Coxeter–Dynkin diagram

Coxeter groups in the plane with equivalent diagrams. Domain mirrors are labeled as edge m1, m2, etc. Vertices are colored by their reflection order. The prismatic group [I^~₁xI^~₂] is shown as a doubling of the C^~₂, but can also be created as rectangular domains from doubling the H^~₂ triangles. The A^~₂ is a doubling of the H^~₂ triangle.

Coxeter groups in the sphere with equivalent diagrams. One fundamental domain is outlined in yellow. Vertices are colored by their reflection order.

Coxeter groups in 3-space with diagrams. Mirrors (triangle faces) are labeled by opposite vertex 0..3. Edges are colored by their reflection order.
C^~₃ fills 1/24 of the cube. B^~₃ fills 1/12 of the cube. A^~₃ fills 1/6 of the cube.

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

Each diagram represents a Coxeter group. Dynkin diagrams are used to classify root systems and therefore Lie algebras.^[1]

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.

Description

Edges of a Coxeter–Dynkin diagram are labeled with a rational number p, representing a dihedral angle of 180°/p. When p = 2 the angle is 90 degrees and the mirrors have no interaction, so the edge can be omitted from the diagram. If an edge is unlabeled, it is assumed to have p = 3, representing an angle of 60 degrees. Two parallel mirrors have an edge 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, nearly all interesting configurations of mirrors include a number of right angles, so the corresponding edges are omitted.

Polytopes and tessellations can be generated using such mirrors and a single generator point: mirror images create new points as reflections, then polytope edges can be defined between points and a mirror image point. Faces can be constructed by cycles of edges created, etc. To specify the generating vertex, one or more nodes are marked with rings, meaning that the vertex is not on the mirror(s) represented by the ringed node(s). (If two or more mirrors are marked, the vertex is equidistant from them.) A mirror is active (creates reflections) only with respect to points not on it. A diagram needs at least one active node to represent a polytope.

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.

Examples

• A single node represents a single mirror. This is called group A₁. If ringed this creates a line segment perpendicular to the mirror, represented as {}.
• Two unattached nodes represent two perpendicular mirrors. If both nodes are ringed, a rectangle can be created, or a square if the point is at equal distance from both mirrors.
• Two nodes attached by an order-n edge can create 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 can be represented by three 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 as a line (with the 2 edges ignored). These will generate uniform tilings.
• Three mirrors can generate uniform polyhedra; including rational numbers gives 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 with a related Coxeter–Dynkin diagram of 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 numbers of segments in each of the three branches.

n	A1+	C2+	D3+	E4−8	F4	H2−4	I2(p)
1	A1 = []
2	A2 = [3]	C2 = [4]				H2 = [5]	I2(p) = [p]
3	A3 = [32]	C3 = [4,3]	D3 = A3 = [30,1,1]			H3 = [5,3]
4	A4 = [33]	C4 = [4,32]	D4 = h[4,3,3] = [31,1,1]	E4 = A4 = [30,2,1]	F4 = [3,4,3]	H4 = [5,3,3]
5	A5 = [34]	C5 = [4,33]	D5 = h[4,33] = [32,1,1]	E5 = B5 = [31,2,1]
6	A6 = [35]	C6 = [4,34]	D6 = h[4,34] = [33,1,1]	E6 = [32,2,1]
7	A7 = [36]	C7 = [4,35]	D7 = h[4,35] = [34,1,1]	E7 = [33,2,1]
8	A8 = [37]	C8 = [4,36]	D8 = h[4,36] = [35,1,1]	E8 = [34,2,1]
9	A9 = [38]	C9 = [4,37]	D9 = h[4,37] = [36,1,1]
10+	..	..	..

1. A_n forms the simplex polytope family.
2. 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).
3. B_n or C_n forms the hypercube/orthoplex polytope family.
4. I₂ⁿ forms the regular polygons. (Also named D₂ⁿ)
5. E₆,E₇,E₈ are the generators of the Gosset Semiregular polytopes
6. F₄ is the 24-cell polychoron family.
7. H₃ is the dodecahedron/icosahedron polyhedron family. (Also named G₃)
8. 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:

1. A^~_n−1 is a cyclic group. (Also named P_n)
2. B^~_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.
3. C^~_n−1 forms the hypercube regular tessellation family {4,3,....,4} family. (Also named R_n)
4. D^~_n−1 (Also named Q_n) Also labeled by a q[] notation as a quarter of the regular one.
5. 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.
6. F^~₄ is the 24-cell {3,4,3,3} regular tessellation. (Also named U₅)
7. H^~₂ is the hexagonal tiling. (Also named V₃)
8. I^~₁ is two parallel mirrors. (Also named W₂)

Hyperbolic infinite Coxeter groups

There are many infinite Coxeter groups whose symmetry can tessellate hyperbolic space. There are infinitely many linear groups for order 2, in the form [p,q], and no compact groups beyond order 5.

The two bifurcating groups have doubled fundamental domains as linear ones. [5,3,4] --> [5,3^1,1], and [5,3,3,4] --> [5,3,3^1,1].

n	Linear	Bifurcating	Cyclic
3	∞: , 2(p+q)<pq ... ... ...	0:	∞: , p+q+r≥10 ... ... ... ...
4	3:	1:	5:
5	3:	1:	1:

See also

• Coxeter group
• Root system
• Satake diagram
• Uniform polytope
  • Wythoff symbol
  • Uniform polyhedron
  • List of uniform polyhedra
  • List of uniform planar tilings
  • Uniform polychoron
  • Convex uniform honeycomb
  • Convex uniform honeycombs in hyperbolic space
• Wythoff construction and Wythoff symbol

Notes

1. ^ Hall, Brian C. (2003), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, ISBN 0-387-40122-9 .

References

• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990)
• 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 [1]
  • (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
  • Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope, and Section 11.3 Representation by graphs)

External links

• Weisstein, Eric W., "Coxeter–Dynkin diagram" from MathWorld.