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]
• {{CDD|node|3|node|2x|node}} =  : Coxeter group [3,2]
• {{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

Branches labeled with 2 in Coxeter-Dynkin diagrams

Nodes, ringed nodes, and holes

I have noticed that in certain Coxeter-Dynkin diagrams, the nodes are connected and labelled with 2 rather than being unconnected to indicated that the dihedral angle is π/2. This is largely unnecessary and possibly misleading as it gives an impression that the diagram represents a more complex polytope as if nodes are connected and labelled with a non-integer or an integer greater than 3: polyhedra generated from and are both more complicated than , but is much simpler.

Large numbers of such examples are found in Point groups in four dimensions - note that the CD diagrams not using special "snub nodes" disconnect the nodes if they would be labelled with 2, whereas the CD diagrams using "snub nodes" found in "chiral subgroups" all connect them and label with 2.

As a result, I am recommending that most CD diagrams connecting and labelling the nodes with 2 when they represent order-2 symmetry should be disconnected. An especially noteworthy result is this, from the Duoprismatic Symmetry section of Point groups in four dimensions. One of the CD diagrams are written as this: () – this should be (). In the page source, the CDD template is using '2x' (connect node and label with 2) rather than '2' (disconnect node). This may look easy, but note that in some CDD templates the '2x' is required such as labelling a node with '2p'.

I am not sure when it was decided that certain 'disconnected' nodes of a CD diagram should be connected and labelled with '2'.

--MULLIGANACEOUS-- (talk) 03:16, 22 January 2019 (UTC)

For ordinary diagrams, no need for explicit 2s, but as you see, for snub polyhedra, the 2 shows nodes are alternated as a set rather than individually. And for subgroups even more important. The generator for is the chain product of all the generators in sequence. The bracket notation and marked up Coxeter diagrams should be seen as interchangable. Tom Ruen (talk) 20:14, 4 April 2020 (UTC)

@Tomruen Nice insight despite the late reply! I read the Coxeter-Dynkin page here, and it did not highlight the difference between situations like compared to . Is there an actual source where the author explicitly marked the edges with 2 when it should be disconnected? Also, the Coxeter-Dynkin page should also have a separate, detailed section which specifically deals with alternated and snub nodes.

--MULLIGANACEOUS-- (talk) 16:47, 25 April 2020 (UTC)

Coxeter notation explains these markups in relation to symmetry notation. Source: Geometries and Transformations by Norman W. Johnson Tom Ruen (talk) 07:44, 27 April 2020 (UTC)

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)