Talk:Coxeter–Dynkin diagram/Graphics element documentation

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