Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a graph with labelled edges. It represents the spatial relations between a collection of mirrors (or reflecting hyperplanes), and describes a kaleidoscopic construction.

The diagram represents a Coxeter group. Each graph node represents a mirror (domain facet) and the label attached to a graph edge encodes the dihedral angle order between two mirrors (on a domain ridge).

In addition, when used to represent a specific uniform polytope, the diagram has rings (circles) around nodes for active mirrors and hollow nodes (holes) to represent alternation.

The diagram is borrowed from the Dynkin diagram.

Description

The diagram can also represent polytopes by adding rings (circles) around nodes. Every diagram needs at least one active node to represent a polytope.

The rings express information on whether a generating point is on or off the mirror. Specifically a mirror is active (creates reflections) only when points are off the mirror, so adding a ring means a point is off the mirror and creates a reflection.

Hollow rings (holes) are also used. A polytope with an alternation operator applied has all the ringed nodes replaced by holes. If all the nodes are holes, the figure is considered a snub.

Edges are labeled with an integer n (or sometimes more generally a rational number p/q) representing a dihedral angle of 180/n. If an edge is unlabeled, it is assumed to be 3. If n=2 the angle is 90 degrees and the mirrors have no interaction, and the edge can be omitted. Two parallel mirrors can be marked with "∞".

In principle, n mirrors can be represented by a complete graph in which all n*(n-1)/2 edges are drawn. In practice interesting configurations of mirrors will include a number of right angles, and the corresponding edges can be omitted.

Polytopes and tessellations can be generating using these mirrors and a single generator point. Mirror images create new points as reflections. Edges can be created between points and a mirror image. Faces can be constructed by cycles of edges created, etc.

Examples

A single node represents a single mirror. This is called group A₁. If ringed this creates a digon or edge perpendicular to the mirror, represented as {} or {2}.
Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can be created, or a square if the point is equal distance from both mirrors.
Two nodes attached by an order-n edge can creates an n-gon if the point is on one mirror, and a 2n-gon if the point is off both mirrors. This forms the I₁(n) group.
Two parallel mirrors can represent an infinite polygon I₁(∞) group, also called I^~₁.
Three mirrors in a triangle form images seen in a traditional kaleidoscope and be represented by 3 nodes connected in a triangle. Repeating examples will have edges labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn in a line with the 2 edge ignored. These will generate uniform tilings.
Three mirrors can generate uniform polyhedrons, including rational numbers is the set of Schwarz triangles.
Three mirrors with one perpendicular to the other two can form the uniform prisms.

In general all regular n-polytopes, represented by Schläfli symbol symbol {p,q,r,...} can have their fundamental domains represented by a set of n mirrors and a related in a Coxeter-Dynkin diagram in a line of nodes and edges labeled by p,q,r...

Finite Coxeter groups

Families of convex uniform polytopes are defined by Coxeter groups.

Notes:

Three different symbols are given for the same groups - as a letter/number, as a bracketed set of numbers, and as the Coxeter diagram.
The bifurcated D_n groups are also given an h[] notation representing the fact it is half or alternated version of the regular C_n groups.
The bifurcated D_n and E_n groups are also labeled by a superscript form [3^a,b,c] where a,b,c are the number of segments in each of the 3 branches.

n	A₁₊	C₂₊	D₃₊	E_4-8	F₄	H_2-4
1	1=[] >
2	2=[3] >C₂=[4] \|			2=[5] >I₂(p)=[p] \|-	3	3=[3²] >C₃=[4,3] \|D₃=A₃=[3^0,1,1] \|	3=[5,3] >
4	4=[3³] >C₄=[4,3²] \|D₄=h[4,3,3]=[3^1,1,1] \|E₄=A₄=[3^0,2,1] \|F₄=[3,4,3] \|H₄=[5,3,3] \|
5	5=[3⁴] >C₅=[4,3³] \|D₅=h[4,3³]=[3^2,1,1] \|E₅=B₅=[3^1,2,1] \|
6	6=[3⁵] >C₆=[4,3⁴] \|D₆=h[4,3⁴]=[3^3,1,1] \|E₆=[3^2,2,1] \|
7	7=[3⁶] >C₇=[4,3⁵] \|D₇=h[4,3⁵]=[3^4,1,1] \|E₇=[3^3,2,1] \|
8	8=[3⁷] >C₈=[4,3⁶] \|D₈=h[4,3⁶]=[3^5,1,1] \|E₈=[3^4,2,1] \|
9	9=[3⁸] >C₉=[4,3⁷] \|D₉=h[4,3⁷]=[3^6,1,1] \|
10+	..	..	..

A_n forms the simplex polytope family.
D_n is the family of demihypercubes, beginning at n=4 with the 16-cell, and n=5 with the demipenteract. (Also named B_n)
B_n or C_n forms the hypercube/orthoplex polytope family.
I_{1ⁿ forms the regular polygons. (Also named D₂ⁿ)}
E₆,E₇,E₈ are the generators of the Gosset Semiregular polytopes
F₄ is the 24-cell polychoron family.
H₃ is the dodecahedron/icosahedron polyhedron family. (Also named G₃₎
H₄ is the 120-cell/600-cell polychoron family. (Also named G₄₎

Infinite Coxeter groups

Families of convex uniform tessellations are defined by Coxeter groups.

Note:

A^~_n-1 is a cyclic group. (Also named P_n)
B^~_n-1 forms the hypercube regular tessellation family {4,3,....,4} family. (Also named R_n)
C^~_n-1 forms the alternated hypercubic tessellation family. (Also named S_n) Also labeled by a h[] notation as a half of the regular one.
D^~_n-1 (Also named Q_n) Also labeled by a q[] notation as a quarter of the regular one.
E^~₆,E^~₇,E^~₈,E^~₉ are Gosset tessellations. (Also named T₇,T₈,T₉,T₁₀) T₁₀ exists in hyperbolic space. Also labeled by a superscript form [3^a,b,c] where a,b,c are the number of segments in each of the 3 branches.
F^~₄ is the 24-cell {3,4,3,3} regular tessellation. (Also named U₅)
H^~₂ is the Hexagonal tiling. (Also named V₃)
I^~₁ is two parallel mirrors. (Also named W₂)

n	A^~₂₊	B^~₂₊	C^~₃₊	D^~₄₊	I^~₁
1					~₁=[∞] >-	2	~₂=h[6,3] >B^~₂=[4,4] \|	~₂=[6,3] >
3	~₃=q[4,3,4] >B^~₃=[4,3,4] \|C^~₃=h[4,3,4] \|
4	~₄ >B₄=[4,3²,4] \|C^~₄=h[4,3²,4] \|D^~₄=q[4,3²,4] \|	~₄=[3,4,3,3] >
5	~₅ >B^~₅=[4,3³,4] \|C^~₅=h[4,3³,4] \|D^~₅=q[4,3³,4] \|
6	~₆ >B^~₆=[4,3⁴,4] \|C^~₆=h[4,3⁴,4] \|D^~₆=q[4,3⁴,4] \|E^~₆=[3^2,2,2] \|
7	~₇ >B^~₇=[4,3⁵,4] \|C^~₇=h[4,3⁵,4] \|D^~₇=q[4,3⁵,4] \|E^~₇=[3^3,3,1] \|
8	~₈ >B^~₈=[4,3⁶,4] \|C^~₈=h[4,3⁶,4] \|D^~₈=q[4,3⁶,4] \|E^~₈=[3^5,2,1] \|
9	~₉ >B^~₉=[4,3⁷,4] \|C^~₉=h[4,3⁷,4] \|D^~₉=q[4,3⁷,4] \|E^~₉=[3^6,2,1] \|
10	...	...	...	...

Hyperbolic infinite Coxeter groups

There are many infinite Coxeter groups whose symmetry can tessellate hyperbolic space, including:

n
3	∞ linear: ...	...	...	...	...
4	3 linear:	1 bifurcating: \| valign=top\|5 cyclic: \|-	5	3 linear:	>1 cyclic:

References

Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6

(Paper 17) Coxeter, The Evolution of Coxeter-Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233-248]

Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Coxeter Regular Polytopes (1963), Macmillian Company