In geometry, a Coxeter–
Contents 1 Description 2 Schläfli matrix 2.1 Rank 2 Coxeter groups 2.2 Geometric visualizations 3 Finite Coxeter groups 4 Application with uniform polytopes 4.1 Example polyhedra and tilings 5 Affine Coxeter groups 6 Hyperbolic Coxeter groups 6.1 Hyperbolic groups in H2 6.1.1 Arithmetic triangle group 6.1.2 Hyperbolic Coxeter polygons above triangles 6.2 Compact (Lannér simplex groups) 6.2.1 Ranks 4–5 6.3 Paracompact (Koszul simplex groups) 6.3.1 Ideal simplices 6.3.2 Ranks 4–10 6.3.2.1 Subgroup relations of paracompact hyperbolic groups 6.4 Hypercompact Coxeter groups (Vinberg polytopes) 6.4.1 Vinberg polytopes with rank n+2 for n dimensional space 6.4.2 Vinberg polytopes with rank n+3 for n dimensional space 6.4.3 Vinberg polytopes with rank n+4 for n dimensional space 7 Lorentzian groups 7.1 Very-extended Coxeter Diagrams 8 Geometric folding 9 Complex reflections 10 See also 11 References 12 Further reading 13 External links Description[edit]
Branches of a Coxeter–
Type Finite Affine Hyperbolic Geometry ... Coxeter [ ] [2] [3] [4] [p] [∞] [∞] [iπ/λ] Order 2 4 6 8 2p ∞
Rank 2
Order p Group Coxeter diagram Schläfli matrix [ 2 a 12 a 21 2 ] displaystyle left[ begin matrix 2&a_ 12 \a_ 21 &2end matrix right] Determinant (4-a21*a12) Finite (Determinant>0) 2 I2(2) = A1xA1 [2] [ 2 0 0 2 ] displaystyle left[ begin smallmatrix 2&0\0&2end smallmatrix right] 4 3 I2(3) = A2 [3] [ 2 − 1 − 1 2 ] displaystyle left[ begin smallmatrix 2&-1\-1&2end smallmatrix right] 3 3/2 [3/2] [ 2 1 1 2 ] displaystyle left[ begin smallmatrix 2&1\1&2end smallmatrix right] 4 I2(4) = B2 [4] [ 2 − 2 − 2 2 ] displaystyle left[ begin smallmatrix 2&- sqrt 2 \- sqrt 2 &2end smallmatrix right] 2 4/3 [4/3] [ 2 2 2 2 ] displaystyle left[ begin smallmatrix 2& sqrt 2 \ sqrt 2 &2end smallmatrix right] 5 I2(5) = H2 [5] [ 2 − ϕ − ϕ 2 ] displaystyle left[ begin smallmatrix 2&-phi \-phi &2end smallmatrix right] ( 5 − 5 ) / 2 displaystyle (5- sqrt 5 )/2 ~1.38196601125 5/4 [5/4] [ 2 ϕ ϕ 2 ] displaystyle left[ begin smallmatrix 2&phi \phi &2end smallmatrix right] 5/2 [5/2] [ 2 1 − ϕ 1 − ϕ 2 ] displaystyle left[ begin smallmatrix 2&1-phi \1-phi &2end smallmatrix right] ( 5 + 5 ) / 2 displaystyle (5+ sqrt 5 )/2 ~3.61803398875 5/3 [5/3] [ 2 ϕ − 1 ϕ − 1 2 ] displaystyle left[ begin smallmatrix 2&phi -1\phi -1&2end smallmatrix right] 6 I2(6) = G2 [6] [ 2 − 3 − 3 2 ] displaystyle left[ begin smallmatrix 2&- sqrt 3 \- sqrt 3 &2end smallmatrix right] 1 6/5 [6/5] [ 2 3 3 2 ] displaystyle left[ begin smallmatrix 2& sqrt 3 \ sqrt 3 &2end smallmatrix right] 8 I2(8) [8] [ 2 − 2 + 2 − 2 + 2 2 ] displaystyle left[ begin smallmatrix 2&- sqrt 2+ sqrt 2 \- sqrt 2+ sqrt 2 &2end smallmatrix right] 2 − 2 displaystyle 2- sqrt 2 ~0.58578643763 10 I2(10) [10] [ 2 − ( 5 + 5 ) / 2 − ( 5 + 5 ) / 2 2 ] displaystyle left[ begin smallmatrix 2&- sqrt (5+ sqrt 5 )/2 \- sqrt (5+ sqrt 5 )/2 &2end smallmatrix right] ( 3 − 5 ) / 2 displaystyle (3- sqrt 5 )/2 ~0.38196601125 12 I2(12) [12] [ 2 − 2 + 3 − 2 + 3 2 ] displaystyle left[ begin smallmatrix 2&- sqrt 2+ sqrt 3 \- sqrt 2+ sqrt 3 &2end smallmatrix right] 2 − 3 displaystyle 2- sqrt 3 ~0.26794919243 p I2(p) [p] [ 2 − 2 cos ( π / p ) − 2 cos ( π / p ) 2 ] displaystyle left[ begin smallmatrix 2&-2cos(pi /p)\-2cos(pi /p)&2end smallmatrix right] 4 sin 2 ( π / p ) displaystyle 4sin ^ 2 (pi /p) Affine (Determinant=0) ∞ I2(∞) = I ~ 1 displaystyle tilde I _ 1 = A ~ 1 displaystyle tilde A _ 1 [∞] [ 2 − 2 − 2 2 ] displaystyle left[ begin smallmatrix 2&-2\-2&2end smallmatrix right] 0 Hyperbolic (Determinant≤0) ∞ [∞] [ 2 − 2 − 2 2 ] displaystyle left[ begin smallmatrix 2&-2\-2&2end smallmatrix right] 0 ∞ [iπ/λ] [ 2 − 2 c o s h ( 2 λ ) − 2 c o s h ( 2 λ ) 2 ] displaystyle left[ begin smallmatrix 2&-2cosh(2lambda )\-2cosh(2lambda )&2end smallmatrix right] − 4 sinh 2 ( 2 λ ) ≤ 0 displaystyle -4sinh ^ 2 (2lambda )leq 0 Geometric visualizations[edit]
The Coxeter–
Coxeter groups in the Euclidean plane with equivalent diagrams. Reflections are labeled as graph nodes R1, R2, etc. and are colored by their reflection order. Reflections at 90 degrees are inactive and therefore suppressed from the diagram. Parallel mirrors are connected by an ∞ labeled branch. The prismatic group I ~ 1 displaystyle tilde I _ 1 x I ~ 1 displaystyle tilde I _ 1 is shown as a doubling of the C ~ 2 displaystyle tilde C _ 2 , but can also be created as rectangular domains from doubling the G ~ 2 displaystyle tilde G _ 2 triangles. The A ~ 2 displaystyle tilde A _ 2 is a doubling of the G ~ 2 displaystyle tilde G _ 2 triangle. Many Coxeter groups in the hyperbolic plane can be extended from the Euclidean cases as a series of hyperbolic solutions. Coxeter groups in 3-space with diagrams. Mirrors (triangle faces) are labeled by opposite vertex 0..3. Branches are colored by their reflection order. C ~ 3 displaystyle tilde C _ 3 fills 1/48 of the cube. B ~ 3 displaystyle tilde B _ 3 fills 1/24 of the cube. A ~ 3 displaystyle tilde A _ 3 fills 1/12 of the cube. Coxeter groups in the sphere with equivalent diagrams. One fundamental domain is outlined in yellow. Domain vertices (and graph branches) are colored by their reflection order. Finite Coxeter groups[edit] See also polytope families for a table of end-node uniform polytopes associated with these groups. 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 Dn groups is half or alternated version of the regular Cn groups. The bifurcated Dn and En groups are also labeled by a superscript form [3a,b,c] where a,b,c are the numbers of segments in each of the three branches. Connected finite Dynkin graphs up to (ranks 1 to 9) Rank Simple Lie groups Exceptional Lie groups A 1 + displaystyle A _ 1+ B 2 + displaystyle B _ 2+ D 2 + displaystyle D _ 2+ E 3 − 8 displaystyle E _ 3-8 F 3 − 4 displaystyle F _ 3-4 G 2 displaystyle G _ 2 H 2 − 4 displaystyle H _ 2-4 I 2 ( p ) displaystyle I _ 2 (p) 1 A1=[ ]
2 A2=[3] B2=[4] D2=A1A1 G2=[6] H2=[5] I2[p] 3 A3=[32] B3=[3,4] D3=A3 E3=A2A1 F3=B3 H3 4 A4=[33] B4=[32,4] D4=[31,1,1] E4=A4 F4 H4 5 A5=[34] B5=[33,4] D5=[32,1,1] E5=D5
6 A6=[35] B6=[34,4] D6=[33,1,1] E6=[32,2,1] 7 A7=[36] B7=[35,4] D7=[34,1,1] E7=[33,2,1] 8 A8=[37] B8=[36,4] D8=[35,1,1] E8=[34,2,1] 9 A9=[38] B9=[37,4] D9=[36,1,1]
10+ .. .. .. .. Application with uniform polytopes[edit] In constructing uniform polytopes, nodes are marked as active by a ring if a generator point is off the mirror, creating a new edge between a generator point and its mirror image. An unringed node represents an inactive mirror that generates no new points. Two orthogonal mirrors can be used to generate a square, , seen here with a red generator point and 3 virtual copies across the mirrors. The generator has to be off both mirrors in this orthogonal case to generate an interior. The ring markup presumes active rings have generators equal distance from all mirrors, while a rectangle can also represent a nonuniform solution. Coxeter–Dynkin diagrams can explicitly enumerate nearly all classes
of uniform polytope and uniform tessellations. Every uniform polytope
with pure reflective symmetry (all but a few special cases have pure
reflectional symmetry) can be represented by a Coxeter–Dynkin
diagram with permutations of markups. Each uniform polytope 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. An unconnected diagram (subgroups separated by order-2
branches, or orthogonal mirrors) requires at least one active node in
each subgraph.
All regular polytopes, represented by
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 branch 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 Ĩ1. 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 branches labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn as a line (with the 2 branches 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. There are 7 reflective uniform constructions within a general triangle, based on 7 topological generator positions within the fundamental domain. Every active mirror generates an edge, with two active mirrors have generators on the domain sides and three active mirrors has the generator in the interior. One or two degrees of freedom can be solved for a unique position for equal edge lengths of the resulting polyhedron or tiling. Example 7 generators on octahedral symmetry, fundamental domain triangle (4 3 2), with 8th snub generation as an alternation The duals of the uniform polytopes are sometimes marked up with a
perpendicular slash replacing ringed nodes, and a slash-hole for hole
nodes of the snubs. For example, represents a rectangle (as two
active orthogonal mirrors), and represents its dual polygon, the
rhombus.
Example polyhedra and tilings[edit]
For example, the B3
Uniform octahedral polyhedra Symmetry: [4,3], (*432) [4,3]+ (432) [1+,4,3] = [3,3] (*332) [3+,4] (3*2) 4,3 t 4,3 r 4,3 r 31,1 t 3,4 t 31,1 3,4 31,1 rr 4,3 s2 3,4 tr 4,3 sr 4,3 h 4,3 3,3 h2 4,3 t 3,3 s 3,4 s 31,1 = = = = or = or = Duals to uniform polyhedra V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35 The same constructions can be made on disjointed (orthogonal) Coxeter groups like the uniform prisms, and can be seen more clearly as tilings of dihedrons and hosohedrons on the sphere, like this [6]×[] or [6,2] family: Uniform hexagonal dihedral spherical polyhedra Symmetry: [6,2], (*622) [6,2]+, (622) [6,2+], (2*3) 6,2 t 6,2 r 6,2 t 2,6 2,6 rr 6,2 tr 6,2 sr 6,2 s 2,6 Duals to uniforms V62 V122 V62 V4.4.6 V26 V4.4.6 V4.4.12 V3.3.3.6 V3.3.3.3 In comparison, the [6,3], family produces a parallel set of 7 uniform tilings of the Euclidean plane, and their dual tilings. There are again 3 alternations and some half symmetry version. Uniform hexagonal/triangular tilings Symmetry: [6,3], (*632) [6,3]+ (632) [6,3+] (3*3) 6,3 t 6,3 r 6,3 t 3,6 3,6 rr 6,3 tr 6,3 sr 6,3 s 3,6 63 3.122 (3.6)2 6.6.6 36 3.4.12.4 4.6.12 3.3.3.3.6 3.3.3.3.3.3 Uniform duals V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 In the hyperbolic plane [7,3], family produces a parallel set of uniform tilings of the Euclidean plane, and their dual tilings. There is only 1 alternation (snub) since all branch orders are odd. Many other hyperbolic families of uniform tilings can be seen at uniform tilings in hyperbolic plane. Uniform heptagonal/triangular tilings Symmetry: [7,3], (*732) [7,3]+, (732) 7,3 t 7,3 r 7,3 t 3,7 3,7 rr 7,3 tr 7,3 sr 7,3 Uniform duals V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7 Affine Coxeter groups[edit]
Families of convex uniform Euclidean tessellations are defined by the
affine Coxeter groups. These groups are identical to the finite groups
with the inclusion of one added node. In letter names they are given
the same letter with a "~" above the letter. The index refers to the
finite group, so the rank is the index plus 1. (
A ~ n − 1 displaystyle tilde A _ n-1 : diagrams of this type are cycles. (Also Pn) C ~ n − 1 displaystyle tilde C _ n-1 is associated with the hypercube regular tessellation 4, 3, ...., 4 family. (Also Rn) B ~ n − 1 displaystyle tilde B _ n-1 related to C by one removed mirror. (Also Sn) D ~ n − 1 displaystyle tilde D _ n-1 related to C by two removed mirrors. (Also Qn) E ~ 6 displaystyle tilde E _ 6 , E ~ 7 displaystyle tilde E _ 7 , E ~ 8 displaystyle tilde E _ 8 . (Also T7, T8, T9) F ~ 4 displaystyle tilde F _ 4 forms the 3,4,3,3 regular tessellation. (Also U5) G ~ 2 displaystyle tilde G _ 2 forms 30-60-90 triangle fundamental domains. (Also V3) I ~ 1 displaystyle tilde I _ 1 is two parallel mirrors. ( = A ~ 1 displaystyle tilde A _ 1 = C ~ 1 displaystyle tilde C _ 1 ) (Also W2) Composite groups can also be defined as orthogonal projects. The most common use A ~ 1 displaystyle tilde A _ 1 , like A ~ 1 2 displaystyle tilde A _ 1 ^ 2 , represents square or rectangular checker board domains in the Euclidean plane. And A ~ 1 G ~ 2 displaystyle tilde A _ 1 tilde G _ 2 represents triangular prism fundamental domains in Euclidean 3-space. Affine Coxeter graphs up to (2 to 10 nodes) Rank A ~ 1 + displaystyle tilde A _ 1+ (P2+) B ~ 3 + displaystyle tilde B _ 3+ (S4+) C ~ 1 + displaystyle tilde C _ 1+ (R2+) D ~ 4 + displaystyle tilde D _ 4+ (Q5+) E ~ n displaystyle tilde E _ n (Tn+1) / F ~ 4 displaystyle tilde F _ 4 (U5) / G ~ 2 displaystyle tilde G _ 2 (V3) 2 A ~ 1 displaystyle tilde A _ 1 =[∞]
C ~ 1 displaystyle tilde C _ 1 =[∞]
3 A ~ 2 displaystyle tilde A _ 2 =[3[3]] * C ~ 2 displaystyle tilde C _ 2 =[4,4] * G ~ 2 displaystyle tilde G _ 2 =[6,3] * 4 A ~ 3 displaystyle tilde A _ 3 =[3[4]] * B ~ 3 displaystyle tilde B _ 3 =[4,31,1] * C ~ 3 displaystyle tilde C _ 3 =[4,3,4] * D ~ 3 displaystyle tilde D _ 3 =[31,1,3−1,31,1] = A ~ 3 displaystyle tilde A _ 3 5 A ~ 4 displaystyle tilde A _ 4 =[3[5]] * B ~ 4 displaystyle tilde B _ 4 =[4,3,31,1] * C ~ 4 displaystyle tilde C _ 4 =[4,32,4] * D ~ 4 displaystyle tilde D _ 4 =[31,1,1,1] * F ~ 4 displaystyle tilde F _ 4 =[3,4,3,3] * 6 A ~ 5 displaystyle tilde A _ 5 =[3[6]] * B ~ 5 displaystyle tilde B _ 5 =[4,32,31,1] * C ~ 5 displaystyle tilde C _ 5 =[4,33,4] * D ~ 5 displaystyle tilde D _ 5 =[31,1,3,31,1] * 7 A ~ 6 displaystyle tilde A _ 6 =[3[7]] * B ~ 6 displaystyle tilde B _ 6 =[4,33,31,1] C ~ 6 displaystyle tilde C _ 6 =[4,34,4] D ~ 6 displaystyle tilde D _ 6 =[31,1,32,31,1] E ~ 6 displaystyle tilde E _ 6 =[32,2,2] 8 A ~ 7 displaystyle tilde A _ 7 =[3[8]] * B ~ 7 displaystyle tilde B _ 7 =[4,34,31,1] * C ~ 7 displaystyle tilde C _ 7 =[4,35,4] D ~ 7 displaystyle tilde D _ 7 =[31,1,33,31,1] * E ~ 7 displaystyle tilde E _ 7 =[33,3,1] * 9 A ~ 8 displaystyle tilde A _ 8 =[3[9]] * B ~ 8 displaystyle tilde B _ 8 =[4,35,31,1] C ~ 8 displaystyle tilde C _ 8 =[4,36,4] D ~ 8 displaystyle tilde D _ 8 =[31,1,34,31,1] E ~ 8 displaystyle tilde E _ 8 =[35,2,1] * 10 A ~ 9 displaystyle tilde A _ 9 =[3[10]] * B ~ 9 displaystyle tilde B _ 9 =[4,36,31,1] C ~ 9 displaystyle tilde C _ 9 =[4,37,4] D ~ 9 displaystyle tilde D _ 9 =[31,1,35,31,1] 11 ... ... ... ... Hyperbolic Coxeter groups[edit] There are many infinite hyperbolic Coxeter groups. Hyperbolic groups are categorized as compact or not, with compact groups having bounded fundamental domains. Compact simplex hyperbolic groups (Lannér simplices) exist as rank 3 to 5. Paracompact simplex groups (Koszul simplices) exist up to rank 10. Hypercompact (Vinberg polytopes) groups have been explored but not been fully determined. In 2006, Allcock proved that there are infinitely many compact Vinberg polytopes for dimension up to 6, and infinitely many finite-volume Vinberg polytopes for dimension up to 19,[4] so a complete enumeration is not possible. All of these fundamental reflective domains, both simplices and nonsimplices, are often called Coxeter polytopes or sometimes less accurately Coxeter polyhedra. Hyperbolic groups in H2[edit] Further information: Uniform tilings in hyperbolic plane
Example right triangles [p,q] [3,7] [3,8] [3,9] [3,∞] [4,5] [4,6] [4,7] [4,8] [∞,4] [5,5] [5,6] [5,7] [6,6] [∞,∞] Example general triangles [(p,q,r)] [(3,3,4)] [(3,3,5)] [(3,3,6)] [(3,3,7)] [(3,3,∞)] [(3,4,4)] [(3,6,6)] [(3,∞,∞)] [(6,6,6)] [(∞,∞,∞)] Two-dimensional hyperbolic triangle groups exist as rank 3 Coxeter diagrams, defined by triangle (p q r) for: 1 p + 1 q + 1 r < 1. displaystyle frac 1 p + frac 1 q + frac 1 r <1. There are infinitely many compact triangular hyperbolic Coxeter groups, including linear and triangle graphs. The linear graphs exist for right triangles (with r=2).[5] Compact hyperbolic Coxeter groups Linear Cyclic ∞ [p,q], : 2(p+q)<pq ... ... ... ∞ [(p,q,r)], : p+q+r>9 ... Paracompact Coxeter groups of rank 3 exist as limits to the compact ones. Linear graphs Cyclic graphs [p,∞] [∞,∞] [(p,q,∞)] [(p,∞,∞)] [(∞,∞,∞)] Arithmetic triangle group[edit] The hyperbolic triangle groups that are also arithmetic groups form a finite subset. By computer search the complete list was determined by Kisao Takeuchi in his 1977 paper Arithmetic triangle groups.[6] There are 85 total, 76 compact and 9 paracompact. Right triangles (p q 2) General triangles (p q r) Compact groups: (76) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Paracompact right triangles: (4) , , , General triangles: (39) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Paracompact general triangles: (5) , , , , (2 3 7), (2 3 8), (2 3 9), (2 3 10), (2 3 11), (2 3 12), (2 3 14), (2 3 16), (2 3 18), (2 3 24), (2 3 30) (2 4 5), (2 4 6), (2 4 7), (2 4 8), (2 4 10), (2 4 12), (2 4 18), (2 5 5), (2 5 6), (2 5 8), (2 5 10), (2 5 20), (2 5 30) (2 6 6), (2 6 8), (2 6 12) (2 7 7), (2 7 14), (2 8 8), (2 8 16), (2 9 18) (2 10 10) (2 12 12) (2 12 24), (2 15 30), (2 18 18) (2 3 ∞) (2,4 ∞) (2,6 ∞) (2 ∞ ∞) (3 3 4), (3 3 5), (3 3 6), (3 3 7), (3 3 8), (3 3 9), (3 3 12), (3 3 15) (3 4 4), (3 4 6), (3 4 12), (3 5 5), (3 6 6), (3 6 18), (3 8 8), (3 8 24), (3 10 30), (3 12 12) (4 4 4), (4 4 5), (4 4 6), (4 4 9), (4 5 5), (4 6 6), (4 8 8), (4 16 16) (5 5 5), (5 5 10), (5 5 15), (5 10 10) (6 6 6), (6 12 12), (6 24 24) (7 7 7) (8 8 8) (9 9 9) (9 18 18) (12 12 12) (15 15 15) (3,3 ∞) (3 ∞ ∞) (4,4 ∞) (6 6 ∞) (∞ ∞ ∞) Hyperbolic Coxeter polygons above triangles[edit] Fundamental domains of quadrilateral groups or [∞,3,∞] [iπ/λ1,3,iπ/λ2] (*3222) or [((3,∞,3)),∞] [((3,iπ/λ1,3)),iπ/λ2] (*3322) or [(3,∞)[2]] [(3,iπ/λ1,3,iπ/λ2)] (*3232) or [(4,∞)[2]] [(4,iπ/λ1,4,iπ/λ2)] (*4242) (*3333) Domains with ideal vertices [iπ/λ1,∞,iπ/λ2] (*∞222) (*∞∞22) [(iπ/λ1,∞,iπ/λ2,∞)] (*2∞2∞) (*∞∞∞∞) (*4444) Other H2 hyperbolic kaleidoscopes can be constructed from higher order polygons. Like triangle groups these kaleidoscopes can be identified by a cyclic sequence of mirror intersection orders around the fundamental domain, as (a b c d ...), or equivalently in orbifold notation as *abcd.... Coxeter–Dynkin diagrams for these polygonal kaleidoscopes can be seen as a degenerate (n-1)-simplex fundamental domains, with a cyclic of branches order a,b,c... and the remaining n*(n-3)/2 branches are labeled as infinite (∞) representing the non-intersecting mirrors. The only nonhyperbolic example is Euclidean symmetry four mirrors in a square or rectangle as , [∞,2,∞] (orbifold *2222). Another branch representation for non-intersecting mirrors by Vinberg gives infinite branches as dotted or dashed lines, so this diagram can be shown as , with the four order-2 branches suppressed around the perimeter. For example, a quadrilateral domain (a b c d) will have two infinite order branches connecting ultraparallel mirrors. The smallest hyperbolic example is , [∞,3,∞] or [iπ/λ1,3,iπ/λ2] (orbifold *3222), where (λ1,λ2) are the distance between the ultraparallel mirrors. The alternate expression is , with three order-2 branches suppressed around the perimeter. Similarly (2 3 2 3) (orbifold *3232) can be represented as and (3 3 3 3), (orbifold *3333) can be represented as a complete graph . The highest quadrilateral domain (∞ ∞ ∞ ∞) is an infinite square, represented by a complete tetrahedral graph with 4 perimeter branches as ideal vertices and two diagonal branches as infinity (shown as dotted lines) for ultraparallel mirrors: . Compact (Lannér simplex groups)[edit] Compact hyperbolic groups are called Lannér groups after Folke Lannér who first studied them in 1950.[7] They only exist as rank 4 and 5 graphs. Coxeter studied the linear hyperbolic coxeter groups in his 1954 paper Regular Honeycombs in hyperbolic space,[8] which included two rational solutions in hyperbolic 4-space: [5/2,5,3,3] = and [5,5/2,5,3] = . Ranks 4–5[edit] Further information: Uniform honeycombs in hyperbolic space The fundamental domain of either of the two bifurcating groups, [5,31,1] and [5,3,31,1], is double that of a corresponding linear group, [5,3,4] and [5,3,3,4] respectively. Letter names are given by Johnson as extended Witt symbols.[9] Compact hyperbolic Coxeter groups Dimension Hd Rank Total count Linear Bifurcating Cyclic H3 4 9 3: B H ¯ 3 displaystyle overline BH _ 3 = [4,3,5]: K ¯ 3 displaystyle overline K _ 3 = [5,3,5]: J ¯ 3 displaystyle overline J _ 3 = [3,5,3]: D H ¯ 3 displaystyle overline DH _ 3 = [5,31,1]: A B ^ 3 displaystyle widehat AB _ 3 = [(33,4)]: A H ^ 3 displaystyle widehat AH _ 3 = [(33,5)]: B B ^ 3 displaystyle widehat BB _ 3 = [(3,4)[2]]: B H ^ 3 displaystyle widehat BH _ 3 = [(3,4,3,5)]: H H ^ 3 displaystyle widehat HH _ 3 = [(3,5)[2]]: H4 5 5 3: H ¯ 4 displaystyle overline H _ 4 = [33,5]: B H ¯ 4 displaystyle overline BH _ 4 = [4,3,3,5]: K ¯ 4 displaystyle overline K _ 4 = [5,3,3,5]: D H ¯ 4 displaystyle overline DH _ 4 = [5,3,31,1]: A F ^ 4 displaystyle widehat AF _ 4 = [(34,4)]: Paracompact (Koszul simplex groups)[edit] An example order-3 apeirogonal tiling, ∞,3 with one green apeirogon and its circumscribed horocycle Paracompact (also called noncompact) hyperbolic Coxeter groups contain
affine subgroups and have asymptotic simplex fundamental domains. The
highest paracompact hyperbolic
Ideal fundamental domains of , [(∞,∞,∞)] seen in the Poincare disk model There are 5 hyperbolic Coxeter groups expressing ideal simplices, graphs where removal of any one node results in an affine Coxeter group. Thus all vertices of this ideal simplex are at infinity.[12] Rank Ideal group Affine subgroups 3 [(∞,∞,∞)] [∞] 4 [4[4]] [4,4] 4 [3[3,3]] [3[3]] 4 [(3,6)[2]] [3,6] 6 [(3,3,4)[2]] [4,3,3,4], [3,4,3,3] , Ranks 4–10[edit] Infinite Euclidean cells like a hexagonal tiling, properly scaled
converge to a single ideal point at infinity, like the hexagonal
tiling honeycomb, 6,3,3 , as shown with this single cell in a
Further information: Paracompact uniform honeycombs There are a total of 58 paracompact hyperbolic Coxeter groups from rank 4 through 10. All 58 are grouped below in five categories. Letter symbols are given by Johnson as Extended Witt symbols, using PQRSTWUV from the affine Witt symbols, and adding LMNOXYZ. These hyperbolic groups are given an overline, or a hat, for cycloschemes. The bracket notation from Coxeter is a linearized representation of the Coxeter group. Hyperbolic paracompact groups Rank Total count Groups 4 23 B R ^ 3 displaystyle widehat BR _ 3 = [(3,3,4,4)]: C R ^ 3 displaystyle widehat CR _ 3 = [(3,43)]: R R ^ 3 displaystyle widehat RR _ 3 = [4[4]]: A V ^ 3 displaystyle widehat AV _ 3 = [(33,6)]: B V ^ 3 displaystyle widehat BV _ 3 = [(3,4,3,6)]: H V ^ 3 displaystyle widehat HV _ 3 = [(3,5,3,6)]: V V ^ 3 displaystyle widehat VV _ 3 = [(3,6)[2]]: P ¯ 3 displaystyle overline P _ 3 = [3,3[3]]: B P ¯ 3 displaystyle overline BP _ 3 = [4,3[3]]: H P ¯ 3 displaystyle overline HP _ 3 = [5,3[3]]: V P ¯ 3 displaystyle overline VP _ 3 = [6,3[3]]: D V ¯ 3 displaystyle overline DV _ 3 = [6,31,1]: O ¯ 3 displaystyle overline O _ 3 = [3,41,1]: M ¯ 3 displaystyle overline M _ 3 = [41,1,1]: R ¯ 3 displaystyle overline R _ 3 = [3,4,4]: N ¯ 3 displaystyle overline N _ 3 = [43]: V ¯ 3 displaystyle overline V _ 3 = [3,3,6]: B V ¯ 3 displaystyle overline BV _ 3 = [4,3,6]: H V ¯ 3 displaystyle overline HV _ 3 = [5,3,6]: Y ¯ 3 displaystyle overline Y _ 3 = [3,6,3]: Z ¯ 3 displaystyle overline Z _ 3 = [6,3,6]: D P ¯ 3 displaystyle overline DP _ 3 = [3[]x[]]: P P ¯ 3 displaystyle overline PP _ 3 = [3[3,3]]: 5 9 P ¯ 4 displaystyle overline P _ 4 = [3,3[4]]: B P ¯ 4 displaystyle overline BP _ 4 = [4,3[4]]: F R ^ 4 displaystyle widehat FR _ 4 = [(32,4,3,4)]: D P ¯ 4 displaystyle overline DP _ 4 = [3[3]x[]]: N ¯ 4 displaystyle overline N _ 4 = [4,3,((4,2,3))]: O ¯ 4 displaystyle overline O _ 4 = [3,4,31,1]: S ¯ 4 displaystyle overline S _ 4 = [4,32,1]: R ¯ 4 displaystyle overline R _ 4 = [(3,4)2]: M ¯ 4 displaystyle overline M _ 4 = [4,31,1,1]: 6 12 P ¯ 5 displaystyle overline P _ 5 = [3,3[5]]: A U ^ 5 displaystyle widehat AU _ 5 = [(35,4)]: A R ^ 5 displaystyle widehat AR _ 5 = [(3,3,4)[2]]: S ¯ 5 displaystyle overline S _ 5 = [4,3,32,1]: O ¯ 5 displaystyle overline O _ 5 = [3,4,31,1]: N ¯ 5 displaystyle overline N _ 5 = [3,(3,4)1,1]: U ¯ 5 displaystyle overline U _ 5 = [33,4,3]: X ¯ 5 displaystyle overline X _ 5 = [3,3,4,3,3]: R ¯ 5 displaystyle overline R _ 5 = [3,4,3,3,4]: Q ¯ 5 displaystyle overline Q _ 5 = [32,1,1,1]: M ¯ 5 displaystyle overline M _ 5 = [4,3,31,1,1]: L ¯ 5 displaystyle overline L _ 5 = [31,1,1,1,1]: 7 3 P ¯ 6 displaystyle overline P _ 6 = [3,3[6]]: Q ¯ 6 displaystyle overline Q _ 6 = [31,1,3,32,1]: S ¯ 6 displaystyle overline S _ 6 = [4,32,32,1]: 8 4 P ¯ 7 displaystyle overline P _ 7 = [3,3[7]]: Q ¯ 7 displaystyle overline Q _ 7 = [31,1,32,32,1]: S ¯ 7 displaystyle overline S _ 7 = [4,33,32,1]: T ¯ 7 displaystyle overline T _ 7 = [33,2,2]: 9 4 P ¯ 8 displaystyle overline P _ 8 = [3,3[8]]: Q ¯ 8 displaystyle overline Q _ 8 = [31,1,33,32,1]: S ¯ 8 displaystyle overline S _ 8 = [4,34,32,1]: T ¯ 8 displaystyle overline T _ 8 = [34,3,1]: 10 3 P ¯ 9 displaystyle overline P _ 9 = [3,3[9]]: Q ¯ 9 displaystyle overline Q _ 9 = [31,1,34,32,1]: S ¯ 9 displaystyle overline S _ 9 = [4,35,32,1]: T ¯ 9 displaystyle overline T _ 9 = [36,2,1]: Subgroup relations of paracompact hyperbolic groups[edit] These trees represents subgroup relations of paracompact hyperbolic groups. Subgroup indices on each connection are given in red.[13] Subgroups of index 2 represent a mirror removal, and fundamental domain doubling. Others can be inferred by commensurability (integer ratio of volumes) for the tetrahedral domains. H3 H4 H5 Hypercompact Coxeter groups (Vinberg polytopes)[edit]
Just like the hyperbolic plane H2 has nontriangular polygonal domains,
higher-dimensional reflective hyperbolic domains also exists. These
nonsimplex domains can be considered degenerate simplices with
non-intersecting mirrors given infinite order, or in a Coxeter
diagram, such branches are given dotted or dashed lines. These
nonsimplex domains are called Vinberg polytopes, after Ernest Vinberg
for his
Dimension Rank Graphs H3 5 , , , , , , , , , , , , , , , , , , , , , , , , , , , Another subgroup [1+,41,1,1] = [∞,4,1+,4,∞] = [∞[6]]. = = . [17] Vinberg polytopes with rank n+3 for n dimensional space[edit] There are a finite number of degenerate fundamental simplices exist up to 8-dimensions. The complete list of Compact Vinberg polytopes with rank n+3 mirrors for n-dimensions has been enumerated by P. Tumarkin in 2004. These groups are labeled by dotted/broken lines for ultraparallel branches. The complete list of non-Compact Vinberg polytopes with rank n+3 mirrors and with one non-simple vertex for n-dimensions has been enumerated by Mike Roberts.[18] For 4 to 8 dimensions, rank 7 to 11 Coxeter groups are counted as 44, 16, 3, 1, and 1 respectively.[19] The highest was discovered by Bugaenko in 1984 in dimension 8, rank 11:[20] Dimensions Rank Cases Graphs H4 7 44 ... H5 8 16 .. H6 9 3 H7 10 1 H8 11 1 Vinberg polytopes with rank n+4 for n dimensional space[edit] There are a finite number of degenerate fundamental simplices exist up to 8-dimensions. Compact Vinberg polytopes with rank n+4 mirrors for n-dimensions has been explored by A. Felikson and P. Tumarkin in 2005.[21] Lorentzian groups[edit] Regular honeycombs with Lorentzian groups 3,3,7 in hyperbolic 3-space, rendered the intersection of the honeycomb with the plane-at-infinity, in the Poincare half-space model. 7,3,3 viewed outside of Poincare ball model. This shows rank 5 Lorentzian groups arranged as subgroups from [6,3,3,3], and [6,3,6,3]. The highly symmetric group , [3[3,3,3]] is an index 120 subgroup of [6,3,3,3]. Lorentzian groups for simplex domains can be defined as graphs beyond
the paracompact hyperbolic forms. These are sometimes called
super-ideal simplices and are also related to a Lorentzian geometry,
named after
Lorentzian Coxeter groups Rank Total count Groups 4 ∞ [3,3,7] ... [∞,∞,∞]: ... [4,3[3]] ... [∞,∞[3]]: ... [5,41,1] ... [∞1,1,1]: ... ... [(5,4,3,3)] ... [∞[4]]: ... ... ... [4[]×[]] ... [∞[]×[]]: ... ... [4[3,3]] ... [∞[3,3]] 5 186 ...[3[3,3,3]]:... 6 66 7 36 [31,1,1,1,1,1]: ... 8 13 [3,3,3[6]]: [3,3[6],3]: [3,3[2+4],3]: [3,3[1+5],3]: [3[ ]e×[3]]: [4,3,3,33,1]: [31,1,3,33,1]: [3,(3,3,4)1,1]: [32,1,3,32,1]: [4,3,3,32,2]: [31,1,3,32,2]: 9 10 [3,3[3+4],3]: [3,3[9]]: [3,3[2+5],3]: [32,1,32,32,1]: [33,1,33,4]: [33,1,3,3,31,1]: [33,3,2]: [32,2,4]: [32,2,33,4]: [32,2,3,3,31,1]: 10 8 [3,3[8],3]: [3,3[3+5],3]: [3,3[9]]: [32,1,33,32,1]: [35,3,1]: [33,1,34,4]: [33,1,33,31,1]: [34,4,1]: 11 4 [32,1,34,32,1]: [32,1,36,4]: [32,1,35,31,1]: [37,2,1]: Very-extended Coxeter Diagrams[edit] One usage includes a very-extended definition from the direct Dynkin diagram usage which considers affine groups as extended, hyperbolic groups over-extended, and a third node as very-extended simple groups. These extensions are usually marked by an exponent of 1,2, or 3 + symbols for the number of extended nodes. This extending series can be extended backwards, by sequentially removing the nodes from the same position in the graph, although the process stops after removing branching node. The E8 extended family is the most commonly shown example extending backwards from E3 and forwards to E11. The extending process can define a limited series of Coxeter graphs that progress from finite to affine to hyperbolic to Lorentzian. The determinant of the Cartan matrices determine where the series changes from finite (positive) to affine (zero) to hyperbolic (negative), and ending as a Lorentzian group, containing at least one hyperbolic subgroup.[26] The noncrystalographic Hn groups forms an extended series where H4 is extended as a compact hyperbolic and over-extended into a lorentzian group. The determinant of the Schläfli matrix by rank are:[27] det(A1n=[2n-1]) = 2n (Finite for all n) det(An=[3n-1]) = n+1 (Finite for all n) det(Bn=[4,3n-2]) = 2 (Finite for all n) det(Dn=[3n-3,1,1]) = 4 (Finite for all n) Determinants of the Schläfli matrix in exceptional series are: det(En=[3n-3,2,1]) = 9-n (Finite for E3(=A2A1), E4(=A4), E5(=D5), E6, E7 and E8, affine at E9 ( E ~ 8 displaystyle tilde E _ 8 ), hyperbolic at E10) det([3n-4,3,1]) = 2(8-n) (Finite for n=4 to 7, affine ( E ~ 7 displaystyle tilde E _ 7 ), and hyperbolic at n=8.) det([3n-4,2,2]) = 3(7-n) (Finite for n=4 to 6, affine ( E ~ 6 displaystyle tilde E _ 6 ), and hyperbolic at n=7.) det(Fn=[3,4,3n-3]) = 5-n (Finite for F3(=B3) to F4, affine at F5 ( F ~ 4 displaystyle tilde F _ 4 ), hyperbolic at F6) det(Gn=[6,3n-2]) = 3-n (Finite for G2, affine at G3 ( G ~ 2 displaystyle tilde G _ 2 ), hyperbolic at G4) Smaller extended series Finite A 2 displaystyle A_ 2 C 2 displaystyle C_ 2 G 2 displaystyle G_ 2 A 3 displaystyle A_ 3 B 3 displaystyle B_ 3 C 3 displaystyle C_ 3 H 4 displaystyle H_ 4 Rank n [3[3],3n-3] [4,4,3n-3] Gn=[6,3n-2] [3[4],3n-4] [4,31,n-3] [4,3,4,3n-4] Hn=[5,3n-2] 2 [3] A2 [4] C2 [6] G2 [2] A12 [4] C2 [5] H2 3 [3[3]] A2+= A ~ 2 displaystyle tilde A _ 2 [4,4] C2+= C ~ 2 displaystyle tilde C _ 2 [6,3] G2+= G ~ 2 displaystyle tilde G _ 2 [3,3]=A3 [4,3] B3 [4,3] C3 [5,3] H3 4 [3[3],3] A2++= P ¯ 3 displaystyle overline P _ 3 [4,4,3] C2++= R ¯ 3 displaystyle overline R _ 3 [6,3,3] G2++= V ¯ 3 displaystyle overline V _ 3 [3[4]] A3+= A ~ 3 displaystyle tilde A _ 3 [4,31,1] B3+= B ~ 3 displaystyle tilde B _ 3 [4,3,4] C3+= C ~ 3 displaystyle tilde C _ 3 [5,3,3] H4 5 [3[3],3,3] A2+++ [4,4,3,3] C2+++ [6,3,3,3] G2+++ [3[4],3] A3++= P ¯ 4 displaystyle overline P _ 4 [4,32,1] B3++= S ¯ 4 displaystyle overline S _ 4 [4,3,4,3] C3++= R ¯ 4 displaystyle overline R _ 4 [5,33] H5= H ¯ 4 displaystyle overline H _ 4 6 [3[4],3,3] A3+++ [4,33,1] B3+++ [4,3,4,3,3] C3+++ [5,34] H6 Det(Mn) 3(3-n) 2(3-n) 3-n 4(4-n) 2(4-n) Middle extended series Finite A 4 displaystyle A_ 4 B 4 displaystyle B_ 4 C 4 displaystyle C_ 4 D 4 displaystyle D_ 4 F 4 displaystyle F_ 4 A 5 displaystyle A_ 5 B 5 displaystyle B_ 5 D 5 displaystyle D_ 5 Rank n [3[5],3n-5] [4,3,3n-4,1] [4,3,3,4,3n-5] [3n-4,1,1,1] [3,4,3n-3] [3[6],3n-6] [4,3,3,3n-5,1] [31,1,3,3n-5,1] 3 [4,3−1,1] B2A1 [4,3] B3 [3−1,1,1,1] A13 [3,4] B3 [4,3,3] C3 4 [33] A4 [4,3,3] B4 [4,3,3] C4 [30,1,1,1] D4 [3,4,3] F4 [4,3,3,3−1,1] B3A1 [31,1,3,3−1,1] A3A1 5 [3[5]] A4+= A ~ 4 displaystyle tilde A _ 4 [4,3,31,1] B4+= B ~ 4 displaystyle tilde B _ 4 [4,3,3,4] C4+= C ~ 4 displaystyle tilde C _ 4 [31,1,1,1] D4+= D ~ 4 displaystyle tilde D _ 4 [3,4,3,3] F4+= F ~ 4 displaystyle tilde F _ 4 [34] A5 [4,3,3,3,3] B5 [31,1,3,3] D5 6 [3[5],3] A4++= P ¯ 5 displaystyle overline P _ 5 [4,3,32,1] B4++= S ¯ 5 displaystyle overline S _ 5 [4,3,3,4,3] C4++= R ¯ 5 displaystyle overline R _ 5 [32,1,1,1] D4++= Q ¯ 5 displaystyle overline Q _ 5 [3,4,33] F4++= U ¯ 5 displaystyle overline U _ 5 [3[6]] A5+= A ~ 5 displaystyle tilde A _ 5 [4,3,3,31,1] B5+= B ~ 5 displaystyle tilde B _ 5 [31,1,3,31,1] D5+= D ~ 5 displaystyle tilde D _ 5 7 [3[5],3,3] A4+++ [4,3,33,1] B4+++ [4,3,3,4,3,3] C4+++ [33,1,1,1] D4+++ [3,4,34] F4+++ [3[6],3] A5++= P ¯ 6 displaystyle overline P _ 6 [4,3,3,32,1] B5++= S ¯ 6 displaystyle overline S _ 6 [31,1,3,32,1] D5++= Q ¯ 6 displaystyle overline Q _ 6 8 [3[6],3,3] A5+++ [4,3,3,33,1] B5+++ [31,1,3,33,1] D5+++ Det(Mn) 5(5-n) 2(5-n) 4(5-n) 5-n 6(6-n) 4(6-n) Some higher extended series Finite A 6 displaystyle A_ 6 B 6 displaystyle B_ 6 D 6 displaystyle D_ 6 E 6 displaystyle E_ 6 A 7 displaystyle A_ 7 B 7 displaystyle B_ 7 D 7 displaystyle D_ 7 E 7 displaystyle E_ 7 E 8 displaystyle E_ 8 Rank n [3[7],3n-7] [4,33,3n-6,1] [31,1,3,3,3n-6,1] [3n-5,2,2] [3[8],3n-8] [4,34,3n-7,1] [31,1,3,3,3,3n-7,1] [3n-5,3,1] En=[3n-4,2,1] 3 [3−1,2,1] E3=A2A1 4 [3−1,2,2] A22 [3−1,3,1] A3A1 [30,2,1] E4=A4 5 [4,3,3,3,3−1,1] B4A1 [31,1,3,3,3−1,1] D4A1 [30,2,2] A5 [30,3,1] A5 [31,2,1] E5=D5 6 [35] A6 [4,34] B6 [31,1,3,3,3] D6 [31,2,2] E6 [4,3,3,3,3,3−1,1] B5A1 [31,1,3,3,3,3−1,1] D5A1 [31,3,1] D6 [32,2,1] E6 * 7 [3[7]] A6+= A ~ 6 displaystyle tilde A _ 6 [4,33,31,1] B6+= B ~ 6 displaystyle tilde B _ 6 [31,1,3,3,31,1] D6+= D ~ 6 displaystyle tilde D _ 6 [32,2,2] E6+= E ~ 6 displaystyle tilde E _ 6 [36] A7 [4,35] B7 [31,1,3,3,3,30,1] D7 [32,3,1] E7 * [33,2,1] E7 * 8 [3[7],3] A6++= P ¯ 7 displaystyle overline P _ 7 [4,33,32,1] B6++= S ¯ 7 displaystyle overline S _ 7 [31,1,3,3,32,1] D6++= Q ¯ 7 displaystyle overline Q _ 7 [33,2,2] E6++= T ¯ 7 displaystyle overline T _ 7 [3[8]] A7+= A ~ 7 displaystyle tilde A _ 7 * [4,34,31,1] B7+= B ~ 7 displaystyle tilde B _ 7 * [31,1,3,3,3,31,1] D7+= D ~ 7 displaystyle tilde D _ 7 * [33,3,1] E7+= E ~ 7 displaystyle tilde E _ 7 * [34,2,1] E8 * 9 [3[7],3,3] A6+++ [4,33,33,1] B6+++ [31,1,3,3,33,1] D6+++ [34,2,2] E6+++ [3[8],3] A7++= P ¯ 8 displaystyle overline P _ 8 * [4,34,32,1] B7++= S ¯ 8 displaystyle overline S _ 8 * [31,1,3,3,3,32,1] D7++= Q ¯ 8 displaystyle overline Q _ 8 * [34,3,1] E7++= T ¯ 8 displaystyle overline T _ 8 * [35,2,1] E9=E8+= E ~ 8 displaystyle tilde E _ 8 * 10 [3[8],3,3] A7+++ * [4,34,33,1] B7+++ * [31,1,3,3,3,33,1] D7+++ * [35,3,1] E7+++ * [36,2,1] E10=E8++= T ¯ 9 displaystyle overline T _ 9 * 11 [37,2,1] E11=E8+++ * Det(Mn) 7(7-n) 2(7-n) 4(7-n) 3(7-n) 8(8-n) 2(8-n) 4(8-n) 2(8-n) 9-n Geometric folding[edit] Finite and affine foldings[28] φA : AΓ --> AΓ' for finite types Γ Γ' Folding description Coxeter–Dynkin diagrams I2(h) Γ(h) Dihedral folding Bn A2n (I,sn) Dn+1, A2n-1 (A3,+/-ε) F4 E6 (A3,±ε) H4 E8 (A4,±ε) H3 D6 H2 A4 G2 A5 (A5,±ε) D4 (D4,±ε) φ: AΓ+ --> AΓ'+ for affine types A ~ n − 1 displaystyle tilde A _ n-1 A ~ k n − 1 displaystyle tilde A _ kn-1 Locally trivial B ~ n displaystyle tilde B _ n D ~ 2 n + 1 displaystyle tilde D _ 2n+1 (I,sn) D ~ n + 1 displaystyle tilde D _ n+1 , D ~ 2 n displaystyle tilde D _ 2n (A3,±ε) C ~ n displaystyle tilde C _ n B ~ n + 1 displaystyle tilde B _ n+1 , C ~ 2 n displaystyle tilde C _ 2n (A3,±ε) C ~ 2 n + 1 displaystyle tilde C _ 2n+1 (I,sn) C ~ n displaystyle tilde C _ n A ~ 2 n + 1 displaystyle tilde A _ 2n+1 (I,sn) & (I,s0) A ~ 2 n displaystyle tilde A _ 2n (A3,ε) & (I,s0) A ~ 2 n − 1 displaystyle tilde A _ 2n-1 (A3,ε) & (A3,ε') C ~ n displaystyle tilde C _ n D ~ n + 2 displaystyle tilde D _ n+2 (A3,-ε) & (A3,-ε') C ~ 2 displaystyle tilde C _ 2 D ~ 5 displaystyle tilde D _ 5 (I,s1) F ~ 4 displaystyle tilde F _ 4 E ~ 6 displaystyle tilde E _ 6 , E ~ 7 displaystyle tilde E _ 7 (A3,±ε) G ~ 2 displaystyle tilde G _ 2 D ~ 6 displaystyle tilde D _ 6 , E ~ 7 displaystyle tilde E _ 7 (A5,±ε) B ~ 3 displaystyle tilde B _ 3 , F ~ 4 displaystyle tilde F _ 4 (B3,±ε) D ~ 4 displaystyle tilde D _ 4 , E ~ 6 displaystyle tilde E _ 6 (D4,±ε) See also:
A few hyperbolic foldings Complex reflections[edit] Coxeter–Dynkin diagrams have been extended to Complex space, Cn where nodes are unitary reflections of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Coxeter writes the complex group, p[q]r, as diagram .[32] A 1-dimensional regular complex polytope in C 1 displaystyle mathbb C ^ 1 is represented as , having p vertices. Its real representation is a regular polygon, p . Its symmetry is p[] or , order p. A unitary operator generator for is seen as a rotation in R 2 displaystyle mathbb R ^ 2 by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is e2πi/p = cos(2π/p) + i sin(2π/p). When p = 2, the generator is eπi = –1, the same as a point reflection in the real plane. In a higher polytope, p or represents a p-edge element, with a 2-edge, or , representing an ordinary real edge between two vertices. Regular complex 1-polytopes Complex 1-polytopes, , represented in the
12 irreducible Shephard groups with their subgroup index relations.[33] Subgroups index 2 relate by removing a real reflection: p[2q]2 --> p[q]p, index 2. p[4]q --> p[q]p, index q. p[4]2 subgroups: p=2,3,4... p[4]2 --> [p], index p p[4]2 --> p[]×p[], index 2 Aa regular complex polygons in C 2 displaystyle mathbb C ^ 2 , has the form p q r or Coxeter diagram . The symmetry group of a regular complex polygon is not called a Coxeter group, but instead a Shephard group, a type of Complex reflection group. The order of p[q]r is 8 / q ⋅ ( 1 / p + 2 / q + 1 / r − 1 ) − 2 displaystyle 8/qcdot (1/p+2/q+1/r-1)^ -2 .[34] The rank 2 Shephard groups are: 2[q]2, p[4]2, 3[3]3, 3[6]2, 3[4]3, 4[3]4, 3[8]2, 4[6]2, 4[4]3, 3[5]3, 5[3]5, 3[10]2, 5[6]2, and 5[4]3 or , , , , , , , , , , , , , of order 2q, 2p2, 24, 48, 72, 96, 144, 192, 288, 360, 600, 1200, and 1800 respectively. The symmetry group p1[q]p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q-1)/2R2 = (R1R2)(q-1)/2R1. When q is odd, p1=p2. The C 3 displaystyle mathbb C ^ 3 group or [1 1 1]p is defined by 3 period 2 unitary reflections R1, R2, R3 : R12 = R12 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = (R1R2R3R1)p = 1. The period p can be seen as a double rotation in real R 4 displaystyle mathbb R ^ 4 . A similar C 3 displaystyle mathbb C ^ 3 group or [1 1 1](p) is defined by 3 period 2 unitary reflections R1, R2, R3 : R12 = R12 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = (R1R2R3R2)p = 1. See also[edit] Coxeter group Schwarz triangle Goursat tetrahedron Dynkin diagram Uniform polytope Wythoff symbol Uniform polyhedron List of uniform polyhedra List of uniform planar tilings Uniform 4-polytope Convex uniform honeycomb Convex uniform honeycombs in hyperbolic space
References[edit] ^ Hall, Brian C. (2003), Lie Groups, Lie Algebras, and
Representations: An Elementary Introduction, Springer,
ISBN 0-387-40122-9
^ Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition,
ISBN 0-486-61480-8, Sec 7.7. page 133, Schläfli's Criterion
^ Lannér F., On complexes with transitive groups of automorphisms,
Medd. Lunds Univ. Mat. Sem. [Comm. Sem. Math. Univ. Lund], 11 (1950),
1–71
^ Allcock, Daniel (11 July 2006). "Infinitely many hyperbolic Coxeter
groups through dimension 19".
Further reading[edit] James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990) Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [10], Googlebooks [11] (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) H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for
Discrete Groups 4th ed, Springer-Verlag. New York. 1980
Norman Johnson, Geometries and Transformations, Chapters 11,12,13,
preprint 2011
N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The
size of a hyperbolic Coxeter simplex, Transformation Groups 1999,
Volume 4, Issue 4, pp 329–353 [12] [13]
Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter
Groups
External links[edit] Wikimedia Commons has media related to Coxeter-Dynkin diagrams. Weisstein, Eric W. "Coxeter–Dynkin diagram". MathWorld. October 1978 discussion on the history of the Coxeter diagrams by Coxeter and Dynkin in Toronto, Canada; Eugene Dynkin Collection of Mathematics Interviews, Cornell Univers |