Talk:Coxeter–Dynkin diagram/Archive 1

message: Coxeter-Dynkins graphics on Wikipedia:
These component elements can be strung together to create linear diagrams. There are three variations are 20, 30, and 48 pixels tall and are not to be intermixed to avoid vertical alignment issues. The elements are variable width, (making them hard to systematically scale with "px" pixel-width codes).
The 20-pixel tall elements are for simple linear and integer labels, with filenames prefixed by "CDW_". The 30-pixel tall elements are for simple linear elements with fractional labels, AND ALSO for 2-level node symbols, with filenames prefixed by "CD_". The 48-pixel elements are for triple level node symbols, with filenames prefixed by as "CDT_".
Note: These are all experimental, and may be improved/simplified/renamed. SVG variations exist for some, but there are some unresolved issues with some of them.
=== NEW ===
Here's a new set (not complete), built with Template:CDD argument sets, allowing 3 vertical levels of nodes, top(a), middle, and bottom(b).  Tom Ruen (talk) 07:38, 4 January 2011 (UTC)
thread_title:  Graphics element documentation 

message: Here's a new set (not complete), built with Template:CDD argument sets, allowing 3 vertical levels of nodes, top(a), middle, and bottom(b).  Tom Ruen (talk) 07:38, 4 January 2011 (UTC)
thread_title:  NEW 

message: There are still    codes on the PNG element images, so I expect there'll be periodic heroes who will come along. If I remove the template, I'll get complains, and if I leave it, its a prod for potentially hasty action. I don't have a solution since SVG is the wiki gods' favorite image child. Tom Ruen (talk) 09:36, 15 March 2010 (UTC)
replies: text: I'd suggest you remove the tags, as they are no longer applicable. If any mistaken evangelists still arise, direct them to this discussion, Talk:Coxeter–Dynkin diagram#SVG vs PNG, and if that doesn't work give us all a shout. Soon they'll end up as crestfallen as I did. -- Cheers, Steelpillow (Talk) 19:29, 15 March 2010 (UTC)
thread_title: <nowiki>{{svg}}</nowiki> tags

message: What I really dislike much more than SVG images (sorry for the pun ;) is the fact that this article uses the names for the diagrams from Coxeter's original book. This may be the "historically" right terminology. But to my knowledge, virtually everybody in the field is using the "modern" terminology. Which was introduced by Bourbaki (or rather, Jacques Tits) and has been in usefor several decades. Experts in the field (like Humphreys, 
I.e. what is called B_n in this article is called D_n elsewhere; B_n and C_n coincide; and more. The [Coxeter group] page lists both for comparison. 
IMO it's fine to mention the old notation in a "history" section, but for practical work, only the modern names should be used -- else beginners will learn the wrong notation if they read up here, and experts will be highly confused by the mixed up names. I am willing to make the required changes, too. BlackFingolfin (talk) 12:43, 16 August 2008 (UTC)
replies: text: I'm in the process updating polytope articles to reference the newer names. Sorry things have to remain confusing for a while until it is completed. Tom Ruen (talk) 16:11, 27 August 2008 (UTC)

text: Hold on, you mentioned Humphreys, whose text I've got in front of me at the momentHumphreys, James E. ; Reflection Groups and Coxeter Groups, CUP. I've noticed that in this book, the labels B~_n and C~_n are the other way round from in this article. Which one's the convention? 137.205.31.198 (talk) 20:46, 14 December 2008 (UTC)

text: Sadly, I've seen references both ways, and I don't know which ought to be considered standard. I'll look again when I have some time, or if you want to look more, feel free to change it. Tom Ruen (talk) 21:53, 14 December 2008 (UTC)

text: Okay, I confirmed that source, and swapped them. Tom Ruen (talk) 22:02, 14 December 2008 (UTC)

text: Just made a similar swap in the Coxeter group article. I don't know how to change PNG files though, so right now there is one picture in each article that disagrees with the changes just made. 137.205.31.198 (talk) 14:12, 16 December 2008 (UTC)

text: I changed the graphic 100px. Hopefully the author won't object! Tom Ruen (talk) 16:36, 16 December 2008 (UTC)
thread_title:  Switch to the modern notation for Coxeter-dynkin diagrams 

message: The Dynkin diagrams are used in singularity theory. Arnold proved the connexion between the simple Lie groups and certain singularity types. As a result, Dynkin diagrams turn up quite a lot in the literature; although I must confess that I don't totally understand. I think it would be a nice addition to the article. Dharma6662000 (talk) 23:32, 19 August 2008 (UTC)
thread_title:  Singularity Theory 

message: The image :File:Coxeter-dynkin plane groups.png contains a couple of errors/inconsistencies in its headings. I don't have access to SVG editing facilities; perhaps someone could update it?
* The heading ' I~2×I~2 (W2×W2)' should read ' I~1×I~1 (W2×W2)'.
* The heading ' B~2 group (R3)' should be updated to ' C~2 group (R3)' in line with Tom Ruen's recent changes to the rest of the article.
Thanks! Robert Stanforth (talk) 22:28, 24 December 2008 (UTC)
replies: text: Corrected! Thanks for the careful look. (I just edited with MSPaint incidentally!) Tom Ruen (talk) 22:35, 24 December 2008 (UTC)
thread_title:  Incorrect headings in lead image 

message: I was looking at the text below the pictures of the fundamental simplexes:
C~3 fills 1/24 of the cube. B~3  fills 1/12 of the cube.
 A~3 fills 1/6 of the cube.
The pictures are beautiful, by the way. They are rarely mentioned in the literature (to my surprise). The numbers below seem to be off though. The C~3 simplex partition of the cube corresponds to the partition of one square face into 8 right triangles. Since a cube has 6 faces the simplex is 1/48 of the cube. Incidentally the fundamental simplex is the intersection of the cube with a Weyl chamber, so 48 agrees with the formula for the Weyl chamber count for C3: 2^3*3!.  —Preceding unsigned comment added by Jhrabows (talk • contribs) 01:34, 29 March 2010 (UTC)
replies: text: Thanks! You're right. I'll change the paragraph. Tom Ruen (talk) 07:40, 29 March 2010 (UTC)
thread_title: Questioning the numbers

message: Exactly why Coxeter's 1935 paper in 'twelve essays', is relegated so low in the references, when it is one of the clearest treatments on the subject, remains a mystery.  In any case, the correct terminology is to use 'nodes' and 'branches' for the points and edges of the graph.  This prevents confusion with the points and edges generated in the polytope. 
Coxeter seems to have invented both the Schläfli symbol, and the Coxeter-Dynkin diagram fairly early in the piece.  However, a great number of others had found the schläfli symbol before him.  His graph dates from the early 1930s, while those of de Witt and of Dynkin date from 1940s.  None the same, it's pretty clear: a point represents a free operator under the relation XX = 1, and the branch is a relation between two non-comuting operators: eg an unmarked branch is ABA=BAB.  Where no branch exists, AB=BA.  This is what happens in both Lie groups and with mirrors.  Mirror groups are, after all a kind of Lie group.
Stott's notation is to use symbols for expansion against the vertex, edge, hedron, ... of a polytope, from which one can derive most of the uniform polyhedra.  For example, a tCO is an e_1 C (ie edge-expanded cube).  Wythoff made the same construction by use of named mirrors, explicitly reversing Stott's c_0 to e_0 (so working from a point, rather than a polytope).  The notation remains unaltered under Coxeter, where the operator precedes the symmetry.
When Coxeter read Wythoff's paper, he already understood the diagram, but realised how to decorate the thing to make polytopes.  Since he already knew how to evaluate subgroups (by removing nodes and evaluating the residue order), the final notation by him is simply Wythoff's "t" notation (the real thing, not the later one) to Schläfli's symbol, eg t_1,2 {3,3,5}.  
It never becomes an inline representation of the graph, such as i use in my polygloss, eg o3x3x5o for o--@--@-5-o.  The main polytope listed in the twelve-essays paper ends up as an awful thing, although i write it as x3x3x3x3x3x3xBx.
--Wendy.krieger (talk) 10:57, 29 March 2010 (UTC)
replies: text: I changed the graph terminology to nodes and branches, as Coxeter uses. It would be nice to document Stott's notation somewhere, unsure where. Tom Ruen (talk) 23:35, 30 March 2010 (UTC)
thread_title:  Mystery. 

message: I don't know what pure reflective symmetry means, but I believe the grand antiprism has the reflective symmetry of a 5x5 duoprism. —Tamfang (talk) 05:14, 4 January 2011 (UTC)
replies: text: The article Grand antiprism says its symmetry [10,2+,10]. So that + implies the existence of 2-fold rotation symmetry, so its reflectional fundamental domain ought to have 2 copies of the generator point. How can this be better worded? Tom Ruen (talk) 07:15, 4 January 2011 (UTC)
replies: text:  "Every uniform polytope which can be generated by reflections of a single vertex"? — Do you want to cover hollow rings here? —Tamfang (talk) 20:01, 4 January 2011 (UTC)
thread_title:  pure reflective symmetry 

message: Unfortunately I don't have the background to add it, but there seems to be a great amount of detail missing on the actual use/meaning of the symbology in the diagrams, used extensively throughout the geometry pages; Mostly notably the ring and hole node symbols. Hoping this note catches the attention of some with the proper skills 74.213.201.51 (talk) 16:40, 22 March 2014 (UTC)
replies: text: There's information in this section Coxeter-Dynkin_diagram#Application_with_uniform_polytopes. Tom Ruen (talk) 19:22, 22 March 2014 (UTC)
thread_title:  Missing information 

message: There is a contradiction in: "The eigenvalues of the Schläfli matrix determines ... affine type (all non-zero, at least one is zero) ... ." (Unless the affine type does not exist!)
Over on Coxeter group, the statement is: "... (non-negative, at least one zero) ...", which I think is right - but I'm not expert enough to do this edit.
Perhaps someone can suggest a reference for this definition which can be added to both pages.
Stephen.G.McAteer (talk) 01:21, 14 January 2015 (UTC)
replies: text: Thanks for the catch. Nonnegative definitely makes more sense. I'll check for some references. Tom Ruen (talk) 01:32, 14 January 2015 (UTC)
thread_title:  Error in definition of affine type of Schläfli matrix 

message: in Hyperbolic Coxeter groups:
: and infinitely many finite-volume Viberg polytopes for dimension up to 19
May be Vinberg? Jumpow (talk) 15:49, 5 October 2015 (UTC)
replies: text: Fixed it. Tom Ruen (talk) 16:07, 5 October 2015 (UTC)
thread_title:  Viberg - misspelling? 

message: The link in further readings to the Coxeter and Dynkin interview was broken, but here is a corrected link
       https://ecommons.cornell.edu/handle/1813/17339
The variant references (3. and 7.) to  Lanner F.  and to Folke Lanner  seems confusing.
thanks, 71.186.130.243 (talk) 21:22, 30 March 2016 (UTC)
thread_title:  two small changes for the references and further readings sections. 

message: Substituted at 01:56, 5 May 2016 (UTC)
thread_title: Assessment comment

message: The articles says that "Branches of a Coxeter–Dynkin diagram are labeled with a rational number p". Is this correct? Shouldn't labels be rational integers (or infinity)?
Fcesi (talk) 13:53, 14 June 2016 (UTC)
replies: text: Noninteger rational numbers are allowed, like 5/2, . I agree this needs explaining somewhere in this article. These groups define the regular star polygons. Coxeter uses these groups for his uniform star polyhedron constructions. The rank 3 point groups are based on Schwarz_triangles. The rank 4 point groups are defined by Goursat tetrahedra, and create the 10 regular star 4-polytopes. Tom Ruen (talk) 22:37, 14 June 2016 (UTC)
thread_title:  Branches labels 

message: I just undid an earlier edit, which was meant to simply amend one statement in the Description section which had garbled syntax.  Unfortunately, for reasons not clear to me, all page content except that section disappeared! :-(  So I've just undone the change that lost the content.  The net effect of both edits is just to correct that garbled grammar.  Sorry if the briefly missing content caused anybody any confusion or other grief.  yoyo (talk) 15:00, 23 June 2016 (UTC)
thread_title:  Disappearing contents 

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