COXETER–DYNKIN DIAGRAM

In geometry, a 'Coxeter–Dynkin diagram' is a graph representing a relational set of mirror (or reflectional hyperplanes) in space for a kaleidoscopic construction.
As a graph itself, the diagram represents Coxeter groups, each graph node represents a mirror (domain facet) and each graph edge represents the dihedral angle order between two mirrors (on a domain ridge).
In addition the graphs have ''rings'' (circles) around nodes for active mirrors representing a specific uniform polytope.
The diagram is borrowed from the Dynkin diagram.

Contents

Description

Examples

Finite Coxeter groups

Infinite Coxeter groups

See also

References

External links

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.
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 D₂ⁿ group.

★ Two parallel mirrors can represent an infinite polygon D₂^∞ group, also called W₂.

★ 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 B_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 B_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+	B4+	C2+	D2p	E6-8	F4	G2-4
1	A1=[] CDW_dot.png
2	A2=[3] CDW_dot.png CDW_3b.png CDW_dot.png		C2=[4] CDW_dot.png CDW_4.png CDW_dot.png	D2p=[p] CDW_dot.png CDW_p.png CDW_dot.png			G2=[5] CDW_dot.png CDW_5.png CDW_dot.png
3	A3=[32] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B3=A3=[30,1,1] CD dot.png CD 3b.png CD_downbranch-00.png	C3=[4,3] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png				G3=[5,3] CDW_dot.png CDW_5.png CDW_dot.png CDW_3b.png CDW_dot.png
4	A4=[33] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B4=h[4,3,3]=[31,1,1] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png	C4=[4,32] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png		E4=A4=[30,2,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD_downbranch-00.png	F4=[3,4,3] CDW_dot.png CDW_3b.png CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png	G4=[5,3,3] CDW_dot.png CDW_5.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png
5	A5=[34] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B5=h[4,33]=[32,1,1] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png	C5=[4,33] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png		E5=B5=[31,2,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png
6	A6=[35] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B6=h[4,34]=[33,1,1] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png	C6=[4,34] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png		E6=[32,2,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png
7	A7=[36] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B7=h[4,35]=[34,1,1] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png	C7=[4,35] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png		E7=[33,2,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png
8	A8=[37] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B8=h[4,36]=[35,1,1] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png	C8=[4,36] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png		E8=[34,2,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png
9	A9=[38] CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png	B9=h[4,37]=[36,1,1] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png	C9=[4,37] CDW_dot.png CDW_4.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png CDW_3b.png CDW_dot.png
10+	..	..	..

Note: (Alternate names as Simple Lie groups also given)
#A_n forms the simplex polytope family. (Same A_n)
#B_n is the family of demihypercubes, beginning at n=4 with the 16-cell, and n=5 with the penteract. (Also named D_n)
#C_n forms the hypercube polytope family. (Same C_n)
#D₂ⁿ forms the regular polygons. (Also named I1ⁿ)
#E₆,E₇,E₈ are the generators of the Gosset Semiregular polytopes (Same E₆,E₇,E₈)
#F₄ is the 24-cell polychoron family. (Same F₄)
#G₃ is the dodecahedron/icosahedron polyhedron family. (Also named H3)
#G₄ is the 120-cell/600-cell polychoron family. (Also named H4)

Infinite Coxeter groups

Families of convex uniform tessellations are defined by Coxeter groups.
Notes:

★ Regular (linear) groups can be given with an equivalent bracket notation.

★ The S_n group can also be labeled by a h[] notation as a ''half'' of the regular one.

★ The Q_n group can also be labeled by a q[] notation as a ''quarter'' of the regular one.

★ The bifurcated T_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	P3+	Q5+	R3+	S4+	T7-9	U5	V3	W2
2								W2=[∞] CDW dot.png CDW infin.png CDW dot.png
3	P3=h[6,3]		R3=[4,4] CDW dot.png CDW 4.png CDW dot.png CDW 4.png CDW dot.png				V3=[6,3] CDW dot.png CDW 6.png CDW dot.png CDW 3b.png CDW dot.png
4	P4=q[4,3,4] CD downbranch-00.png CD downbranch-33.png CD downbranch-00.png		R4=[4,3,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S4=h[4,3,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD 4.png CD dot.png
5	P5 CD downbranch-00.png CD downbranch-33.png	Q5=q[4,32,4] CD downbranch-00.png CD 3b.png CD dot.png	R5=[4,32,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S5=h[4,32,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 4.png CD dot.png		U5=[3,4,3,3] CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png
6	P6 CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-00.png	Q6=q[4,33,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png	R6=[4,33,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S6=h[4,33,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 4.png CD dot.png
7	P7 CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png	Q7=q[4,34,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png	R7=[4,34,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S7=h[4,34,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 4.png CD dot.png	T7=[32,2,2] CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD 3b.png CD dot.png
8	P8 CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-00.png	Q8=q[4,35,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png	R8=[4,35,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S8=h[4,35,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 4.png CD dot.png	T8=[33,3,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png
9	P9 CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png	Q9=q[4,36,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png	R9=[4,36,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S9=h[4,36,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 4.png CD dot.png	T9=[35,2,1] CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png
10	P10 CD downbranch-00.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-open.png CD downbranch-33.png CD downbranch-00.png	Q10=q[4,37,4] CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png	R10=[4,37,4] CDW dot.png CDW 4.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 3b.png CDW dot.png CDW 4.png CDW dot.png	S10=h[4,37,4] CD dot.png CD 3b.png CD_downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 4.png CD dot.png
11	...	...	...	...

Note: (Alternate names as Simple Lie groups also given)
#P_n is a cyclic group. (Also named ~A_n-1)
#Q_n (Also named ~D_n-1)
#R_n forms the hypercube {4,3,....} regular tessellation family. (Also named ~B_n-1)
#S_n forms the alternated hypercubic tessellation family. (Also named ~C_n-1)
#T₇,T₈,T₉ are Gosset tessellations. (Also named ~E₆,~E₇,~E₇)
#U₅ is the 24-cell {3,4,3,3} regular tessellation. (Also named ~F₄)
#V₃ is the Hexagonal tiling. (Also named ~H₂)
#W₂ is two parallel mirrors. (Also named ~I₁)

See also

★ Coxeter group

★ Root system

★ Uniform polytope

★
★ Wythoff symbol

★
★ Uniform polyhedron

★
★ List of uniform polyhedra

★
★ List of uniform planar tilings

★
★ Uniform polychoron

★
★ Convex uniform honeycomb

★ Wythoff construction and Wythoff symbol

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 [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

★

★ ''Regular Polytopes, Root Lattices, and Quasicrystals'', R. Bruce King PDF