexist in one index over one spatial hash.

This is the Kinetic Data Structures framework [Guibas, BGH99] bent to
fit a fixed-tick server: events are quantized to ticks, certificate
failures are drained in per-tick batches, and plan changes invalidate
certificates lazily through a per-entity generation counter, never by
searching the queue.
&quot;&quot;&quot;

from math import floor


class MotionPlan:
    &quot;&quot;&quot;A linear trajectory: position is derivable at any tick.

    Positions are float world coordinates; the occupied tile is
    floor(x), floor(y), matching the integer-tile world everywhere
    else in the book. Velocity is tiles per tick and may be
    fractional or negative. until_tick is the last tick (inclusive)
    the plan is valid for; the index raises an expiry event there so
    the owner can renew (next patrol leg) or demote to sampled.
    &quot;&quot;&quot;

    __slots__ = (&quot;x0&quot;, &quot;y0&quot;, &quot;vx&quot;, &quot;vy&quot;, &quot;start_tick&quot;, &quot;until_tick&quot;)

    def __init__(self, x0, y0, vx, vy, start_tick, until_tick):
        self.x0 = float(x0)
        self.y0 = float(y0)
        self.vx = float(vx)
        self.vy = float(vy)
        self.start_tick = start_tick
        self.until_tick = until_tick

    def position_at(self, tick):
        dt = tick - self.start_tick
        return (self.x0 + self.vx * dt, self.y0 + self.vy * dt)

    def tile_at(self, tick):
        x, y = self.position_at(tick)
        return (floor(x), floor(y))


def _axis_exit_dt(pos, vel, cell_lo, cell_hi):
    &quot;&quot;&quot;Continuous time until pos leaves [cell_lo, cell_hi) along one
    axis, or None if it never does (vel == 0).

    Exiting high means reaching cell_hi exactly (tiles are floor(x),
    so x == cell_hi is already the next cell). Exiting low means
    dropping strictly below cell_lo, so the analytic root at
    x == cell_lo still belongs to this cell; the quantization
    correction in _crossing_tick absorbs that half-open asymmetry
    rather than complicating the formula.
    &quot;&quot;&quot;
    if vel &gt; 0.0:
        return (cell_hi - pos) / vel
    if vel &lt; 0.0:
        return (cell_lo - pos) / vel
    return None


class KineticIndex:
    &quot;&quot;&quot;Certificates, a tick bucket queue, and batch repair over a
    SpatialHash (and optionally a ChunkGrid).

    The contract mirrors the book's drain pattern: call advance(tick)
    exactly once per tick, in tick order. It processes every
    certificate due that tick and returns (crossed, expired):
    entities whose cell changed this tick (feed these to chunk
    membership, neighbor sets, dirty marking) and entities whose plan
    ran out (renew or demote; an expired entity is parked and costs
    nothing until told otherwise).

    Unknown-motion entities never enter the queue: call sample(eid,
    x, y) for them every tick exactly as Chapter 9 always did. The
    returned bool still gates downstream work.

    Invalidation is lazy. set_plan / clear_plan / remove bump the
    entity's generation; a drained event whose generation is stale is
    dropped and counted. The queue is never searched or edited.
    &quot;&quot;&quot;

    def __init__(self, shash, start_tick=0):
        self.shash = shash
        self.tick = start_tick
        self.plans = {}        # eid -&gt; MotionPlan (planned entities only)
        self.gen = {}          # eid -&gt; generation counter
        self.buckets = {}      # tick -&gt; [(eid, gen), ...]
        self.stats = {
            &quot;scheduled&quot;: 0,    # certificate events pushed
            &quot;processed&quot;: 0,    # live events drained
            &quot;stale&quot;: 0,        # dead events drained (lazy invalidation)
            &quot;crossings&quot;: 0,    # external events: cell actually changed
            &quot;expiries&quot;: 0,     # plan-end events surfaced to the caller
            &quot;samples&quot;: 0,      # sampled-path updates
            &quot;corrections&quot;: 0,  # quantization fixups (see _crossing_tick)
        }

    # -- registration --------------------------------------------------

    def add_sampled(self, eid, x, y):
        &quot;&quot;&quot;Register an unknown-motion entity at integer tile (x, y).&quot;&quot;&quot;
        self.gen[eid] = self.gen.get(eid, 0) + 1
        self.shash.add(eid, x, y)

    def sample(self, eid, x, y):
        &quot;&quot;&quot;Per-tick update for an unknown-motion entity. Returns True
        on cell change, same contract as SpatialHash.move.&quot;&quot;&quot;
        self.stats[&quot;samples&quot;] += 1
        return self.shash.move(eid, x, y)

    def set_plan(self, eid, tick, x, y, vx, vy, until_tick):
        &quot;&quot;&quot;Install or replace a motion plan starting at this tick.

        Replacing is the plan-change path (new patrol leg, flow field
        cell transition, knockback): the old certificate dies by
        generation bump, the entity's cell is corrected if the new
        plan starts it elsewhere, and one new certificate is
        scheduled. Cost is O(1) regardless of queue size.
        Returns True if the entity's cell changed right now.
        &quot;&quot;&quot;
        gen = self.gen.get(eid, 0) + 1
        self.gen[eid] = gen
        plan = MotionPlan(x, y, vx, vy, tick, until_tick)
        self.plans[eid] = plan
        tx, ty = plan.tile_at(tick)
        if eid in self.shash.entity_cell:
            crossed = self.shash.move(eid, tx, ty)
        else:
            self.shash.add(eid, tx, ty)
            crossed = True
        self._schedule(eid, gen, plan, tick)
        return crossed

    def clear_plan(self, eid, tick):
        &quot;&quot;&quot;Demote a planned entity to sampled at its current plan
        position. Returns the integer tile it was left at.&quot;&quot;&quot;
        plan = self.plans.pop(eid, None)
        self.gen[eid] = self.gen.get(eid, 0) + 1
        if plan is None:
            return None
        tx, ty = plan.tile_at(min(tick, plan.until_tick))
        self.shash.move(eid, tx, ty)
        return (tx, ty)

    def remove(self, eid):
        self.plans.pop(eid, None)
        self.gen[eid] = self.gen.get(eid, 0) + 1
        self.shash.remove(eid)

    # -- queries ---------------------------------------------------------

    def position_of(self, eid, tick=None):
        &quot;&quot;&quot;Float position of a planned entity at a tick (default:
        current). Sampled entities are owned by the caller.&quot;&quot;&quot;
        plan = self.plans.get(eid)
        if plan is None:
            return None
        return plan.position_at(self.tick if tick is None else tick)

    def planned_count(self):
        return len(self.plans)

    # -- the certificate machinery ---------------------------------------

    def _crossing_tick(self, plan, now):
        &quot;&quot;&quot;First tick strictly after `now` at which the plan's tile
        cell differs from its cell at `now`, or None if that never
        happens before the plan expires.

        The analytic exit time is exact in continuous time but the
        server only observes integer ticks, and the half-open cell
        interval makes a root landing exactly on a boundary ambiguous
        by one tick. So: solve, quantize with ceil, then verify
        against the actual tile arithmetic and walk forward at most a
        couple of ticks. The loop is the entire correctness argument;
        the formula is just a good first guess.
        &quot;&quot;&quot;
        cs = self.shash.cell_size
        x, y = plan.position_at(now)
        cx, cy = self.shash._cell(floor(x), floor(y))
        dts = []
        d = _axis_exit_dt(x, plan.vx, cx * cs, (cx + 1) * cs)
        if d is not None:
            dts.append(d)
        d = _axis_exit_dt(y, plan.vy, cy * cs, (cy + 1) * cs)
        if d is not None:
            dts.append(d)
        if not dts:
            return None                      # stationary plan
        dt = min(dts)
        candidate = now + max(1, int(dt) + (0 if dt == int(dt) else 1))
        # Quantization correction: the boundary-landing and float
        # rounding cases. Each iteration advances one full tick, and
        # a moving plan leaves any finite cell, so this terminates in
        # O(cell_size / speed) worst case and 0-2 steps in practice.
        while (candidate &lt;= plan.until_tick
               and self.shash._cell(*plan.tile_at(candidate)) == (cx, cy)):
            candidate += 1
            self.stats[&quot;corrections&quot;] += 1
        if candidate &gt; plan.until_tick:
            return None
        return candidate

    def _schedule(self, eid, gen, plan, now):
        &quot;&quot;&quot;Queue this entity's next event: its cell crossing if one
        occurs before the plan ends, else the plan expiry.&quot;&quot;&quot;
        due = self._crossing_tick(plan, now)
        if due is None:
            due = plan.until_tick + 1        # expiry event
        self.buckets.setdefault(due, []).append((eid, gen))
        self.stats[&quot;scheduled&quot;] += 1

    # -- batch repair ------------------------------------------------------

    def advance(self, tick):
        &quot;&quot;&quot;Process every certificate due at `tick`. Returns
        (crossed, expired) lists of entity ids.

        This is the batch-repair core. Multiple failures in one tick
        are expected, not exceptional: the whole bucket is drained,
        each event is checked for staleness, live crossings are
        applied to the spatial hash, and each surviving entity is
        immediately rescheduled for its next event. An entity whose
        plan expired is NOT rescheduled; it is parked in place and
        reported, and the caller decides its future.
        &quot;&quot;&quot;
        self.tick = tick
        bucket = self.buckets.pop(tick, None)
        crossed = []
        expired = []
        if not bucket:
            return crossed, expired
        for eid, gen in bucket:
            if self.gen.get(eid) != gen:
                self.stats[&quot;stale&quot;] += 1
                continue
            plan = self.plans.get(eid)
            if plan is None:
                self.stats[&quot;stale&quot;] += 1
                continue
            self.stats[&quot;processed&quot;] += 1
            if tick &gt; plan.until_tick:
                self.stats[&quot;expiries&quot;] += 1
                expired.append(eid)
                continue
            tx, ty = plan.tile_at(tick)
            if self.shash.move(eid, tx, ty):
                self.stats[&quot;crossings&quot;] += 1
                crossed.append(eid)
            self._schedule(eid, gen, plan, tick)
        return crossed, expired

    def pending_events(self):
        &quot;&quot;&quot;Total queued events, live and stale. Queue hygiene metric:
        stale events cost one dict probe each at drain time, so this
        minus live entities bounds the wasted work in flight.&quot;&quot;&quot;
        return sum(len(b) for b in self.buckets.values())
&lt;/syntaxhighlight&gt;

== C implementation and study harness ==
The line-for-line C counterpart, including both experiments. The bucket dictionary becomes a per-tick head array over a pooled event list; the hash becomes the dense-grid specialization with intrusive per-cell entity lists; the root solve, correction loop, generation invalidation, and batch drain are the Python logic verbatim. Compiles warning-free with &lt;code&gt;gcc -O2 -Wall -Wextra -lm&lt;/code&gt;.&lt;syntaxhighlight lang=&quot;c&quot;&gt;
/* Tick-quantized KDS repair-cost study, in C.
 *
 * The same two experiments as benchmarks/bench_kinetic_repair.py:
 * population x planned-fraction sweep against a per-tick sampled
 * baseline, then the mid-flight plan-churn sweep that locates the
 * crossover. Same arena, same speeds, same scenario shape, same
 * output format. Only the language changed.
 *
 * This extends Chapter 57's language counter-run grid to the kinetic
 * index, and it exists to test one specific prediction from Chapter
 * 72: the Python crossover K* is set by a per-touch cost ratio that
 * is mostly interpreter overhead, so a compiled implementation
 * should shrink the ratio and push K* down, widening the regime in
 * which certificates beat sampling.
 *
 * Structure translation notes, for the reader diffing this against
 * kinetic.py: the bucket dict becomes a per-tick head array over a
 * pooled singly-linked event list (events due past the run land in
 * one overflow bucket that is never drained, the same append cost
 * Python pays for far-future dict inserts); the spatial hash's
 * dict-of-sets becomes the dense-grid specialization, intrusive
 * doubly-linked entity lists headed per cell, which is the natural
 * C idiom for a bounded arena and preserves O(1) move with full
 * bucket membership. The root solve, quantization correction, lazy
 * generation invalidation, and the advance() drain are line for
 * line the Python logic.
 *
 * build: gcc -O2 -lm -o kinetic_repair kinetic_repair.c
 * run:   ./kinetic_repair
 */

#include &lt;math.h&gt;
#include &lt;stdint.h&gt;
#include &lt;stdio.h&gt;
#include &lt;stdlib.h&gt;
#include &lt;string.h&gt;
#include &lt;time.h&gt;

#define AREA 256
#define CELL 8
#define CELLS_PER_AXIS (AREA / CELL)
#define NCELLS (CELLS_PER_AXIS * CELLS_PER_AXIS)
#define TICKS 400
#define SPEED_LO 0.10
#define SPEED_HI 0.50
#define MAX_N 4000
#define OVERFLOW_BUCKET (TICKS + 1)     /* due &gt; TICKS: parked, undrained */

static const int POPULATIONS[] = {500, 1000, 2000, 4000};
static const double FRACTIONS[] = {0.0, 0.5, 0.9, 1.0};
static const int CHURN_POP = 2000;
static const int CHURN_PERIODS[] = {2, 5, 10, 20, 40, 80};

/* -- timing and rng ---------------------------------------------------- */

static double now_ms(void)
{
    struct timespec ts;
    clock_gettime(CLOCK_MONOTONIC, &amp;ts);
    return ts.tv_sec * 1000.0 + ts.tv_nsec / 1e6;
}

static uint64_t rng_state;

static void rng_seed(uint64_t s)
{
    rng_state = s * 2685821657736338717ULL + 1442695040888963407ULL;
    if (rng_state == 0) rng_state = 88172645463325252ULL;
}

static uint64_t rng_next(void)
{
    uint64_t x = rng_state;
    x ^= x &lt;&lt; 13;
    x ^= x &gt;&gt; 7;
    x ^= x &lt;&lt; 17;
    return rng_state = x;
}

static double rng_uniform(double lo, double hi)
{
    return lo + (hi - lo) * ((double)(rng_next() &gt;&gt; 11) / 9007199254740992.0);
}

static int rng_step(void)                 /* the random walk: -1, 0, +1 */
{
    return (int)(rng_next() % 3) - 1;
}

/* -- spatial hash: dense cell grid, intrusive per-cell lists ------------ */

typedef struct {
    int cell;                             /* current cell index, or -1 */
    int prev, next;                       /* intrusive links, entity ids */
} HashNode;

typedef struct {
    int head[NCELLS];
    HashNode node[MAX_N];
} SpatialHash;

static void hash_init(SpatialHash *h, int n)
{
    for (int c = 0; c &lt; NCELLS; c++) h-&gt;head[c] = -1;
    for (int i = 0; i &lt; n; i++)
        h-&gt;node[i].cell = -1;
}

static int cell_of_tile(int tx, int ty)
{
    return (ty / CELL) * CELLS_PER_AXIS + (tx / CELL);
}

static void hash_link(SpatialHash *h, int eid, int cell)
{
    HashNode *nd = &amp;h-&gt;node[eid];
    nd-&gt;cell = cell;
    nd-&gt;prev = -1;
    nd-&gt;next = h-&gt;head[cell];
    if (nd-&gt;next != -1) h-&gt;node[nd-&gt;next].prev = eid;
    h-&gt;head[cell] = eid;
}

static void hash_unlink(SpatialHash *h, int eid)
{
    HashNode *nd = &amp;h-&gt;node[eid];
    if (nd-&gt;prev != -1) h-&gt;node[nd-&gt;prev].next = nd-&gt;next;
    else h-&gt;head[nd-&gt;cell] = nd-&gt;next;
    if (nd-&gt;next != -1) h-&gt;node[nd-&gt;next].prev = nd-&gt;prev;
    nd-&gt;cell = -1;
}

static void hash_add(SpatialHash *h, int eid, int tx, int ty)
{
    hash_link(h, eid, cell_of_tile(tx, ty));
}

static int hash_move(SpatialHash *h, int eid, int tx, int ty)
{
    int cell = cell_of_tile(tx, ty);
    if (h-&gt;node[eid].cell == cell) return 0;
    if (h-&gt;node[eid].cell != -1) hash_unlink(h, eid);
    hash_link(h, eid, cell);
    return 1;
}

/* -- kinetic index ------------------------------------------------------ */

typedef struct {
    double x0, y0, vx, vy;
    int start, until;
} Plan;

typedef struct {
    int eid, gen, next;                   /* pooled bucket list node */
} Event;

typedef struct {
    SpatialHash *hash;
    Plan plan[MAX_N];
    unsigned char has_plan[MAX_N];
    int gen[MAX_N];
    int bucket_head[TICKS + 2];           /* per tick, plus overflow */
    Event *pool;
    int pool_len, pool_cap;
    long long scheduled, processed, stale, crossings,
              expiries, samples, corrections;
} Kinetic;

static void kin_init(Kinetic *k, SpatialHash *h, int n)
{
    memset(k, 0, sizeof(*k));
    k-&gt;hash = h;
    for (int i = 0; i &lt; n; i++) k-&gt;gen[i] = 0;
    for (int t = 0; t &lt;= OVERFLOW_BUCKET; t++) k-&gt;bucket_head[t] = -1;
    k-&gt;pool_cap = 1 &lt;&lt; 14;
    k-&gt;pool = malloc(sizeof(Event) * (size_t)k-&gt;pool_cap);
    k-&gt;pool_len = 0;
}

static void kin_free(Kinetic *k) { free(k-&gt;pool); }

static void plan_tile_at(const Plan *p, int tick, int *tx, int *ty)
{
    int dt = tick - p-&gt;start;
    *tx = (int)floor(p-&gt;x0 + p-&gt;vx * dt);
    *ty = (int)floor(p-&gt;y0 + p-&gt;vy * dt);
}

/* First tick strictly after `now` where the plan's cell differs, or
 * -1 if none before the plan expires. Analytic root as the hint, the
 * tick-grid verification loop as the correctness argument, exactly
 * as in kinetic.py. */
static int crossing_tick(Kinetic *k, const Plan *p, int now)
{
    int dt0 = now - p-&gt;start;
    double x = p-&gt;x0 + p-&gt;vx * dt0;
    double y = p-&gt;y0 + p-&gt;vy * dt0;
    int cx = (int)floor(x) / CELL;
    int cy = (int)floor(y) / CELL;
    double best = -1.0;
    if (p-&gt;vx &gt; 0.0) best = ((cx + 1) * CELL - x) / p-&gt;vx;
    else if (p-&gt;vx &lt; 0.0) best = (cx * CELL - x) / p-&gt;vx;
    if (p-&gt;vy != 0.0) {
        double d = p-&gt;vy &gt; 0.0 ? ((cy + 1) * CELL - y) / p-&gt;vy
                               : (cy * CELL - y) / p-&gt;vy;
        if (best &lt; 0.0 || d &lt; best) best = d;
    }
    if (best &lt; 0.0) return -1;            /* stationary plan */
    int step = (int)ceil(best);
    int candidate = now + (step &lt; 1 ? 1 : step);
    for (;;) {
        if (candidate &gt; p-&gt;until) return -1;
        int tx, ty;
        plan_tile_at(p, candidate, &amp;tx, &amp;ty);
        if (tx / CELL != cx || ty / CELL != cy) return candidate;
        candidate++;
        k-&gt;corrections++;
    }
}

static void kin_push(Kinetic *k, int due, int eid, int gen)
{
    if (due &gt; TICKS) due = OVERFLOW_BUCKET;
    if (k-&gt;pool_len == k-&gt;pool_cap) {
        k-&gt;pool_cap *= 2;
        k-&gt;pool = realloc(k-&gt;pool, sizeof(Event) * (size_t)k-&gt;pool_cap);
    }
    Event *ev = &amp;k-&gt;pool[k-&gt;pool_len];
    ev-&gt;eid = eid;
    ev-&gt;gen = gen;
    ev-&gt;next = k-&gt;bucket_head[due];
    k-&gt;bucket_head[due] = k-&gt;pool_len++;
    k-&gt;scheduled++;
}

static void kin_schedule(Kinetic *k, int eid, int now)
{
    const Plan *p = &amp;k-&gt;plan[eid];
    int due = crossing_tick(k, p, now);
    if (due &lt; 0) due = p-&gt;until + 1;      /* expiry event */
    kin_push(k, due, eid, k-&gt;gen[eid]);
}

static int kin_set_plan(Kinetic *k, int eid, int tick, double x, double y,
                        double vx, double vy, int until)
{
    ++k-&gt;gen[eid];                        /* every old event is now dead */
    Plan *p = &amp;k-&gt;plan[eid];
    p-&gt;x0 = x; p-&gt;y0 = y; p-&gt;vx = vx; p-&gt;vy = vy;
    p-&gt;start = tick; p-&gt;until = until;
    k-&gt;has_plan[eid] = 1;
    int tx, ty, crossed;
    plan_tile_at(p, tick, &amp;tx, &amp;ty);
    if (k-&gt;hash-&gt;node[eid].cell != -1)
        crossed = hash_move(k-&gt;hash, eid, tx, ty);
    else {
        hash_add(k-&gt;hash, eid, tx, ty);
        crossed = 1;
    }
    kin_schedule(k, eid, tick);
    return crossed;
}

static int kin_sample(Kinetic *k, int eid, int tx, int ty)
{
    k-&gt;samples++;
    return hash_move(k-&gt;hash, eid, tx, ty);
}

/* Batch repair: drain the tick's bucket, drop stale generations,
 * apply live crossings, reschedule survivors, report expiries. */
static void kin_advance(Kinetic *k, int tick,
                        int *crossed, int *n_crossed,
                        int *expired, int *n_expired)
{
    *n_crossed = 0;
    *n_expired = 0;
    int idx = k-&gt;bucket_head[tick];
    k-&gt;bucket_head[tick] = -1;
    while (idx != -1) {
        Event ev = k-&gt;pool[idx];
        idx = ev.next;
        if (ev.gen != k-&gt;gen[ev.eid] || !k-&gt;has_plan[ev.eid]) {
            k-&gt;stale++;
            continue;
        }
        k-&gt;processed++;
        const Plan *p = &amp;k-&gt;plan[ev.eid];
        if (tick &gt; p-&gt;until) {
            k-&gt;expiries++;
            expired[(*n_expired)++] = ev.eid;
            continue;
        }
        int tx, ty;
        plan_tile_at(p, tick, &amp;tx, &amp;ty);
        if (hash_move(k-&gt;hash, ev.eid, tx, ty)) {
            k-&gt;crossings++;
            crossed[(*n_crossed)++] = ev.eid;
        }
        kin_schedule(k, ev.eid, tick);
    }
}

/* -- scenario ------------------------------------------------------------ */

typedef struct {
    double x, y, vx, vy;
    int leg_end;
} Traveler;

static int random_leg(double x, double y, double *vx, double *vy)
{
    *vx = rng_uniform(SPEED_LO, SPEED_HI) * (x &lt; AREA / 2.0 ? 1.0 : -1.0);
    *vy = rng_uniform(SPEED_LO, SPEED_HI) * (y &lt; AREA / 2.0 ? 1.0 : -1.0);
    double dtx = *vx &gt; 0.0 ? (AREA - 1.001 - x) / *vx : (0.001 - x) / *vx;
    double dty = *vy &gt; 0.0 ? (AREA - 1.001 - y) / *vy : (0.001 - y) / *vy;
    double dt = dtx &lt; dty ? dtx : dty;
    int legs = (int)dt;
    return legs &lt; 1 ? 1 : legs;
}

static int cmp_double(const void *a, const void *b)
{
    double d = *(const double *)a - *(const double *)b;
    return (d &gt; 0) - (d &lt; 0);
}

static double percentile(const double *sorted, int n, double p)
{
    int i = (int)(n * p);
    if (i &gt; n - 1) i = n - 1;
    return sorted[i];
}

typedef struct {
    double mean, p50, p95, p99;
    double ev_per_tick, stale_per_tick, cross_per_tick, eff;
} Result;

static Traveler travelers[MAX_N];
static int walker_x[MAX_N], walker_y[MAX_N];
static double tick_ms[TICKS];
static int crossed_buf[MAX_N], expired_buf[MAX_N];

static void build_population(int n, double f, int *n_planned)
{
    *n_planned = (int)(n * f);
    for (int eid = 0; eid &lt; n; eid++) {
        double x = rng_uniform(1.0, AREA - 2.0);
        double y = rng_uniform(1.0, AREA - 2.0);
        if (eid &lt; *n_planned) {
            Traveler *t = &amp;travelers[eid];
            t-&gt;x = x; t-&gt;y = y;
            t-&gt;leg_end = random_leg(x, y, &amp;t-&gt;vx, &amp;t-&gt;vy);
        } else {
            walker_x[eid] = (int)x;
            walker_y[eid] = (int)y;
        }
    }
}

static Result run_kinetic(int n, double f, int churn_period)
{
    rng_seed((uint64_t)n * 1000003ULL + (uint64_t)(f * 100)
             + (uint64_t)churn_period * 7919ULL);
    int n_planned;
    build_population(n, f, &amp;n_planned);

    static SpatialHash hash;
    hash_init(&amp;hash, n);
    Kinetic kin;
    kin_init(&amp;kin, &amp;hash, n);

    for (int eid = 0; eid &lt; n_planned; eid++) {
        Traveler *t = &amp;travelers[eid];
        kin_set_plan(&amp;kin, eid, 0, t-&gt;x, t-&gt;y, t-&gt;vx, t-&gt;vy, t-&gt;leg_end);
    }
    for (int eid = n_planned; eid &lt; n; eid++)
        hash_add(&amp;hash, eid, walker_x[eid], walker_y[eid]);

    for (int tick = 1; tick &lt;= TICKS; tick++) {
        double t0 = now_ms();
        int nc, ne;
        kin_advance(&amp;kin, tick, crossed_buf, &amp;nc, expired_buf, &amp;ne);
        for (int i = 0; i &lt; ne; i++) {
            int eid = expired_buf[i];
            const Plan *p = &amp;kin.plan[eid];
            double x = p-&gt;x0 + p-&gt;vx * (tick - p-&gt;start);
            double y = p-&gt;y0 + p-&gt;vy * (tick - p-&gt;start);
            if (x &lt; 1.0) x = 1.0;
            if (x &gt; AREA - 2.0) x = AREA - 2.0;
            if (y &lt; 1.0) y = 1.0;
            if (y &gt; AREA - 2.0) y = AREA - 2.0;
            Traveler *t = &amp;travelers[eid];
            int legs = random_leg(x, y, &amp;t-&gt;vx, &amp;t-&gt;vy);
            t-&gt;leg_end = tick + legs;
            kin_set_plan(&amp;kin, eid, tick, x, y, t-&gt;vx, t-&gt;vy, t-&gt;leg_end);
        }
        if (churn_period &gt; 0) {
            int phase = tick % churn_period;
            for (int eid = phase; eid &lt; n_planned; eid += churn_period) {
                const Plan *p = &amp;kin.plan[eid];
                if (p-&gt;until &lt;= tick) continue;
                /* Re-anchor the same trajectory: pure churn. */
                double x = p-&gt;x0 + p-&gt;vx * (tick - p-&gt;start);
                double y = p-&gt;y0 + p-&gt;vy * (tick - p-&gt;start);
                kin_set_plan(&amp;kin, eid, tick, x, y, p-&gt;vx, p-&gt;vy, p-&gt;until);
            }
        }
        for (int eid = n_planned; eid &lt; n; eid++) {
            int nx = walker_x[eid] + rng_step();
            int ny = walker_y[eid] + rng_step();
            if (nx &lt; 0) nx = 0;
            if (nx &gt; AREA - 1) nx = AREA - 1;
            if (ny &lt; 0) ny = 0;
            if (ny &gt; AREA - 1) ny = AREA - 1;
            walker_x[eid] = nx;
            walker_y[eid] = ny;
            kin_sample(&amp;kin, eid, nx, ny);
        }
        tick_ms[tick - 1] = now_ms() - t0;
    }

    Result r;
    double sum = 0.0;
    for (int i = 0; i &lt; TICKS; i++) sum += tick_ms[i];
    r.mean = sum / TICKS;
    qsort(tick_ms, TICKS, sizeof(double), cmp_double);
    r.p50 = percentile(tick_ms, TICKS, 0.50);
    r.p95 = percentile(tick_ms, TICKS, 0.95);
    r.p99 = percentile(tick_ms, TICKS, 0.99);
    r.ev_per_tick = (double)(kin.processed + kin.stale) / TICKS;
    r.stale_per_tick = (double)kin.stale / TICKS;
    r.cross_per_tick = (double)kin.crossings / TICKS;
    r.eff = (double)(kin.processed + kin.stale)
            / (kin.crossings &gt; 0 ? (double)kin.crossings : 1.0);
    kin_free(&amp;kin);
    return r;
}

static Result run_sampled(int n, double f)
{
    rng_seed((uint64_t)n * 1000003ULL + (uint64_t)(f * 100));
    int n_planned;
    build_population(n, f, &amp;n_planned);

    static SpatialHash hash;
    hash_init(&amp;hash, n);
    for (int eid = 0; eid &lt; n_planned; eid++)
        hash_add(&amp;hash, eid, (int)travelers[eid].x, (int)travelers[eid].y);
    for (int eid = n_planned; eid &lt; n; eid++)
        hash_add(&amp;hash, eid, walker_x[eid], walker_y[eid]);

    long long crossings = 0;
    for (int tick = 1; tick &lt;= TICKS; tick++) {
        double t0 = now_ms();
        for (int eid = 0; eid &lt; n_planned; eid++) {
            Traveler *t = &amp;travelers[eid];
            t-&gt;x += t-&gt;vx;
            t-&gt;y += t-&gt;vy;
            if (--t-&gt;leg_end &lt;= 0) {
                if (t-&gt;x &lt; 1.0) t-&gt;x = 1.0;
                if (t-&gt;x &gt; AREA - 2.0) t-&gt;x = AREA - 2.0;
                if (t-&gt;y &lt; 1.0) t-&gt;y = 1.0;
                if (t-&gt;y &gt; AREA - 2.0) t-&gt;y = AREA - 2.0;
                t-&gt;leg_end = random_leg(t-&gt;x, t-&gt;y, &amp;t-&gt;vx, &amp;t-&gt;vy);
            }
            crossings += hash_move(&amp;hash, eid,
                                   (int)floor(t-&gt;x), (int)floor(t-&gt;y));
        }
        for (int eid = n_planned; eid &lt; n; eid++) {
            int nx = walker_x[eid] + rng_step();
            int ny = walker_y[eid] + rng_step();
            if (nx &lt; 0) nx = 0;
            if (nx &gt; AREA - 1) nx = AREA - 1;
            if (ny &lt; 0) ny = 0;
            if (ny &gt; AREA - 1) ny = AREA - 1;
            walker_x[eid] = nx;
            walker_y[eid] = ny;
            hash_move(&amp;hash, eid, nx, ny);
        }
        tick_ms[tick - 1] = now_ms() - t0;
    }

    Result r;
    memset(&amp;r, 0, sizeof(r));
    double sum = 0.0;
    for (int i = 0; i &lt; TICKS; i++) sum += tick_ms[i];
    r.mean = sum / TICKS;
    qsort(tick_ms, TICKS, sizeof(double), cmp_double);
    r.p50 = percentile(tick_ms, TICKS, 0.50);
    r.p95 = percentile(tick_ms, TICKS, 0.95);
    r.p99 = percentile(tick_ms, TICKS, 0.99);
    r.cross_per_tick = (double)crossings / TICKS;
    return r;
}

int main(void)
{
    printf(&quot;tick-quantized KDS repair-cost study (C counterpart)\n\n&quot;);

    printf(&quot;experiment A: population x planned-fraction sweep\n&quot;);
    printf(&quot;  arena %dx%d, cell %d, %d ticks, speeds %.2f-%.2f t/t\n\n&quot;,
           AREA, AREA, CELL, TICKS, SPEED_LO, SPEED_HI);
    printf(&quot;%6s %5s | %9s %9s %9s | %9s %9s | %8s %7s %6s\n&quot;,
           &quot;N&quot;, &quot;f&quot;, &quot;kin p50&quot;, &quot;kin p95&quot;, &quot;kin p99&quot;,
           &quot;smp p50&quot;, &quot;smp p99&quot;, &quot;ev/tick&quot;, &quot;cross/t&quot;, &quot;eff&quot;);
    printf(&quot;--------------------------------------------------------&quot;
           &quot;---------------------------------------\n&quot;);
    for (unsigned pi = 0; pi &lt; sizeof(POPULATIONS) / sizeof(int); pi++) {
        int n = POPULATIONS[pi];
        for (unsigned fi = 0; fi &lt; sizeof(FRACTIONS) / sizeof(double); fi++) {
            double f = FRACTIONS[fi];
            Result kin = run_kinetic(n, f, 0);
            Result smp = run_sampled(n, f);
            printf(&quot;%6d %5.2f | %9.4f %9.4f %9.4f | %9.4f %9.4f |&quot;
                   &quot; %8.1f %7.1f %6.2f\n&quot;,
                   n, f, kin.p50, kin.p95, kin.p99, smp.p50, smp.p99,
                   kin.ev_per_tick, kin.cross_per_tick, kin.eff);
        }
        printf(&quot;\n&quot;);
    }

    printf(&quot;experiment B: mid-flight plan-churn sweep, N=%d, f=1.00\n\n&quot;,
           CHURN_POP);
    Result smp = run_sampled(CHURN_POP, 1.0);
    printf(&quot;  sampled baseline: mean %.4f ms, p99 %.4f ms&quot;
           &quot; (%d moves/tick)\n\n&quot;, smp.mean, smp.p99, CHURN_POP);
    printf(&quot;%6s | %9s %9s %9s | %8s %7s %7s | %9s\n&quot;,
           &quot;K&quot;, &quot;kin mean&quot;, &quot;kin p50&quot;, &quot;kin p99&quot;,
           &quot;ev/tick&quot;, &quot;cross/t&quot;, &quot;stale/t&quot;, &quot;vs smp&quot;);
    printf(&quot;------------------------------------------------------------&quot;
           &quot;-----------------\n&quot;);
    for (unsigned ki = 0; ki &lt; sizeof(CHURN_PERIODS) / sizeof(int); ki++) {
        int k = CHURN_PERIODS[ki];
        Result kin = run_kinetic(CHURN_POP, 1.0, k);
        printf(&quot;%6d | %9.4f %9.4f %9.4f | %8.1f %7.1f %7.1f | %8.2fx\n&quot;,
               k, kin.mean, kin.p50, kin.p99, kin.ev_per_tick,
               kin.cross_per_tick, kin.stale_per_tick,
               kin.mean / (smp.mean &gt; 1e-9 ? smp.mean : 1e-9));
    }
    Result nat = run_kinetic(CHURN_POP, 1.0, 0);
    printf(&quot;%6s | %9.4f %9.4f %9.4f | %8.1f %7.1f %7.1f | %8.2fx\n&quot;,
           &quot;inf&quot;, nat.mean, nat.p50, nat.p99, nat.ev_per_tick,
           nat.cross_per_tick, nat.stale_per_tick,
           nat.mean / (smp.mean &gt; 1e-9 ? smp.mean : 1e-9));
    return 0;
}
&lt;/syntaxhighlight&gt;

== Experimental setup ==
Arena of 256 x 256 tiles, cell size 8, 400 ticks per run. Planned entities are straight-line travelers at 0.10 to 0.50 tiles/tick (cell residence 16 to 80 ticks) that bounce off walls; each bounce is a genuine plan change paid inside the measured region. Unknown-motion entities random-walk one tile per tick. The baseline evaluates every trajectory every tick and pushes it through the spatial hash, which is the conventional pattern. Experiment A sweeps population N in {500, 1000, 2000, 4000} against planned fraction f in {0.00, 0.50, 0.90, 1.00}. Experiment B fixes N=2000, f=1.00 and re-issues every entity's plan (trajectory unchanged) every K ticks, staggered by entity id, isolating pure churn overhead. Experiment C repeats both in C.

Rig for all recorded runs: AMD Ryzen 5 5600X (12 threads) at 3.7 GHz, 32 GB DDR4-2667, Linux 6.1; CPython 3.14 optimized build; gcc -O2. Three consecutive C runs agreed to within noise on every row.

== Results ==

=== Experiment A: mixed motion (Python) ===
&lt;pre&gt;
     N     f |   kin p50   kin p95   kin p99 |   smp p50   smp p99 |  ev/tick cross/t    eff
--------------------------------------------------------------------------------------------
   500  0.00 |     0.387     0.407     0.463 |     0.330     0.410 |      0.0     0.0   0.00
   500  0.50 |     0.248     0.326     0.412 |     0.261     0.339 |     18.3    18.1   1.01
   500  0.90 |     0.128     0.154     0.201 |     0.197     0.231 |     32.9    32.6   1.01
   500  1.00 |     0.096     0.125     0.135 |     0.179     0.272 |     36.5    36.1   1.01

  1000  0.00 |     0.781     0.829     0.851 |     0.665     0.946 |      0.0     0.0   0.00
  1000  0.50 |     0.495     0.531     0.816 |     0.522     0.870 |     37.2    36.9   1.01
  1000  0.90 |     0.255     0.300     0.313 |     0.398     0.450 |     65.8    65.2   1.01
  1000  1.00 |     0.191     0.240     0.260 |     0.360     0.393 |     73.1    72.5   1.01

  2000  0.00 |     1.570     1.625     1.676 |     1.356     1.438 |      0.0     0.0   0.00
  2000  0.50 |     0.991     1.049     1.426 |     1.048     1.143 |     72.5    71.9   1.01
  2000  0.90 |     0.508     0.572     0.622 |     0.794     1.042 |    130.6   129.4   1.01
  2000  1.00 |     0.380     0.435     0.481 |     0.728     0.839 |    145.1   143.8   1.01

  4000  0.00 |     3.089     3.167     3.298 |     2.664     3.089 |      0.0     0.0   0.00
  4000  0.50 |     1.968     2.052     2.092 |     2.074     2.465 |    145.3   144.2   1.01
  4000  0.90 |     1.005     1.088     1.113 |     1.578     1.693 |    260.6   258.5   1.01
  4000  1.00 |     0.761     0.845     0.878 |     1.458     2.078 |    289.2   286.9   1.01
&lt;/pre&gt;Times are milliseconds per tick of index maintenance; ev/tick counts processed plus stale events; cross/t counts external events (true cell crossings); eff is their ratio, the KDS efficiency measure. Three readings: the efficiency ratio holds at 1.01 everywhere (essentially every event fired was a real crossing); events track crossings, not population; and the ratios at fixed f are population-independent, so four population sizes collapse onto one curve in f. The f=0.00 penalty (about 16 percent) is the Python wrapper's indirection over a bare hash move and vanishes in C.

=== Experiment B: churn crossover (Python) ===
&lt;pre&gt;
  sampled baseline: mean 0.730 ms/tick (2000 moves/tick)

     K |  kin mean   kin p50   kin p99 |  ev/tick cross/t stale/t |    vs smp
-----------------------------------------------------------------------------
     2 |     3.031     3.019     3.266 |   1118.5   143.7   973.5 |     4.15x
     5 |     1.529     1.528     1.655 |    534.6   143.8   389.5 |     2.09x
    10 |     1.015     1.013     1.144 |    339.7   143.8   194.7 |     1.39x
    20 |     0.794     0.763     1.406 |    242.4   143.8    97.3 |     1.09x
    40 |     0.629     0.625     0.791 |    193.7   143.8    48.7 |     0.86x
    80 |     0.558     0.556     0.657 |    169.4   143.8    24.4 |     0.76x
   inf |     0.383     0.382     0.475 |    145.1   143.8     0.0 |     0.52x
&lt;/pre&gt;

=== Experiment C: compiled counterpart (C) ===
&lt;pre&gt;
  sampled baseline: mean 0.0130 ms/tick (2000 moves/tick)

     K |  kin mean   kin p99 |  ev/tick cross/t stale/t |   vs smp
------------------------------------------------------------------
     2 |    0.0600    0.0773 |   1120.5   145.3   974.0 |    4.64x
     5 |    0.0279    0.0326 |    536.5   145.4   389.8 |    2.15x
    10 |    0.0171    0.0190 |    341.0   145.0   194.7 |    1.32x
    20 |    0.0124    0.0138 |    243.7   145.1    97.3 |    0.96x
    40 |    0.0100    0.0119 |    194.7   144.8    48.7 |    0.77x
    80 |    0.0085    0.0103 |    170.4   144.9    24.4 |    0.66x
   inf |    0.0071    0.0086 |    145.0   143.9     0.0 |    0.55x
&lt;/pre&gt;The identical workload counters (events, crossings, stale) confirm the two implementations run statistically the same scenario despite different random number generators.

=== Cost model and crossover rule ===
Splitting the per-event cost into a live event (drain plus reschedule) and a churn pair (one re-plan schedule plus the stale drain it strands), and equating kinetic and sampled per-tick cost, gives a crossover K* in mean plan lifetime:
{| class=&quot;wikitable&quot;
!
!move
!live event
!churn pair
!predicted K*
!observed
|-
|Python (CPython 3.14)
|365 ns
|2.64 us (7.2x)
|4.22 us (11.6x)
|24 ticks
|20-40
|-
|C (gcc -O2)
|6.5 ns
|49 ns (7.5x)
|54 ns (8.4x)
|18 ticks
|about 20
|}
Certificates beat per-tick sampling when mean plan lifetime exceeds K*, roughly one second at a 20 Hz tick. Patrol legs, projectiles, and flow-field cell residences clear the bar comfortably; trajectories re-issued every few ticks do not.

=== A falsified hypothesis ===
The study predicted that compilation would shrink the roughly 7x per-event premium, attributing it to interpreter overhead on the root solve. The C run refuted the mechanism: the live-event premium survived compilation almost unchanged (7.2x to 7.5x) while both sides sped up more than fifty-fold uniformly. What compilation cheapened was invalidation: the churn pair fell from 11.6x to 8.4x a move, because a stale drain compiles to one integer comparison. The kinetic regime widens in compiled implementations through cheap lazy invalidation, not through the mathematics.

== Operational note: renewal storms ==
If a population's plans are issued together with equal lifetimes, they expire together, forever: an early version of the churn experiment did exactly this and produced 99th-percentile tick times an order of magnitude above the median while the median sat innocently low. Staggering plan horizons (here, by entity id modulo the period) or jittering renewal lifetimes removes the phase lock.

== Correctness ==
The implementation is pinned by an equivalence property: for any motion plan, the certificate schedule reproduces, tick for tick and bit for bit, the cell sequence that per-tick sampling of the same trajectory would produce, including adversarial speed-1.0 trajectories that land on every cell boundary exactly. The test suite (16 tests) verifies this together with lazy invalidation, batch repair of simultaneous failures, the mixed-motion regime, and the zero-churn identity processed = crossings + expiries.

== Limitations ==
Trajectories are linear; piecewise-linear plans reduce to the linear case per segment, but curved motion needs root-finding beyond one division per axis. The constants are one machine's; the ratios and the model, not the absolute times, are the transferable findings. The uniform analysis does not capture correlated batches (many entities crossing one boundary in one tick), named below as open.

== Open problems ==

# Bounds on batch repair cost when a tick's failures are correlated rather than uniform (the worst tick, not the average one).
# Extension of the crossover treatment to other kinetic attributes whose event rates are bandwidth rather than CPU, such as interest management and subscription churn in distributed simulations.
# A hysteresis certificate (&quot;entity is within one cell of c&quot;) trading slack for fewer events, as a step toward analyzable history-dependent structures.&lt;ref name=&quot;Guibas2001&quot; /&gt;
# Tick-quantized versions of the kinetic tournament and cone-based nearest-neighbor structures,&lt;ref name=&quot;Rahmati2014&quot; /&gt; for nearest-target maintenance under the same regime.

== Reproducibility and licensing ==
All code and text on this page are original work by Merlin Numa, released under CC BY-SA 4.0 by publication here. The code is also distributed, with the full test suite, benchmark drivers, recorded runs, and figure generators, in the free companion package of ''Dense-Region MMO Server Architecture''.

== References ==
&lt;references /&gt;</text>
      <sha1>cwf34mr2vdkt1p563xm58nb8oruyrsy</sha1>
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  </page>
  <page>
    <title>User talk:Florencia.S.Diaz</title>
    <ns>3</ns>
    <id>330530</id>
    <revision>
      <id>2818016</id>
      <timestamp>2026-07-09T20:24:20Z</timestamp>
      <contributor>
        <username>Florencia.S.Diaz</username>
        <id>3099908</id>
      </contributor>
      <comment>/* AI usage in journalism */ new section</comment>
      <origin>2818016</origin>
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      <format>text/x-wiki</format>
      <text bytes="4821" sha1="c4mutxtuv3vsfmh9l7r4q2jo55qib96" xml:space="preserve">== AI usage in journalism ==

'''Overview'''

Artificial Intelligence (AI) has become present in everyday life and has also transformed the field of journalism. Because AI can generate content instantly, many people have recognised its potential to simplify various tasks. And with that, people have realised AI’s capacity for writing assistance in journalism. However, the use of AI in news production has also raised concerns regarding accuracy, reliability and misinformation. This occurs because, as Broussard (2019) mentions, some people consider what they read on the Internet to be a legitimate news outlet. Understanding both the functions and implications of AI is essential for evaluating its impact on modern news production.

(The IFTTT team) describes AI, more specifically ChatGPT, as the future at our fingertips. This reflects the fact that ChatGPT can serve as a tool that helps people organise and process information more efficiently. AI systems can assist in content creation, writing support, editorial tasks, and entertainment. However, these systems do not operate through independent thought. They rely on mathematical models and algorithms that process patterns in large datasets. During training, AI models learn from human feedback, which allows later versions to produce responses that are more accurate and useful.

Benefits of AI in Journalism

According to the IFTTT Team (2024), AI systems such as ChatGPT are applied to routine newsroom tasks. They can be used to transcribe interviews, organise large amounts of information, and generate summaries of lengthy documents. AI tools are also employed to analyse data more quickly and to identify patterns that might otherwise remain unnoticed.

These AI functions can be used as a support tool in journalism, assisting with repetitive or technical processes while journalists continue to focus on investigation and analysis.

Ethical concerns

As mentioned before, AI systems like ChatGPT are trained on massive amounts of text, including books, articles and conversations (IFTTT, 2024). This raises questions about copyright, originality, and ownership of ideas.

As Porlezza and Schapals (2024) explain, transparency, accountability, and bias are among the most frequently discussed ethical challenges associated with AI in journalism. Researchers have argued that AI systems may operate as “black boxes”, making it difficult to understand how decisions are made and who is responsible for errors.

In addition, algorithmic systems can reinforce existing biases and contribute to the creation of information bubbles that limit exposure to diverse perspectives.

Broussard’s Criticism

Meredith Broussard argues that AI tools are not neutral and should not be trusted blindly in journalism. In Artificial Unintelligence, she explains that AI reflects the biases of its creators and the data it is trained on (Broussard, 2018). Therefore, AI can reproduce stereotypes, reinforce social inequalities, and amplify misinformation. She identifies this over-reliance on technology as a form of “technological solutionism”, the assumption that technological solutions are always superior to human ones (Broussard, 2019). In her view, this bias leads newsrooms to adopt AI for tasks where human judgement remains essential.

Avoid in-text citations of authors or books.

Broussard also mentions that journalism must remain human-centred, with critical thinking and ethical responsibility. Using the metaphor of a riding mower that cannot cut sharp corners, she illustrates that automation handles routine work but fails at edge cases, which require human intervention to complete properly (Broussard, 2019). In the same way, AI can help with tasks like summarising or organising information, but it cannot replace the investigative role of journalists.

- Broussard's chapters must be referenced separately.

- For the IFTTT source, check item 17 from Chapter 10 in the APA Guide.

References:

Broussard, M. (2018). ''Artificial unintelligence: How computers misunderstand the world'' (Chaps. 1–2). MIT Press. &lt;nowiki&gt;https://doi.org/10.7551/mitpress/11022.001.0001&lt;/nowiki&gt;

Broussard, M. (2019). Letting go of technochauvinism. ''Public Books''. &lt;nowiki&gt;https://www.publicbooks.org/letting-go-of-technochauvinism/&lt;/nowiki&gt;

IFTTT Team. (2024). How does ChatGPT work? ''IFTTT Blog.'' &lt;nowiki&gt;https://ifttt.com/explore/how-does-chat-gpt-work&lt;/nowiki&gt;

Porlezza, C., &amp; Schapals, A. K. (2024). AI ethics in journalism (studies): An evolving field between research and practice. ''Journalism Studies'', ''2''(3). &lt;nowiki&gt;https://doi.org/10.1177/27523543241288818&lt;/nowiki&gt; [[User:Florencia.S.Diaz|Florencia.S.Diaz]] ([[User talk:Florencia.S.Diaz|discuss]] • [[Special:Contributions/Florencia.S.Diaz|contribs]]) 20:24, 9 July 2026 (UTC)</text>
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  <page>
    <title>User talk:FlorenciaSoledadDiaz</title>
    <ns>3</ns>
    <id>330531</id>
    <revision>
      <id>2818021</id>
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      <contributor>
        <username>FlorenciaSoledadDiaz</username>
        <id>3099917</id>
      </contributor>
      <comment>/* AI usage in journalism */ new section</comment>
      <origin>2818021</origin>
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      <format>text/x-wiki</format>
      <text bytes="5232" sha1="8rjsg8mz60l8z7hisoohrc83ob7195s" xml:space="preserve">== AI usage in journalism ==

&lt;u&gt;Overview&lt;/u&gt; 

Artificial Intelligence (AI) has become present in everyday life and has also transformed the field of journalism. Because AI can generate content instantly, many people have recognised its potential to simplify various tasks. And with that, people have realised AI’s capacity for writing assistance in journalism. However, the use of AI in news production has also raised concerns regarding accuracy, reliability and misinformation. This occurs because some people consider what they read on the Internet to be a legitimate news outlet. Understanding both the functions and implications of AI is part of evaluating its role in modern news production, as these systems are increasingly used in media organizations.  

&lt;u&gt;AI’s Function&lt;/u&gt; 

AI, more specifically ChatGPT, has been described as the future at our fingertips. This reflects the fact that ChatGPT can serve as a tool that helps people organise and process information more efficiently. AI systems can assist in content creation, writing support, editorial tasks, and entertainment. However, these systems do not operate through independent thought. They rely on mathematical models and algorithms that process patterns in large datasets. During training, AI models learn from human feedback, which allows later versions to produce responses that are more accurate and useful. These tools are designed to handle specific tasks while users remain responsible for reviewing the output.  

&lt;u&gt;Benefits of AI in Journalism&lt;/u&gt; 

AI systems such as ChatGPT are applied to routine newsroom tasks. They can be used to transcribe interviews, organise large amounts of information, and generate summaries of lengthy documents. AI tools are also employed to analyse data more quickly and to identify patterns that might otherwise remain unnoticed.   

These AI functions can be used as a support tool in journalism, assisting with repetitive or technical processes while journalists continue to focus on investigation and analysis. This allows newsrooms to direct time and resources toward reporting and verification tasks that require human attention.  

&lt;u&gt;Ethical concerns&lt;/u&gt; 

This usage in journalism raises questions about copyright, originality, and ownership of ideas; after all, transparency, accountability, and bias are among the most frequently discussed ethical challenges associated with AI in journalism. Researchers have argued that AI systems may operate as “black boxes”, making it difficult to understand how decisions are made and who is responsible for errors. 

In addition, algorithmic systems can reinforce existing biases and contribute to the creation of information bubbles that limit exposure to diverse perspectives. These issues are part of ongoing discussions about how AI should be regulated and monitored in professional settings.  

&lt;u&gt;Broussard’s Criticism&lt;/u&gt; 

Some scholars argue that AI tools are not neutral and should not be trusted blindly in journalism. AI reflects the biases of its creators and the data it is trained on. Therefore, AI can reproduce stereotypes, reinforce social inequalities, and amplify misinformation. This over-reliance on technology has been described as a form of &quot;technochauvinism&quot;, the assumption that technological solutions are always superior to human ones. This bias leads newsrooms to adopt AI for tasks where human judgement remains essential. 

It has also been argued that journalism should prioritize human-centered values such as critical thinking and ethical responsibility. Using the metaphor of a riding mower that cannot cut sharp corners, one scholar illustrates that automated tools can handle routine work but fail on edge cases that require human intervention to complete properly. In the same way, AI can help with tasks like summarizing or organizing information, but it is generally agreed that it cannot replace the investigative role of journalists. The metaphor serves to show that automation works well for predictable tasks but has limitations in situations that require judgement and contextual awareness. 

'''References'''

Broussard, M. (2018). Chapter 1: Hello, Reader. ''Artificial unintelligence: How computers misunderstand the world''. MIT Press.  

&lt;nowiki&gt;https://doi.org/10.7551/mitpress/11022.001.0001&lt;/nowiki&gt; 

Broussard, M. (2018). Chapter 2: Hello, World. Artificial unintelligence: ''How computers misunderstand the world''. MIT Press.  

&lt;nowiki&gt;https://doi.org/10.7551/mitpress/11022.001.0001&lt;/nowiki&gt;  

Broussard, M. (2019, June 17). Letting go of technochauvinism. ''Public Books''. &lt;nowiki&gt;https://www.publicbooks.org/letting-go-of-technochauvinism/&lt;/nowiki&gt;  

IFTTT Team. (2024, August 2). How does ChatGPT work? ''IFTTT Blog''. 

&lt;nowiki&gt;https://ifttt.com/explore/how-does-chat-gpt-work&lt;/nowiki&gt;  

Porlezza, C., &amp; Schapals, A. K. (2024). AI ethics in journalism (studies): An evolving field between research and practice. ''Journalism Studies'', ''2''(3). 

&lt;nowiki&gt;https://doi.org/10.1177/27523543241288818&lt;/nowiki&gt; [[User:FlorenciaSoledadDiaz|FlorenciaSoledadDiaz]] ([[User talk:FlorenciaSoledadDiaz|discuss]] • [[Special:Contributions/FlorenciaSoledadDiaz|contribs]]) 22:10, 9 July 2026 (UTC)</text>
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      <comment>Created page with &quot;{{subst:iusc|User:Daniel Quinlan/Scripts/Clock.js}}&quot;</comment>
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      <comment>I have created a new page related to Artificial Intelligence.</comment>
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      <text bytes="778" sha1="1bwbreshhqrwy36ty6v6cbb7rm8ixrz" xml:space="preserve">= Artificial Intelligence and the Development of Cognitive Skills =
=== Overview ===
Although the term Artificial Intelligence (AI) was first introduced by John McCarthy at a conference in the twentieth century, it has gained widespread popularity only in recent years with the rapid development of AI systems. These technologies are integrated into people's everyday life for content creation, writing assistance, programming, education, professional development, entertainment, and many other tasks. Alongside AI's practical applications, researchers have examined its influence on cognitive development, particularly on learning, memory and critical thinking. Current studies recognise both the educational benefits and the cognitive limitations related to the overuse of AI.</text>
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      <comment>I added 4 headings and 4 paragraphs. I added five references to sources.</comment>
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      <text bytes="5895" sha1="kigw3iqes0z6irdbvo7idnp1ogjd3o3" xml:space="preserve">= '''Artificial Intelligence and the Development of Cognitive Skills''' =

== '''Overview''' ==
Although the term '''[[Artificial Intelligence]]''' (AI) was first introduced by John McCarthy at a conference in the twentieth century, it has gained widespread popularity only in recent years with the rapid development of '''AI systems'''. These technologies are integrated into people’s everyday life for content creation, writing assistance, programming, education, professional development, entertainment, and many other tasks. Alongside AI’s practical applications, researchers have examined its influence on '''cognitive development''', particularly on learning, memory and critical thinking. Current studies recognise both the educational benefits and the cognitive limitations related to the overuse of AI.

== '''How AI works and its Usefulness''' ==
Computer technicians have defined AI systems such as '''ChatGPT''' (Generative Pre-trained Transformer) as tools based on '''[[Natural Language Processing]]''' (NLP), which enables technological devices to manipulate, interpret, and generate human languages. This process involves breaking down sentences into words and phrases in order to understand the context of a conversation better and provide users with a meaningful response. Its [[Algorithms|'''algorithm''']] splits the input into subwords and creates a vector for each section. Because these models can be customised to meet users’ specific needs, they are capable of assisting when summarising texts, translating languages or supporting professional work. AI’s focus is on engaging, satisfying and giving a better experience.&lt;ref&gt;{{Cite web|url=https://ifttt.com/explore/how-does-chat-gpt-work|title=How Does ChatGPT Work?|last=IFTTT|website=IFTTT|language=en|access-date=2026-07-10}}&lt;/ref&gt;

== '''AI Systems and Human Reasoning''' ==
Recent research in the field of [[Journalism|'''journalism''']] has raised concerns about how AI works. AI systems operate through '''mathematical probability''' rather than human reasoning. &lt;ref&gt;{{Cite book|title=Artificial Unintelligence: How Computers Misunderstand the World|last=Broussard|first=Meredith|publisher=The MIT Press|year=2018|isbn=9780262038003|location=London, England|pages=33-41|chapter=Hello, AI}}&lt;/ref&gt; Their responses are based on '''statistical information''' identified during training instead of personal experiences or ethical judgements. This training uses billions of examples from online forums, books or movies. Consequently, experts are concerned about inaccurate information, as AI systems do not verify the truthfulness of their answers. Researchers recommend that users compare AI-generated information with reliable sources, particularly in academic contexts. Therefore, it has been argued that AI successfully performs language-related tasks but does not disseminate information according to social, cultural, and moral considerations as humans do.&lt;ref&gt;{{Cite book|title=Artificial Unintelligence: How Computers Misunderstand the World|last=Broussard|first=Meredith|publisher=The MIT Press|year=2018|isbn=9780262038003|location=London, England|pages=146-157|chapter=Popular Doesn't Mean Good}}&lt;/ref&gt;

== '''Impact on Cognitive Skills''' ==
Medical professionals have shed light on AI’s influence on human cognitive skills. Research suggests that over-reliance on AI for tasks such as writing and problem-solving may reduce opportunities to practise [[Cognitive processes|'''cognitive processes''']]. This over-reliance has been described as '''cognitive debt''', referring to the gradual reduction of mental effort when intellectual tasks are delegated to AI systems.&lt;ref&gt;{{Cite journal|last=Chen|first=Youjie|last2=Wang|first2=Yingying|last3=Wüstenberg|first3=Torsten|last4=Kizilcec|first4=Rene F.|last5=Fan|first5=Yiwen|last6=Li|first6=Yanfei|last7=Lu|first7=Bin|last8=Yuan|first8=Meng|last9=Zhang|first9=Junlai|date=2025-07-11|title=Effects of generative artificial intelligence on cognitive effort and task performance: study protocol for a randomized controlled experiment among college students|url=https://doi.org/10.1186/s13063-025-08950-3|journal=Trials|language=en|volume=26|issue=1|pages=244|doi=10.1186/s13063-025-08950-3|issn=1745-6215|pmc=12255134|pmid=40646586}}&lt;/ref&gt; For instance, in the educational area, skills such as critical thinking and analytical reasoning receive less practice. Critical thinking involves evaluating evidence, questioning information and comparing multiple sources before reaching a conclusion.

== '''Educational Implications''' ==
Experts indicate that AI’s educational impact depends on how it is used. When it is employed as a learning aid rather than a replacement for independent thinking, AI supports personalised learning, provides feedback, and assists users during the writing process. However, researchers also emphasise the importance of maintaining active '''cognitive engagement''' via analysing and verifying trustworthy sources critically. For example, a renowned journalist has recently observed an AI’s limitation in the field of healthcare. Physicians use AI systems as assisting aids to help them detect diseases, as AI tools are useful in identifying patterns in medical images. But, their performances depend on the quality and type of data on which they have been trained, meaning that inaccurate data may lead to incorrect predictions.&lt;ref&gt;{{Cite news|url=https://www.wired.com/story/artificial-intelligence-cancer-detection/|title=An AI Told Me I Had Cancer|last=Broussard|first=Meredith|date=2023-03-15|work=Wired|access-date=2026-07-10|language=en-US|issn=1059-1028}}&lt;/ref&gt; For this reason, many educational institutions advocate the responsible use of AI, encouraging students to learn from these technological tools. As much as critical thinking is fostered, students’ agencies are developed.

=== Reference List ===</text>
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      <text bytes="3389" sha1="gmeiqpm2897ii8p1tekfdvg2xrhmuza" xml:space="preserve">= Probability Dilation Theory / Dilation Vector Field =

This page is a subpage of [[Probability Dilation Theory]].

== Introduction ==

Probability Dilation Theory (PDT) is naturally formulated as an iterative transformation on probability measures. When the dilation operator depends smoothly on a parameter, the resulting evolution defines a vector field on the manifold of probability distributions.

This vector field provides a differential-geometric description of PDT and forms the basis for studying probability flows, stability, and continuous-time dynamics.

== Smooth Dilation Families ==

Let

&lt;math&gt;
T_\lambda
&lt;/math&gt;

denote a family of probability dilation operators depending smoothly on a parameter

&lt;math&gt;
\lambda.
&lt;/math&gt;

The corresponding probability distribution is

&lt;math&gt;
P_\lambda=T_\lambda(P).
&lt;/math&gt;

As

&lt;math&gt;
\lambda
&lt;/math&gt;

varies, the distributions trace a curve on the statistical manifold.

== Definition: Dilation Vector Field ==

The '''dilation vector field''' is defined by

&lt;math&gt;
V_P
=
\left.
\frac{d}{d\lambda}
T_\lambda(P)
\right|_{\lambda=0}.
&lt;/math&gt;

This vector represents the instantaneous direction in which the probability measure evolves under infinitesimal dilation.

== Exponential Dilation ==

Suppose the dilation field is written as

&lt;math&gt;
D_\lambda(x)
=
e^{\lambda\phi(x)},
&lt;/math&gt;

where

&lt;math&gt;
\phi(x)
&lt;/math&gt;

is a real-valued generator.

The corresponding PDT transformation is

&lt;math&gt;
P_\lambda(x)
=
\frac{
e^{\lambda\phi(x)}P(x)
}
{
\int_\Omega
e^{\lambda\phi(x)}
\,dP(x)
}.
&lt;/math&gt;

Differentiating with respect to

&lt;math&gt;
\lambda
&lt;/math&gt;

at

&lt;math&gt;
\lambda=0
&lt;/math&gt;

gives

&lt;math&gt;
V_P(x)
=
\left(
\phi(x)
-
\mathbb E_P[\phi]
\right)
P(x).
&lt;/math&gt;

Thus the direction of probability flow depends upon the deviation of

&lt;math&gt;
\phi(x)
&lt;/math&gt;

from its expectation.

== Normalization ==

The subtraction of the expectation

&lt;math&gt;
\mathbb E_P[\phi]
&lt;/math&gt;

ensures that

&lt;math&gt;
\int_\Omega V_P\,dP=0,
&lt;/math&gt;

so that probability normalization is preserved throughout the flow.

== Fisher Geometry ==

When the statistical manifold is equipped with the Fisher–Rao metric, the dilation vector field becomes a tangent vector on a Riemannian manifold.

Its magnitude is given by

&lt;math&gt;
\|V_P\|_{FR}
=
\sqrt{
\langle
V_P,
V_P
\rangle_{FR}
}.
&lt;/math&gt;

This provides an intrinsic measure of the speed of probability evolution under PDT.

== Fixed Points ==

A probability distribution

&lt;math&gt;
P^*
&lt;/math&gt;

is a fixed point of PDT if

&lt;math&gt;
T_D(P^*)
=
P^*.
&lt;/math&gt;

Equivalently,

&lt;math&gt;
V_{P^*}
=
0.
&lt;/math&gt;

Thus fixed points correspond to equilibrium points of the dilation vector field.

== Interpretation ==

The dilation vector field transforms PDT from an iterative procedure into a geometric dynamical system.

Rather than viewing PDT as a sequence of discrete probability updates, one may regard it as generating continuous trajectories on the manifold of probability measures.

This interpretation provides a natural framework for studying stability, convergence, geodesics, and gradient flows.

== See Also ==

* [[Probability Dilation Theory]]
* [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]]
* [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]]
* [[Probability Dilation Theory/Convergence and Fixed Points]]
* [[Probability Dilation Theory/Measure-Theoretic Foundations]]</text>
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      <text bytes="4121" sha1="m0s0u0ccbee5k8qf029ve4ost0j7o7x" xml:space="preserve">= Probability Dilation Theory / Dilation Flows =

This page is a subpage of [[Probability Dilation Theory]].

== Introduction ==

Probability Dilation Theory (PDT) is naturally formulated as an iterative transformation on probability measures. When the dilation operator depends smoothly on a continuous parameter, the resulting evolution defines a probability flow on the statistical manifold.

The concept of a dilation flow provides a unified framework for studying continuous-time evolution, stability, convergence, and information geometry.

== Probability Flows ==

Let

&lt;math&gt;
P_t
&lt;/math&gt;

denote a continuously evolving probability measure parameterized by time

&lt;math&gt;
t.
&lt;/math&gt;

The evolution of the system may be written as

&lt;math&gt;
P_t=\Phi_t(P_0),
&lt;/math&gt;

where

&lt;math&gt;
\Phi_t
&lt;/math&gt;

is the dilation flow generated by PDT.

The family

&lt;math&gt;
\{\Phi_t\}_{t\ge0}
&lt;/math&gt;

describes the trajectory of a probability distribution through the statistical manifold.

== Flow Equation ==

Suppose the dilation vector field is

&lt;math&gt;
V_P.
&lt;/math&gt;

The continuous evolution of PDT is governed by

&lt;math&gt;
\frac{dP_t}{dt}=V_{P_t}.
&lt;/math&gt;

This equation is the continuous analogue of the discrete PDT iteration

&lt;math&gt;
P_{n+1}=T_D(P_n).
&lt;/math&gt;

== Exponential Dilation ==

For exponential dilation fields

&lt;math&gt;
D_\lambda(x)=e^{\lambda\phi(x)},
&lt;/math&gt;

the vector field is

&lt;math&gt;
V_P(x)
=
(\phi(x)-\mathbb E_P[\phi])P(x).
&lt;/math&gt;

Therefore the probability flow satisfies

&lt;math&gt;
\frac{dP_t(x)}{dt}
=
(\phi(x)-\mathbb E_{P_t}[\phi])P_t(x).
&lt;/math&gt;

This equation preserves probability normalization throughout the evolution.

== Orbits ==

Given an initial probability measure

&lt;math&gt;
P_0,
&lt;/math&gt;

its orbit under PDT is

&lt;math&gt;
\mathcal O(P_0)
=
\{
\Phi_t(P_0):t\ge0
\}.
&lt;/math&gt;

Different initial distributions generate different probability trajectories.

== Equilibrium Points ==

A probability distribution

&lt;math&gt;
P^*
&lt;/math&gt;

is an equilibrium point if

&lt;math&gt;
V_{P^*}=0.
&lt;/math&gt;

Equivalently,

&lt;math&gt;
\Phi_t(P^*)
=
P^*
&lt;/math&gt;

for every

&lt;math&gt;
t.
&lt;/math&gt;

Thus equilibrium points coincide with fixed points of the dilation operator.

== Stability ==

An equilibrium is said to be stable if nearby probability distributions remain close under the dilation flow.

Determining conditions for stability remains an open problem in PDT.

Possible tools include:

* Fisher geometry
* Wasserstein geometry
* Lyapunov methods
* entropy evolution

== Relation to Euler Methods ==

Euler's method approximates the continuous flow by

&lt;math&gt;
P_{n+1}
=
P_n
+
hV_{P_n},
&lt;/math&gt;

where

&lt;math&gt;
h
&lt;/math&gt;

is the time step.

Thus the discrete PDT iteration may be interpreted as an approximation to an underlying continuous probability flow.

== Information Geometry ==

When equipped with the Fisher–Rao metric, the statistical manifold becomes a Riemannian manifold.

The dilation flow then becomes a trajectory on this curved manifold.

Its speed is measured by

&lt;math&gt;
\|V_P\|_{FR},
&lt;/math&gt;

while the geometry determines the intrinsic distance between successive probability measures.

== Interpretation ==

Probability Dilation Theory may therefore be viewed as a dynamical system on the manifold of probability measures.

The dilation operator generates a vector field, the vector field generates a flow, and the flow determines the evolution of probability distributions through information-geometric space.

== Open Questions ==

Several mathematical questions remain open.

* Do all PDT operators generate smooth flows?
* Are dilation flows geodesic with respect to the Fisher metric?
* Does entropy act as a Lyapunov function?
* Under what conditions do unique equilibrium points exist?
* Which dilation fields generate globally stable flows?

== See Also ==

* [[Probability Dilation Theory]]
* [[Probability Dilation Theory/Dilation Vector Field]]
* [[Probability Dilation Theory/Fisher Geometry and Dilation Flows]]
* [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]]
* [[Probability Dilation Theory/Convergence and Fixed Points]]
* [[Probability Dilation Theory/Worked Example]]</text>
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      <comment>Created page with &quot;/**  * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson  * links on a Wikiversity course page, shows a visual progress bar, and  * persists completion state permanently via a userjs- prefixed user  * option (survives across devices/sessions, since it's stored server-side  * on the user's account, not just in the browser).  *  * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js  * Load via common.js:  *   mw.loader.load('...&quot;</comment>
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 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (survives across devices/sessions, since it's stored server-side
 * on the user's account, not just in the browser).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
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		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	/**
	 * Save completion state for this course, both locally (mw.user.options
	 * cache, instant) and server-side (persists across sessions/devices).
	 */
	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json ); // update local cache immediately
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data ); // persist server-side
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function init() {
		// Lesson links = links within the content area pointing to subpages
		// of this course (same pattern as chapter-style navigation).
		var $links = $( '#mw-content-text a[href]' ).filter( function () {
			var href = $( this ).attr( 'href' ) || '';
			var title = mw.util.getParamValue( 'title', href ) ||
				decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );
			return title.indexOf( pageName + '/' ) === 0;
		} );

		if ( !$links.length ) { return; } // no detectable lessons on this page

		var savedDone = readSavedState();
		var total = $links.length;

		var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
		var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
		var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
		var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
		$track.append( $fill );
		$barWrap.append( $label, $track );

		var $heading = $( '#firstHeading' );
		if ( $heading.length ) {
			$heading.after( $barWrap );
		} else {
			$( '#mw-content-text' ).before( $barWrap );
		}

		function countDone() {
			return $links.filter( function () {
				return $( this ).prev( '.cpt-checkbox' ).is( ':checked' );
			} ).length;
		}

		function refreshBar() {
			updateProgressBar( $fill, $label, countDone(), total );
		}

		$links.each( function () {
			var $link = $( this );
			var href = $link.attr( 'href' );
			var isDone = savedDone.indexOf( href ) !== -1;

			var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
			$link.toggleClass( 'cpt-done-text', isDone );
			$link.before( $checkbox );

			$checkbox.on( 'change', function () {
				var checked = $checkbox.is( ':checked' );
				$link.toggleClass( 'cpt-done-text', checked );

				var current = readSavedState();
				var idx = current.indexOf( href );
				if ( checked &amp;&amp; idx === -1 ) {
					current.push( href );
				} else if ( !checked &amp;&amp; idx !== -1 ) {
					current.splice( idx, 1 );
				}
				saveState( current );
				refreshBar();
			} );
		} );

		refreshBar();
	}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	} );

}() );</text>
      <sha1>l4d7dh10ipgvo4jv58rh5lo20wdeg7u</sha1>
    </revision>
    <revision>
      <id>2818047</id>
      <parentid>2818045</parentid>
      <timestamp>2026-07-10T09:26:01Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <origin>2818047</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="4932" sha1="9uzhnu0jev7kpjtjd024ff8i0ganrts" xml:space="preserve">/**
 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (survives across devices/sessions, since it's stored server-side
 * on the user's account, not just in the browser).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
/* global mw */
( function () {
	'use strict';

	if ( mw.config.get( 'wgAction' ) !== 'view' ) { return; }
	if ( mw.config.get( 'wgNamespaceNumber' ) !== 0 ) { return; } // course/content namespace only

	var pageName = mw.config.get( 'wgPageName' ).replace( /_/g, ' ' );
	// Option key must start with &quot;userjs-&quot; per MediaWiki's option API rules.
	// Keyed per-course so different courses track separately.
	var optionKey = 'userjs-cpt-' + pageName.replace( /[^a-zA-Z0-9]/g, '_' ).slice( 0, 60 );

	mw.util.addCSS(
		'#cpt-bar-wrap{margin:1em 0;border:1px solid #a2a9b1;border-radius:4px;padding:10px 14px;background:var(--background-color-interactive,#f8f9fa);font-size:13px;}' +
		'#cpt-bar-label{margin-bottom:6px;color:#202122;font-weight:600;}' +
		'#cpt-bar-track{height:10px;border-radius:5px;background:#eaecf0;overflow:hidden;}' +
		'#cpt-bar-fill{height:100%;background:#00af89;width:0%;transition:width .3s ease;}' +
		'.cpt-checkbox{margin-right:6px;transform:scale(1.1);cursor:pointer;vertical-align:middle;}' +
		'.cpt-done-text{text-decoration:line-through;color:#54595d;}'
	);

	var api = new mw.Api();

	/**
	 * Read saved completion state for this course.
	 * @return {Array&lt;string&gt;}  array of completed lesson hrefs
	 */
	function readSavedState() {
		var raw = mw.user.options.get( optionKey );
		if ( !raw ) { return []; }
		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	/**
	 * Save completion state for this course, both locally (mw.user.options
	 * cache, instant) and server-side (persists across sessions/devices).
	 */
	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json ); // update local cache immediately
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data ); // persist server-side
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function init() {
	var $links = $( '#mw-content-text a[href]' ).filter( function () {
		var href = $( this ).attr( 'href' ) || '';
		var title = mw.util.getParamValue( 'title', href ) ||
			decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );
		return title.indexOf( pageName + '/' ) === 0;
	} );

	// De-duplicate: keep only the FIRST link to each unique lesson page.
	var seen = {};
	$links = $links.filter( function () {
		var href = $( this ).attr( 'href' );
		if ( seen[ href ] ) { return false; }
		seen[ href ] = true;
		return true;
	} );

	if ( !$links.length ) { return; }
 // no detectable lessons on this page

		var savedDone = readSavedState();
		var total = $links.length;

		var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
		var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
		var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
		var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
		$track.append( $fill );
		$barWrap.append( $label, $track );

		var $heading = $( '#firstHeading' );
		if ( $heading.length ) {
			$heading.after( $barWrap );
		} else {
			$( '#mw-content-text' ).before( $barWrap );
		}

		function countDone() {
			return $links.filter( function () {
				return $( this ).prev( '.cpt-checkbox' ).is( ':checked' );
			} ).length;
		}

		function refreshBar() {
			updateProgressBar( $fill, $label, countDone(), total );
		}

		$links.each( function () {
			var $link = $( this );
			var href = $link.attr( 'href' );
			var isDone = savedDone.indexOf( href ) !== -1;

			var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
			$link.toggleClass( 'cpt-done-text', isDone );
			$link.before( $checkbox );

			$checkbox.on( 'change', function () {
				var checked = $checkbox.is( ':checked' );
				$link.toggleClass( 'cpt-done-text', checked );

				var current = readSavedState();
				var idx = current.indexOf( href );
				if ( checked &amp;&amp; idx === -1 ) {
					current.push( href );
				} else if ( !checked &amp;&amp; idx !== -1 ) {
					current.splice( idx, 1 );
				}
				saveState( current );
				refreshBar();
			} );
		} );

		refreshBar();
	}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	} );

}() );</text>
      <sha1>9uzhnu0jev7kpjtjd024ff8i0ganrts</sha1>
    </revision>
    <revision>
      <id>2818048</id>
      <parentid>2818047</parentid>
      <timestamp>2026-07-10T09:29:28Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <origin>2818048</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="5147" sha1="k324es7aaxxw88stpzsdm8abzvr1r08" xml:space="preserve">/**
 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (stored server-side on the account, survives across devices).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
/* global mw */
( function () {
	'use strict';

	if ( mw.config.get( 'wgAction' ) !== 'view' ) { return; }
	if ( mw.config.get( 'wgNamespaceNumber' ) !== 0 ) { return; } // course/content namespace only

	// Subpages whose title (after the course name) contains any of these
	// words are treated as NOT lessons (overview/admin pages, not content).
	// Edit this list per-course if a course uses different non-lesson names.
	var EXCLUDE_KEYWORDS = [ 'overview', 'syllabus', 'participants', 'timetable',
		'bibliography', 'discussion', 'talk', 'sandbox' ];

	var pageName = mw.config.get( 'wgPageName' ).replace( /_/g, ' ' );
	var optionKey = 'userjs-cpt-' + pageName.replace( /[^a-zA-Z0-9]/g, '_' ).slice( 0, 60 );

	mw.util.addCSS(
		'#cpt-bar-wrap{margin:1em 0;border:1px solid #a2a9b1;border-radius:4px;padding:10px 14px;background:var(--background-color-interactive,#f8f9fa);font-size:13px;}' +
		'#cpt-bar-label{margin-bottom:6px;color:#202122;font-weight:600;}' +
		'#cpt-bar-track{height:10px;border-radius:5px;background:#eaecf0;overflow:hidden;}' +
		'#cpt-bar-fill{height:100%;background:#00af89;width:0%;transition:width .3s ease;}' +
		'.cpt-checkbox{margin-right:6px;transform:scale(1.1);cursor:pointer;vertical-align:middle;}' +
		'.cpt-done-text{text-decoration:line-through;color:#54595d;}'
	);

	var api = new mw.Api();

	function readSavedState() {
		var raw = mw.user.options.get( optionKey );
		if ( !raw ) { return []; }
		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json );
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data );
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function isExcluded( subTitle ) {
		var lower = subTitle.toLowerCase();
		return EXCLUDE_KEYWORDS.some( function ( kw ) { return lower.indexOf( kw ) !== -1; } );
	}

	function init() {
		var $links = $( '#mw-content-text a[href]' ).filter( function () {
			var href = $( this ).attr( 'href' ) || '';
			var title = mw.util.getParamValue( 'title', href ) ||
				decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );
			if ( title.indexOf( pageName + '/' ) !== 0 ) { return false; }
			var subTitle = title.slice( pageName.length + 1 );
			return !isExcluded( subTitle );
		} );

		// De-duplicate: keep only the FIRST link to each unique lesson page.
		var seen = {};
		$links = $links.filter( function () {
			var href = $( this ).attr( 'href' );
			if ( seen[ href ] ) { return false; }
			seen[ href ] = true;
			return true;
		} );

		if ( !$links.length ) { return; }

		var savedDone = readSavedState();
		var total = $links.length;

		var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
		var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
		var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
		var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
		$track.append( $fill );
		$barWrap.append( $label, $track );

		// Insert at the top of #content (stable across skins), not inside
		// #firstHeading (which can be a flex container and break layout).
		var $content = $( '#content' );
		if ( $content.length ) {
			$content.prepend( $barWrap );
		} else {
			$( '#mw-content-text' ).before( $barWrap );
		}

		function countDone() {
			return $links.filter( function () {
				return $( this ).prev( '.cpt-checkbox' ).is( ':checked' );
			} ).length;
		}

		function refreshBar() {
			updateProgressBar( $fill, $label, countDone(), total );
		}

		$links.each( function () {
			var $link = $( this );
			var href = $link.attr( 'href' );
			var isDone = savedDone.indexOf( href ) !== -1;

			var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
			$link.toggleClass( 'cpt-done-text', isDone );
			$link.before( $checkbox );

			$checkbox.on( 'change', function () {
				var checked = $checkbox.is( ':checked' );
				$link.toggleClass( 'cpt-done-text', checked );

				var current = readSavedState();
				var idx = current.indexOf( href );
				if ( checked &amp;&amp; idx === -1 ) {
					current.push( href );
				} else if ( !checked &amp;&amp; idx !== -1 ) {
					current.splice( idx, 1 );
				}
				saveState( current );
				refreshBar();
			} );
		} );

		refreshBar();
	}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	} );

}() );</text>
      <sha1>k324es7aaxxw88stpzsdm8abzvr1r08</sha1>
    </revision>
    <revision>
      <id>2818049</id>
      <parentid>2818048</parentid>
      <timestamp>2026-07-10T09:31:58Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <origin>2818049</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="4919" sha1="bibhwe9or038pcur3e5yvb57oehf5h6" xml:space="preserve">/**
 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (stored server-side on the account, survives across devices).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
/* global mw */
( function () {
	'use strict';

	if ( mw.config.get( 'wgAction' ) !== 'view' ) { return; }
	if ( mw.config.get( 'wgNamespaceNumber' ) !== 0 ) { return; } // course/content namespace only

	// Subpages whose title (after the course name) contains any of these
	// words are treated as NOT lessons (overview/admin pages, not content).
	// Edit this list per-course if a course uses different non-lesson names.
	var EXCLUDE_KEYWORDS = [ 'overview', 'syllabus', 'participants', 'timetable',
		'bibliography', 'discussion', 'talk', 'sandbox' ];

	var pageName = mw.config.get( 'wgPageName' ).replace( /_/g, ' ' );
	var optionKey = 'userjs-cpt-' + pageName.replace( /[^a-zA-Z0-9]/g, '_' ).slice( 0, 60 );

	mw.util.addCSS(
		'#cpt-bar-wrap{margin:1em 0;border:1px solid #a2a9b1;border-radius:4px;padding:10px 14px;background:var(--background-color-interactive,#f8f9fa);font-size:13px;}' +
		'#cpt-bar-label{margin-bottom:6px;color:#202122;font-weight:600;}' +
		'#cpt-bar-track{height:10px;border-radius:5px;background:#eaecf0;overflow:hidden;}' +
		'#cpt-bar-fill{height:100%;background:#00af89;width:0%;transition:width .3s ease;}' +
		'.cpt-checkbox{margin-right:6px;transform:scale(1.1);cursor:pointer;vertical-align:middle;}' +
		'.cpt-done-text{text-decoration:line-through;color:#54595d;}'
	);

	var api = new mw.Api();

	function readSavedState() {
		var raw = mw.user.options.get( optionKey );
		if ( !raw ) { return []; }
		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json );
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data );
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function isExcluded( subTitle ) {
		var lower = subTitle.toLowerCase();
		return EXCLUDE_KEYWORDS.some( function ( kw ) { return lower.indexOf( kw ) !== -1; } );
	}

	function init() {
	var seenTitles = {};
	var items = [];

	$( '#mw-content-text a[href]' ).each( function () {
		var $link = $( this );
		var href = $link.attr( 'href' ) || '';
		var title = mw.util.getParamValue( 'title', href ) ||
			decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );

		if ( title.indexOf( pageName + '/' ) !== 0 ) { return; }
		var subTitle = title.slice( pageName.length + 1 );
		if ( isExcluded( subTitle ) ) { return; }
		if ( seenTitles[ title ] ) { return; } // dedupe by normalized title, not raw href

		seenTitles[ title ] = true;
		items.push( { $link: $link, href: href, title: title } );
	} );

	if ( !items.length ) { return; }

	var savedDone = readSavedState();
	var total = items.length;

	var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
	var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
	var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
	var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
	$track.append( $fill );
	$barWrap.append( $label, $track );

	var $content = $( '#content' );
	if ( $content.length ) {
		$content.prepend( $barWrap );
	} else {
		$( '#mw-content-text' ).before( $barWrap );
	}

	function countDone() {
		return items.filter( function ( item ) {
			return item.$checkbox.is( ':checked' );
		} ).length;
	}

	function refreshBar() {
		updateProgressBar( $fill, $label, countDone(), total );
	}

	items.forEach( function ( item ) {
		var isDone = savedDone.indexOf( item.title ) !== -1;

		var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
		item.$link.toggleClass( 'cpt-done-text', isDone );
		item.$link.before( $checkbox );
		item.$checkbox = $checkbox;

		$checkbox.on( 'change', function () {
			var checked = $checkbox.is( ':checked' );
			item.$link.toggleClass( 'cpt-done-text', checked );

			var current = readSavedState();
			var idx = current.indexOf( item.title );
			if ( checked &amp;&amp; idx === -1 ) {
				current.push( item.title );
			} else if ( !checked &amp;&amp; idx !== -1 ) {
				current.splice( idx, 1 );
			}
			saveState( current );
			refreshBar();
		} );
	} );

	refreshBar();
}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	} );

}() );</text>
      <sha1>bibhwe9or038pcur3e5yvb57oehf5h6</sha1>
    </revision>
    <revision>
      <id>2818050</id>
      <parentid>2818049</parentid>
      <timestamp>2026-07-10T09:33:34Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <origin>2818050</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="4866" sha1="ranhh47x6oxu7nm7zfz9ebvqa3gkrx0" xml:space="preserve">/**
 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (stored server-side on the account, survives across devices).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
/* global mw */
( function () {
	'use strict';

	if ( mw.config.get( 'wgAction' ) !== 'view' ) { return; }
	if ( mw.config.get( 'wgNamespaceNumber' ) !== 0 ) { return; } // course/content namespace only

	// Subpages whose title (after the course name) contains any of these
	// words are treated as NOT lessons (overview/admin pages, not content).
	var EXCLUDE_KEYWORDS = [ 'overview', 'syllabus', 'participants', 'timetable',
		'bibliography', 'discussion', 'talk', 'sandbox' ];

	var pageName = mw.config.get( 'wgPageName' ).replace( /_/g, ' ' );
	var optionKey = 'userjs-cpt-' + pageName.replace( /[^a-zA-Z0-9]/g, '_' ).slice( 0, 60 );

	mw.util.addCSS(
		'#cpt-bar-wrap{margin:1em 0;border:1px solid #a2a9b1;border-radius:4px;padding:10px 14px;background:var(--background-color-interactive,#f8f9fa);font-size:13px;}' +
		'#cpt-bar-label{margin-bottom:6px;color:#202122;font-weight:600;}' +
		'#cpt-bar-track{height:10px;border-radius:5px;background:#eaecf0;overflow:hidden;}' +
		'#cpt-bar-fill{height:100%;background:#00af89;width:0%;transition:width .3s ease;}' +
		'.cpt-checkbox{margin-right:6px;transform:scale(1.1);cursor:pointer;vertical-align:middle;}' +
		'.cpt-done-text{text-decoration:line-through;color:#54595d;}'
	);

	var api = new mw.Api();

	function readSavedState() {
		var raw = mw.user.options.get( optionKey );
		if ( !raw ) { return []; }
		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json );
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data );
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function isExcluded( subTitle ) {
		var lower = subTitle.toLowerCase();
		return EXCLUDE_KEYWORDS.some( function ( kw ) { return lower.indexOf( kw ) !== -1; } );
	}

	function init() {
		var seenTitles = {};
		var items = [];

		$( '#mw-content-text a[href]' ).each( function () {
			var $link = $( this );
			var href = $link.attr( 'href' ) || '';
			var title = mw.util.getParamValue( 'title', href ) ||
				decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );

			if ( title.indexOf( pageName + '/' ) !== 0 ) { return; }
			var subTitle = title.slice( pageName.length + 1 );
			if ( isExcluded( subTitle ) ) { return; }
			if ( seenTitles[ title ] ) { return; } // dedupe by normalized title

			seenTitles[ title ] = true;
			items.push( { $link: $link, title: title } );
		} );

		if ( !items.length ) { return; }

		var savedDone = readSavedState();
		var total = items.length;

		var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
		var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
		var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
		var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
		$track.append( $fill );
		$barWrap.append( $label, $track );

		var $content = $( '#content' );
		if ( $content.length ) {
			$content.prepend( $barWrap );
		} else {
			$( '#mw-content-text' ).before( $barWrap );
		}

		function countDone() {
			return items.filter( function ( item ) {
				return item.$checkbox.is( ':checked' );
			} ).length;
		}

		function refreshBar() {
			updateProgressBar( $fill, $label, countDone(), total );
		}

		items.forEach( function ( item ) {
			var isDone = savedDone.indexOf( item.title ) !== -1;

			var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
			item.$link.toggleClass( 'cpt-done-text', isDone );
			item.$link.before( $checkbox );
			item.$checkbox = $checkbox;

			$checkbox.on( 'change', function () {
				var checked = $checkbox.is( ':checked' );
				item.$link.toggleClass( 'cpt-done-text', checked );

				var current = readSavedState();
				var idx = current.indexOf( item.title );
				if ( checked &amp;&amp; idx === -1 ) {
					current.push( item.title );
				} else if ( !checked &amp;&amp; idx !== -1 ) {
					current.splice( idx, 1 );
				}
				saveState( current );
				refreshBar();
			} );
		} );

		refreshBar();
	}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	}</text>
      <sha1>ranhh47x6oxu7nm7zfz9ebvqa3gkrx0</sha1>
    </revision>
    <revision>
      <id>2818051</id>
      <parentid>2818050</parentid>
      <timestamp>2026-07-10T09:35:42Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <origin>2818051</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="4877" sha1="jwh1idxzbtwav2a9wg6i5fw6tedg6xg" xml:space="preserve">/**
 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (stored server-side on the account, survives across devices).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
/* global mw */
( function () {
	'use strict';

	if ( mw.config.get( 'wgAction' ) !== 'view' ) { return; }
	if ( mw.config.get( 'wgNamespaceNumber' ) !== 0 ) { return; } // course/content namespace only

	// Subpages whose title (after the course name) contains any of these
	// words are treated as NOT lessons (overview/admin pages, not content).
	var EXCLUDE_KEYWORDS = [ 'overview', 'syllabus', 'participants', 'timetable',
		'bibliography', 'discussion', 'talk', 'sandbox' ];

	var pageName = mw.config.get( 'wgPageName' ).replace( /_/g, ' ' );
	var optionKey = 'userjs-cpt-' + pageName.replace( /[^a-zA-Z0-9]/g, '_' ).slice( 0, 60 );

	mw.util.addCSS(
		'#cpt-bar-wrap{margin:1em 0;border:1px solid #a2a9b1;border-radius:4px;padding:10px 14px;background:var(--background-color-interactive,#f8f9fa);font-size:13px;}' +
		'#cpt-bar-label{margin-bottom:6px;color:#202122;font-weight:600;}' +
		'#cpt-bar-track{height:10px;border-radius:5px;background:#eaecf0;overflow:hidden;}' +
		'#cpt-bar-fill{height:100%;background:#00af89;width:0%;transition:width .3s ease;}' +
		'.cpt-checkbox{margin-right:6px;transform:scale(1.1);cursor:pointer;vertical-align:middle;}' +
		'.cpt-done-text{text-decoration:line-through;color:#54595d;}'
	);

	var api = new mw.Api();

	function readSavedState() {
		var raw = mw.user.options.get( optionKey );
		if ( !raw ) { return []; }
		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json );
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data );
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function isExcluded( subTitle ) {
		var lower = subTitle.toLowerCase();
		return EXCLUDE_KEYWORDS.some( function ( kw ) { return lower.indexOf( kw ) !== -1; } );
	}

	function init() {
		var seenTitles = {};
		var items = [];

		$( '#mw-content-text a[href]' ).each( function () {
			var $link = $( this );
			var href = $link.attr( 'href' ) || '';
			var title = mw.util.getParamValue( 'title', href ) ||
				decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );

			if ( title.indexOf( pageName + '/' ) !== 0 ) { return; }
			var subTitle = title.slice( pageName.length + 1 );
			if ( isExcluded( subTitle ) ) { return; }
			if ( seenTitles[ title ] ) { return; } // dedupe by normalized title

			seenTitles[ title ] = true;
			items.push( { $link: $link, title: title } );
		} );

		if ( !items.length ) { return; }

		var savedDone = readSavedState();
		var total = items.length;

		var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
		var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
		var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
		var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
		$track.append( $fill );
		$barWrap.append( $label, $track );

		var $content = $( '#content' );
		if ( $content.length ) {
			$content.prepend( $barWrap );
		} else {
			$( '#mw-content-text' ).before( $barWrap );
		}

		function countDone() {
			return items.filter( function ( item ) {
				return item.$checkbox.is( ':checked' );
			} ).length;
		}

		function refreshBar() {
			updateProgressBar( $fill, $label, countDone(), total );
		}

		items.forEach( function ( item ) {
			var isDone = savedDone.indexOf( item.title ) !== -1;

			var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
			item.$link.toggleClass( 'cpt-done-text', isDone );
			item.$link.before( $checkbox );
			item.$checkbox = $checkbox;

			$checkbox.on( 'change', function () {
				var checked = $checkbox.is( ':checked' );
				item.$link.toggleClass( 'cpt-done-text', checked );

				var current = readSavedState();
				var idx = current.indexOf( item.title );
				if ( checked &amp;&amp; idx === -1 ) {
					current.push( item.title );
				} else if ( !checked &amp;&amp; idx !== -1 ) {
					current.splice( idx, 1 );
				}
				saveState( current );
				refreshBar();
			} );
		} );

		refreshBar();
	}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	} );

}() );</text>
      <sha1>jwh1idxzbtwav2a9wg6i5fw6tedg6xg</sha1>
    </revision>
    <revision>
      <id>2818052</id>
      <parentid>2818051</parentid>
      <timestamp>2026-07-10T09:42:17Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <origin>2818052</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="5345" sha1="5e8aocknvdd9qis4r93lstmg3325hij" xml:space="preserve">/**
 * Course Progress Tracker — adds &quot;mark as done&quot; checkboxes next to lesson
 * links on a Wikiversity course page, shows a visual progress bar, and
 * persists completion state permanently via a userjs- prefixed user
 * option (stored server-side on the account, survives across devices).
 *
 * Save at: en.wikiversity.org/wiki/User:Gauri_Guptaa/courseProgressTracker.js
 * Load via common.js:
 *   mw.loader.load('/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript');
 *
 * @author  [[User:Gauri_Guptaa]]
 * @license CC0-1.0
 */
/* global mw */
( function () {
	'use strict';

	if ( mw.config.get( 'wgAction' ) !== 'view' ) { return; }
	if ( mw.config.get( 'wgNamespaceNumber' ) !== 0 ) { return; } // course/content namespace only

	// Subpages whose title (after the course name) contains any of these
	// words are treated as NOT lessons (overview/admin pages, not content).
	var EXCLUDE_KEYWORDS = [ 'overview', 'syllabus', 'participants', 'timetable',
		'bibliography', 'discussion', 'talk', 'sandbox' ];

	var pageName = mw.config.get( 'wgPageName' ).replace( /_/g, ' ' );
	var optionKey = 'userjs-cpt-' + pageName.replace( /[^a-zA-Z0-9]/g, '_' ).slice( 0, 60 );

	mw.util.addCSS(
		'#cpt-bar-wrap{margin:1em 0;border:1px solid #a2a9b1;border-radius:4px;padding:10px 14px;background:var(--background-color-interactive,#f8f9fa);font-size:13px;}' +
		'#cpt-bar-label{margin-bottom:6px;color:#202122;font-weight:600;}' +
		'#cpt-bar-track{height:10px;border-radius:5px;background:#eaecf0;overflow:hidden;}' +
		'#cpt-bar-fill{height:100%;background:#00af89;width:0%;transition:width .3s ease;}' +
		'.cpt-checkbox{margin-right:6px;transform:scale(1.1);cursor:pointer;vertical-align:middle;}' +
		'.cpt-done-text{text-decoration:line-through;color:#54595d;}'
	);

	var api = new mw.Api();

	function readSavedState() {
		var raw = mw.user.options.get( optionKey );
		if ( !raw ) { return []; }
		try {
			return JSON.parse( raw );
		} catch ( e ) { return []; }
	}

	function saveState( doneList ) {
		var json = JSON.stringify( doneList );
		mw.user.options.set( optionKey, json );
		var data = {};
		data[ optionKey ] = json;
		api.saveOptions( data );
	}

	function updateProgressBar( $fill, $label, done, total ) {
		var pct = total ? Math.round( ( done / total ) * 100 ) : 0;
		$fill.css( 'width', pct + '%' );
		$label.text( 'Course progress: ' + done + ' / ' + total + ' lessons (' + pct + '%)' );
	}

	function isExcluded( subTitle ) {
		var lower = subTitle.toLowerCase();
		return EXCLUDE_KEYWORDS.some( function ( kw ) { return lower.indexOf( kw ) !== -1; } );
	}

function init() {
	var seenLessonNumbers = {};
	var items = [];

	$( '#mw-content-text a[href]' ).each( function () {
		var $link = $( this );
		var href = $link.attr( 'href' ) || '';
		var title = mw.util.getParamValue( 'title', href ) ||
			decodeURIComponent( href.replace( /^.*\/wiki\//, '' ) ).replace( /_/g, ' ' );

		if ( title.indexOf( pageName + '/' ) !== 0 ) { return; }
		var subTitle = title.slice( pageName.length + 1 );

		// Only treat pages whose subtitle contains &quot;Lesson &lt;number&gt;&quot; as real
		// lessons — this automatically skips redirect/shortcut pages (e.g.
		// &quot;World War I/United States&quot;) and merges quote-mark variants of
		// the same lesson (e.g. straight vs curly quotes) via the number.
		var m = subTitle.match( /Lesson\s*(\d+)/i );
		if ( !m ) { return; }
		var lessonNum = m[ 1 ];
		if ( seenLessonNumbers[ lessonNum ] ) { return; } // dedupe by lesson number

		seenLessonNumbers[ lessonNum ] = true;
		items.push( { $link: $link, title: title, num: parseInt( lessonNum, 10 ) } );
	} );

	items.sort( function ( a, b ) { return a.num - b.num; } ); // keep lesson order 1,2,3...

	if ( !items.length ) { return; }

		var savedDone = readSavedState();
		var total = items.length;

		var $barWrap = $( '&lt;div id=&quot;cpt-bar-wrap&quot;&gt;' );
		var $label = $( '&lt;div id=&quot;cpt-bar-label&quot;&gt;' );
		var $track = $( '&lt;div id=&quot;cpt-bar-track&quot;&gt;' );
		var $fill = $( '&lt;div id=&quot;cpt-bar-fill&quot;&gt;' );
		$track.append( $fill );
		$barWrap.append( $label, $track );

		var $content = $( '#content' );
		if ( $content.length ) {
			$content.prepend( $barWrap );
		} else {
			$( '#mw-content-text' ).before( $barWrap );
		}

		function countDone() {
			return items.filter( function ( item ) {
				return item.$checkbox.is( ':checked' );
			} ).length;
		}

		function refreshBar() {
			updateProgressBar( $fill, $label, countDone(), total );
		}

		items.forEach( function ( item ) {
			var isDone = savedDone.indexOf( item.title ) !== -1;

			var $checkbox = $( '&lt;input type=&quot;checkbox&quot; class=&quot;cpt-checkbox&quot;&gt;' ).prop( 'checked', isDone );
			item.$link.toggleClass( 'cpt-done-text', isDone );
			item.$link.before( $checkbox );
			item.$checkbox = $checkbox;

			$checkbox.on( 'change', function () {
				var checked = $checkbox.is( ':checked' );
				item.$link.toggleClass( 'cpt-done-text', checked );

				var current = readSavedState();
				var idx = current.indexOf( item.title );
				if ( checked &amp;&amp; idx === -1 ) {
					current.push( item.title );
				} else if ( !checked &amp;&amp; idx !== -1 ) {
					current.splice( idx, 1 );
				}
				saveState( current );
				refreshBar();
			} );
		} );

		refreshBar();
	}

	mw.loader.using( [ 'mediawiki.api', 'mediawiki.util' ] ).then( function () {
		$( init );
	} );

}() );</text>
      <sha1>5e8aocknvdd9qis4r93lstmg3325hij</sha1>
    </revision>
  </page>
  <page>
    <title>User:Gauri Guptaa/common.js</title>
    <ns>2</ns>
    <id>330538</id>
    <revision>
      <id>2818046</id>
      <timestamp>2026-07-10T09:20:50Z</timestamp>
      <contributor>
        <username>Gauri Guptaa</username>
        <id>3008577</id>
      </contributor>
      <comment>Created page with &quot;// [[User:Gauri_Guptaa/courseProgressTracker.js]] — mark lessons done + progress bar mw.loader.load( '/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript' );&quot;</comment>
      <origin>2818046</origin>
      <model>javascript</model>
      <format>text/javascript</format>
      <text bytes="202" sha1="lvdhc14nzu76gi4gtsjohpm0ckt16k0" xml:space="preserve">// [[User:Gauri_Guptaa/courseProgressTracker.js]] — mark lessons done + progress bar
mw.loader.load( '/w/index.php?title=User:Gauri_Guptaa/courseProgressTracker.js&amp;action=raw&amp;ctype=text/javascript' );</text>
      <sha1>lvdhc14nzu76gi4gtsjohpm0ckt16k0</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:Dmdjdu</title>
    <ns>3</ns>
    <id>330539</id>
    <revision>
      <id>2818054</id>
      <timestamp>2026-07-10T10:48:13Z</timestamp>
      <contributor>
        <username>Dmdjdu</username>
        <id>3066290</id>
      </contributor>
      <comment>/* */</comment>
      <origin>2818054</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="220" sha1="ea7rsw85bcel5crnrvygssf1yl055hb" xml:space="preserve">== Hey there!  ==

Congratulations! If you found this page, you also found a Wikimedia editor. Wanna talk to me about something? Feel free! [[User:Dmdjdu|Dmdjdu]] ([[User talk:Dmdjdu#top|talk]]) 10:48, 10 July 2026 (UTC)</text>
      <sha1>ea7rsw85bcel5crnrvygssf1yl055hb</sha1>
    </revision>
  </page>
</mediawiki>