Talk:Coxeter–Dynkin diagram

Graphics element documentation

Coxeter-Dynkins graphics on Wikipedia:

These component elements can be strung together to create linear diagrams. They are 23 pixels tall. The elements are variable width, (making them hard to systematically scale with "px" pixel-width codes). This set allows two sorts of nonlinear graphs, a central linear one that can branch up and down, and two rows top/bottom (a and b), that can be vertically connected and looped. They are largely complete for the finite and affine groups, but the triangle groups can't be labeled in general.

The small dots represent the graph nodes of a Coxeter group, while the ringed (circled) and ringed (hollow) nodes are used in the generation of Coxeter-Dynkin diagrams representing the uniform polytopes of Coxeter.

Graph symbols

Nodes and graphs

Labeled nodes and branches

| | |

Branches

...

...

Double branches

Ultraparallel branches (dotted lines)

Labels

Markup symbols

Rings and holes

Node removal and operators

Branched rings and holes

Subgroups

Unitary markups

Complex node elements

Examples

The can be used to string a large number of these symbols together, with the CDel_ prefix implicit, and elements separated by pipes.

Examples:

• {{CDD|node|3|node|2|node}} =  : Coxeter group [3,2] as a cross product.
• {{CDD|node|3|node|2x|node}} =  : Coxeter group [3,2] as a single unit. The notation is otherwise notated as a broken line.
• {{CDD|nodes|split2|node|3|node}} =
• {{CDD|node|3|node|split1|nodes|3ab|nodes}} =
• {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}} =
• {{CDD|nodeb|3b|nodeb|3b|branch|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb}} =
• {{CDD|nodes|3ab|nodes|3ab|nodes|split5a|nodes}} =
• {{CDD|label3-2|branch|split2|node|5|node}} =
• 
• {{CDD|node_n1|3|node_n2|4|node_n3|3|node_n4}} =
• {{CDD|node_c1|3|node_c2|4|node_c2|3|node_c1}} = = [[3,4,3]]
  • {{CDD|label4|branch_c2|3ab|nodeab_c1}} = = [[3,4,3]]
• , , , ,
• , , , ,
• or , or , - or representation of atomic "holes", for half symmetry [1⁺,4,3,3], index 2, quarter symmetry: [1⁺,4,3,4,1⁺], index 4, and half symmetry [(4,3,4,2⁺)], index 2.
• , , - [3⁺,4,3] symmetry, index 2, [3⁺,4,3⁺] symmetry, index 4, and [3,4,3]⁺ symmetry, index 2.
• ht₀ht₂ht₃{4^1,1,1}=, h{4^1,1,1}= = , = , ht_0,1ht₂ht₃{4,4^1,1}=, s{4^1,1,1}=
• , , , , , , ,
• 
• 
• = = [4⁺,4⁺] = [(4⁺)^1,1] = [(4⁺)²]
• = [(4⁺)^1,1,1]
• = [4⁺,4⁺,4⁺] = [(4⁺)³]
• = [(4⁺)^[3]]
• = [(4⁺)^[4]]
• = [(4,(4,3,4)⁺)] = [4^[4]]⁺ ?
• = [(4⁺,(4,3,4)⁺)]

Example extended markups

Template	Diagram	Description
{{CDD|node|4|node|3|node}}		Coxeter group [4,3], order 48
{{CDD|node_n0|4|node_n1|3|node_n2}}		[4,3] with indexed mirrors.
{{CDD|node_g|3hg|node_g|4g|node_g}}		Half group [3,4]+, Chiral octahedral symmetry, order 24
{{CDD|node_h2|3|node_h2|3|node_h2}}
{{CDD|node_h0|4|node|3|node}}		Half group [1+,4,3] = [3,3], order 24
{{CDD|node_g|3hg|node_g|4|node}}		Half group [3+,4], pyritohedral symmetry, order 24
{{CDD|node_g|3hg|node_g|4|node_h0}}		Quarter group [3+,4,1+] = [3,3]+, chiral tetrahedral symmetry, order 12
{{CDD|node_g|3sg|node_g|4|node}}		Radical index 6 subgroup [3*,4] = [2,2], order 8
{{CDD|node_g|3sg|node_g|4|node_h0}}		Radical index 12 subgroup [3*,4,1+] = [2,2]+, order 4
{{CDD|node_1|4|node|3|node}}		Cube
{{CDD|node_h|4|node|3|node}}		Half cube, tetrahedron
{{CDD|node_h1|4|node|3|node}}
{{CDD|node_h2|4|node|3|node}}
{{CDD|node_h3|4|node|3|node}}		Both half cubes, stella octangula
{{CDD|node_h12|4|node|3|node}}
{{CDD|node_h|3|node_h|4|node}}		Snub octahedron, icosahedron
{{CDD|node_h2|3|node_h2|4|node}}
{{CDD|node_h3|3|node_h3|4|node}}		Two snub octahedra, compound of two icosahedra
{{CDD|node_h|3|node_h|4|node_h}}		Snub cube
{{CDD|node_h2|3|node_h2|4|node_h2}}
{{CDD|node_h3|3|node_h3|4|node_h3}}		Compound of two snub cubes

Article is a train wreck

This article obviously represents a great deal of work by extremely knowledgeable editors.

Unfortunately, it is virtually incomprehensible in its current form.

The main problem is that various uses of the Coxeter-Dynkin diagram are mentioned willy-nilly throughout the article. The article does not progress in a logical sequence but instead mashes together information about the various applications of the diagrams, making no effort to develop the subject progressively.

Here is what would make the article much, much better:

1. Separate out the various uses of the diagram: a) a family of hyperplanes, b) a group generated by reflections in those planes, c) a polytope with specific symmetry.

2. Devote separate sections of the article to first describe the family of planes, then describe the reflection group, and only then describe how a Coxeter-Dynkin diagram can describe a polytope with specific symmetry.

3. And only after these three steps should the article present the various classifications related to the diagrams. Make sure that the classification for groups is separate from the classification for polytopes. Because these classifications are useless to a reader who cannot understand what they are reading, because the article is developed in an illogical manner.65.141.95.170 (talk) 19:33, 3 November 2020 (UTC)

I agree with the assessment of this article as a train-wreck. I've added the {{Gibberish}} to the article. I would say that I think the description of a group from a diagram is actually quite simple (although you wouldn't be able to tell from this article), and the article could be ordered b -> a -> c. I think it's just a matter of preference. Aside from that I think a lot of the content in this article really just needs to be moved elsewhere. There's a lot of discussion about Coxeter groups which a good deal of only seems tangential to Coxeter-Dynkin diagrams. Some of this should be moved to the Coxeter group page and linked with {{See also}} or similar, some of it should probably just be removed. I added {{Unfocused}} for this reason. Looking at the size of the article and the citations, I also believe that there is likely some amount WP:OR, but the article is currently so incomprehensible it is hard to tell where. In my opinion this article basically needs to be gutted and reworked. AquitaneHungerForce (talk) 15:59, 9 December 2022 (UTC)

Leaving this mess of eyesore banners at the top for over a year isn't accomplishing anything. Unless someone has a concrete plan or at least is in ongoing discussion, these banners should be removed.

If you think the description of a group from a diagram is "quite simple" are you interested in writing and sourcing that simple explanation, and inserting it into the article?

If anyone is looking for a source with a clear explanation, I haven't watched them but John Baez has some lectures on YouTube which might be helpful: summary page, notes. In my experience his other writing has been clear and accessible. –jacobolus (t) 20:06, 26 January 2024 (UTC)

There is a nice historical discussion at: Coxeter, H.S.M., 1994. "The evolution of Coxeter-Dynkin diagrams". In Polytopes: Abstract, Convex and Computational, pp. 21–42. Springer. doi:10.1007/978-94-011-0924-6. –jacobolus (t) 02:22, 28 January 2024 (UTC)

Well, there are really two solutions. I think what is best is to remove excess, rather than rewrite entirely the article (anyone volunteering? big task). There's about 30% excess information that could simply be shortened and say all the same. However, I believe I am not in favor of removing the expanse of Coxeter-Dynkin diagrams that this article actually does okay in describing and good-enough in enumerating (but which can still be marktedly improved), which is the movement between compact, paracompact, hypercompact, and cocompact hyperbolic groups and diagrams. To really understand these, one needs to flow in precisely that formation, since the cocompact groups are the groups that are in some sense the most abstract, but also some of the simplest, that reach the highest dimension (E10:E11). So for me, I think we can reduce a lot of the jargon that is colorful, and get to the point. Because these various hyperbolic groups are really strictly defined and worked with mainly through their Dynkin diagrams, it seems inadequate to remove these. A partial rewrite and cleanup is due, I'll put time into it in a week or so, I hope.

Also, we should keep in mind that this article is actually complex, and full of symbols and diagrams, meaning that the reader will need to come more or less prepared and willing to invest her/himself into reading the article. This being said, it can still be simplified, and maybe some images can be removed too, I don't know which. Radlrb (talk) 12:44, 30 January 2023 (UTC)

numbers and hyphens

I removed hyphens from phrases like "in 7-dimensions". Some of these changes were reverted with this comment:

incorrect, because it is supposed to read 7-dimensions; not 7 dimensions. The point is that it reaches dimension-7, not that they run up through a total of 7 dimensions.

"supposed" begs the question, and the rest I do not understand at all. A two-car garage can hold two cars, and a 7-dimensional space has 7 dimensions: is that controversial? A space of fewer than eight dimensions has up to seven dimensions; is that controversial? —Tamfang (talk) 17:58, 30 January 2023 (UTC)

Now I see that most instances of "7-dimensions" have been reworded to sidestep the question. Incidentally I think n+1 is ugly. If that digit is in math format, why not all of them (dimension 4)? —Tamfang (talk) 18:09, 30 January 2023 (UTC)

Saying moves through seven dimensions could mean it goes from dimension 3 through 9, and not 1 though 7. Hence the disambiguation and why I needed to correct your edit. No harm intended. I only changed that section, feel free to change the rest. Also, these edits were made before you posted this, so no-one was sidestepping anything. I improved it thereafter beyond what we both had corrected. Radlrb (talk) 23:41, 30 January 2023 (UTC)

I hope we don't have to pander to the popular scifi usage of "dimension" to mean "universe". —Tamfang (talk) 00:36, 31 January 2023 (UTC)

Correct, we also should not pander to silly frustrations because our edits were corrected with proper reasoning. Radlrb (talk) 01:01, 31 January 2023 (UTC)

If that's alluding to something, I'm sure I cannot say what. —Tamfang (talk) 01:37, 31 January 2023 (UTC)

For me, time to just drop it. Happy editing! Radlrb (talk) 01:38, 31 January 2023 (UTC)

The Banners

Hello @Jacobolus: let's discuss the banners. In my view this is pretty simple: Reading from WP:WNTRMT, it says not to remove a template if the issue has not been addressed. The issues with this page have not been addressed, so the banners should remain. I disagree that they harm the reader experience, but that seems besides the point, since the issues have not been addressed. AquitaneHungerForce (talk) 01:14, 27 January 2024 (UTC)

The "maintenance" banners are among the worst things that ever happened to Wikipedia: they are hostile to readers without any benefit. The appropriate place to put meta-discussion about article quality, criticisms of article style, todo lists for editors, etc. is this talk page, not the very top of the article body. Readers do not have trouble discerning when something has only a few sources, or was terribly written, or doesn't have any pictures, or whatever. When they are used temporarily backed by some ongoing discussion about fixing whatever someone took issue with, or as self notices by someone who is in the process of fixing something they think is broken, they are only mildly objectionable. In this use case, we're talking about banners staying for a matter of days to weeks, maybe months at the outside. When they get littered around and left on articles for years (or more typically, literally decades) then they have overstayed their welcome and the harm caused far outweighs any conceivable trivial benefit. WP:WNTRMT was forced into existence by a bunch of petty bureaucrats with extra time on their hands, because the only people who found it worth dumping huge amounts of effort into debating the topic were the same people who love littering banners everywhere, and everyone else would rather spend their time improving the encyclopedia instead of having pointless debates about banner litter. Fortunately, over time the crew of banner-happy vandals has slowly diminished and these banners are gradually getting cleaned up (i.e. removed, typically without any specific article change involved), restoring wiki articles to a less ugly condition. Cf. User:Jorge Stolfi/Templates that I sorely miss.

I'd be happy to contribute to this article, though I don't really fully understand all of the trickier types of Coxeter diagrams. If you have concrete suggestions, please start discussion topics or go make article changes. –jacobolus (t) 01:37, 27 January 2024 (UTC)

However strong your feelings are here, I am much more partial to the current policy. It seems like it would be more productive to take this up on WP:WNTRMT with some sort of RFC or whatever, rather than on individual pages.

I do want to improve this article, so here's a concrete plan:

1. I did take the time after your first edit to go through and remove the sections that were blatantly off-topic, which should help address the focus issues.
2. The next step would be to stitch up the wound created by such a large removal, by integrating the parts and removing other bits that are mixed in with other content.
3. Based on my experience cleaning up related articles largely influenced by the same set of editors this article probably has a bunch of WP:OR and some WP:NEO. So the next step would be to go through the text and determine exactly what can be referenced, and remove what can't. This would be pretty tedious and time consuming
4. At this point things should be cut down to the bone, it should be on-topic and well sourced. The next step would be to fill in the information that is necessary to have the article be understandable. e.g. make sure concepts are explained before they are used.
5. At this point all three of the issues should be resolved, and the article can slowly be improved and filled in by normal editing.

I have some time this weekend which I can devote to editing Wikipedia. I will attempt to follow this plan, I will probably be able to resolve the focus issues and some the sourcing problems, but it's unlikely I will finish this in the time I have. Would you agree to having the banners restored and resuming this discussion on Monday? AquitaneHungerForce (talk) 01:59, 27 January 2024 (UTC)

more productive to take this up – no it wouldn't. Challenging the banner fans on their home turf is utterly pointless, since they have a status quo that justifies their behavior and a critical tribal mass to keep it in place. It requires an unbelievably disproportionate amount of effort (like weeks of full-time effort by a team of committed partisans, or more) to dislodge this kind of institution once it has gotten its hooks into a community, which is why it almost never happens. The best anyone can do is quietly work to counter it around the margins by removing outdated banners, as I did here.

banners restored and resuming this discussion on Monday? I'm not going to get into an edit war with you, but what purpose would the banners serve? The only practical effect is to visually assault and distract/confuse whichever readers happen to attempt to read this article between now and whenever someone next removes them. Keeping the banners there is not going to make you more or less likely to remember your article criticisms from more than a year ago, is not going to make the rest of the article any more or less legible, and it's not like there's a team of experts banging down the door to fix this article on the basis of the extra banner (if so, they would have done so a long time ago). –jacobolus (t) 02:59, 27 January 2024 (UTC)

I've tried to work with you here, but this last message reveals to me that you have been dishonest. You don't really care whether the article is being worked on, you don't care about a concrete plan. You simply don't like the banners, and are willing to make whatever arguments serve that end. I will reinstate the banners per policy, and I will fix the article at my leisure. AquitaneHungerForce (talk) 03:10, 27 January 2024 (UTC)

Diagrams for Rank 2 Coxeter groups

The table in this section states that "Mirror lines are colored to correspond to Coxeter diagram nodes." But I fail to see how they correspond in the pair for [3], where the red and green rays don't seem to be mirror lines at all:

[3]

Thanks, Albie's relation of misfortune (talk) 15:25, 15 June 2024 (UTC)

The diagram only shows a portion of the mirror line, if you extend the line indefinitely you get the full mirror. It's just that for odd order pairs then the green and red lines coincide. AquitaneHungerForce (talk) 17:19, 15 June 2024 (UTC)

But wouldn't the green and red mirror lines that coincide represent one in the same reflection? Albie's relation of misfortune (talk) 21:58, 15 June 2024 (UTC)

Yes the coinciding mirrors have the same reflection. This just means that the generating mirrors are in the same conjugacy class. For example in the dihedral group of 6 elements generated by reflections a and b, $a=(ba)b(ba)^{-1}$. In the even case you can color the mirrors by which generator they come from (the generator which they are conjugate to) but in the odd one you need to do something like this coloring. AquitaneHungerForce (talk) 17:46, 13 February 2025 (UTC)

Mirrors

The section Schläfli matrix begins as follows:

"Every Coxeter diagram has a corresponding Schläfli matrix (so named after Ludwig Schläfli), A, with matrix elements a_i,j = a_j,i = −2 cos(π/p_i,j) where p_i,j is the branch order between mirrors i and j; that is, π/p_i,j is the dihedral angle between mirrors i and j."

But up to this point in the article, the word "mirror" has not been defined or even mentioned.

I hope someone knowledgeable about this subject can fill in this missing information.

"Mirror" here means the space of points fixed by a generator. Each generator can be seen as a reflection about that space. AquitaneHungerForce (talk) 17:38, 13 February 2025 (UTC)