User:Tomruen/List of Coxeter groups
This test article is my attempt to construct Coxeter groups by rank, including products. I subdivide them by their finite order, infinite, or hyperbolic infinite.
I name them by their group names, Coxeter's bracket notation names, and their Coxeter-Dynkin diagram graphs. I also give an example polytope or tessellation that has this symmetry.
Finite Definite (spherical)
[edit]Rank 1
[edit](1-space)
Rank 2
[edit](2-space)
(2-space prismatic)
- A1xA1: [ ]x[ ] = (rectangle)
Rank 3
[edit](3-space)
- A3: [3,3] (tetrahedron)
- B3: [4,3] (cube/octahedron)
- H3: [5,3] (dodecahedron/icosahedron)
(3-space prismatic)
- I2(p)xA1: [p]x[ ] (p-gonal prism)
- A1xA1xA1: [ ]x[ ]x[ ] = (cube/cuboid)
Rank 4
[edit](4-space)
- A4: [3,3,3] (5-cell)
- B4: [4,3,3] (tesseract/16-cell)
- D4: [31,1,1] (24-cell)
- F4: [3,4,3] (24-cell)
- H4: [5,3,3] (120-cell/600-cell)
(4-space prismatic)
- A3xA1: [3,3]x[ ] (tetrahedral prism)
- B3xA1: [4,3]x[ ] (tesseract/orthoplex)
- H3xA1: [5,3]x[ ] (dodecahedral prism)
- I2(p)xA1xA1: [p]x[ ]x[ ] = (p-4 duoprism)
- A1xA1xA1xA1: [ ]x[ ]x[ ]x[ ] = (tesseract/orthoplex)
- I2(p)xI2(q) [p]x[q] (p-q duoprism)
Rank 5
[edit](5-space)
- A5: [3,3,3,3] (hexateron)
- B5: [4,3,3,3] (Penteract)
- D5: [32,1,1] (Demipenteract)
(5-space prismatic)
- A4xA1: [3,3,3]x[ ]
- B4xA1: [4,3,3]x[ ] -
- F4xA1: [3,4,3]x[ ] -
- H4xA1: [5,3,3]x[ ] -
- D4xA1: [31,1,1]x[ ] -
- A3xI2p: [3,3]x[p] -
- B3xI2p: [4,3]x[p] -
- H3xI2p: [5,3]x[p] -
- I2pxI2qxA1: [p]x[q]x[ ] -
Rank 6
[edit](6-space)
- A6: [3,3,3,3,3] (6-simplex)
- B6: [4,3,3,3,3] (6-orthoplex/6-hypercube)
- D6: [33,1,1] (6-orthoplex)
- E6: [32,2,1] (1_22 polytope)
Uniform prism
There are 6 categorical uniform prisms based the uniform 5-polytopes.
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A5×A1 | [3,3,3,3] × [ ] | |
2 | B5×A1 | [4,3,3,3] × [ ] | |
3 | D5×A1 | [32,1,1] × [ ] |
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
4 | A3×I2(p)×A1 | [3,3] × [p] × [ ] | |
5 | B3×I2(p)×A1 | [4,3] × [p] × [ ] | |
6 | H3×I2(p)×A1 | [5,3] × [p] × [ ] |
Uniform duoprism
There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower dimensional uniform polytopes. Five are formed as the product of a uniform polychoron with a regular polygon, and six are formed by the product of two uniform polyhedra:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A4×I2(p) | [3,3,3] × [p] | |
2 | B4×I2(p) | [4,3,3] × [p] | |
3 | F4×I2(p) | [3,4,3] × [p] | |
4 | H4×I2(p) | [5,3,3] × [p] | |
5 | D4×I2(p) | [31,1,1] × [p] |
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
6 | A3×A3 | [3,3] × [3,3] | |
7 | A3×B3 | [3,3] × [4,3] | |
8 | A3×H3 | [3,3] × [5,3] | |
9 | B3×B3 | [4,3] × [4,3] | |
10 | B3×H3 | [4,3] × [5,3] | |
11 | H3×A3 | [5,3] × [5,3] |
Uniform triprisms
There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope.
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | I2(p)×I2(q)×I2(r) | [p] × [q] × [r] |
Rank 7
[edit](7-space)
- A7: [3,3,3,3,3,3] (8-simplex)
- B7: [4,3,3,3,3,3] (8-orthoplex/8-hypercube)
- D7: [34,1,1] (8-orthoplex)
- E7: [33,2,1] (3_21 polytope)
There are 16 uniform prismatic families based on the uniform 6-polytopes.
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A6×A1 | [35] × [ ] | |
2 | B6×A1 | [4,34] × [ ] | |
3 | D6×A1 | [33,1,1] × [ ] | |
4 | E6×A1 | [32,2,1] × [ ] | |
5 | A4×I2(p)×A1 | [3,3,3] × [p] × [ ] | |
6 | B4×I2(p)×A1 | [4,3,3] × [p] × [ ] | |
7 | F4×I2(p)×A1 | [3,4,3] × [p] × [ ] | |
8 | H4×I2(p)×A1 | [5,3,3] × [p] × [ ] | |
9 | D4×I2(p)×A1 | [31,1,1] × [p] × [ ] | |
10 | A3×A3×A1 | [3,3] × [3,3] × [ ] | |
11 | A3×B3×A1 | [3,3] × [4,3] × [ ] | |
12 | A3×H3×A1 | [3,3] × [5,3] × [ ] | |
13 | B3×B3×A1 | [4,3] × [4,3] × [ ] | |
14 | B3×H3×A1 | [4,3] × [5,3] × [ ] | |
15 | H3×A3×A1 | [5,3] × [5,3] × [ ] | |
16 | I2(p)×I2(q)×I2(r)×A1 | [p] × [q] × [r] × [ ] |
There are 18 uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A5×I2(p) | [3,3,3] × [p] | |
2 | B5×I2(p) | [4,3,3] × [p] | |
3 | D5×I2(p) | [32,1,1] × [p] | |
4 | A4×A3 | [3,3,3] × [3,3] | |
5 | A4×B3 | [3,3,3] × [4,3] | |
6 | A4×H3 | [3,3,3] × [5,3] | |
7 | B4×A3 | [4,3,3] × [3,3] | |
8 | B4×B3 | [4,3,3] × [4,3] | |
9 | B4×H3 | [4,3,3] × [5,3] | |
10 | H4×A3 | [5,3,3] × [3,3] | |
11 | H4×B3 | [5,3,3] × [4,3] | |
12 | H4×H3 | [5,3,3] × [5,3] | |
13 | F4×A3 | [3,4,3] × [3,3] | |
14 | F4×B3 | [3,4,3] × [4,3] | |
15 | F4×H3 | [3,4,3] × [5,3] | |
16 | D4×A3 | [31,1,1] × [3,3] | |
17 | D4×B3 | [31,1,1] × [4,3] | |
18 | D4×H3 | [31,1,1] × [5,3] |
There are 3 uniform triaprismatic families based on Cartesian products of uniform polyhedrons and two regular polygons.
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A3×I2(p)×I2(q) | [3,3] × [p] × [q] | |
2 | B3×I2(p)×I2(q) | [4,3] × [p] × [q] | |
3 | H3×I2(p)×I2(q) | [5,3] × [p] × [q] |
Rank 8
[edit](8-space)
- A8: [3,3,3,3,3,3,3] (8-simplex)
- B8: [4,3,3,3,3,3,3] (8-orthoplex/8-hypercube)
- D8: [35,1,1] (8-orthoplex)
- E8: [34,2,1] (4_21 polytope)
Four are based on the uniform 7-polytopes:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A7×A1 | [3,3,3,3,3,3] × [ ] | |
2 | B7×A1 | [4,3,3,3,3,3] × [ ] | |
3 | D7×A1 | [34,1,1] × [ ] | |
4 | E7×A1 | [33,2,1] × [ ] |
Three are based on the uniform 6-polytopes and regular polygons:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
5 | A5×I2(p)×A1 | [3,3,3] × [p] × [ ] | |
6 | B5×I2(p)×A1 | [4,3,3] × [p] × [ ] | |
7 | D5×I2(p)×A1 | [32,1,1] × [p] × [ ] |
Fifteen are based on the product of the uniform polychora and uniform polyhedra:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
8 | A4×A3×A1 | [3,3,3] × [3,3] × [ ] | |
9 | A4×B3×A1 | [3,3,3] × [4,3] × [ ] | |
10 | A4×H3×A1 | [3,3,3] × [5,3] × [ ] | |
11 | B4×A3×A1 | [4,3,3] × [3,3] × [ ] | |
12 | B4×B3×A1 | [4,3,3] × [4,3] × [ ] | |
13 | B4×H3×A1 | [4,3,3] × [5,3] × [ ] | |
14 | H4×A3×A1 | [5,3,3] × [3,3] × [ ] | |
15 | H4×B3×A1 | [5,3,3] × [4,3] × [ ] | |
16 | H4×H3×A1 | [5,3,3] × [5,3] × [ ] | |
17 | F4×A3×A1 | [3,4,3] × [3,3] × [ ] | |
18 | F4×B3×A1 | [3,4,3] × [4,3] × [ ] | |
19 | F4×H3×A1 | [3,4,3] × [5,3] × [ ] | |
20 | D4×A3×A1 | [31,1,1] × [3,3] × [ ] | |
21 | D4×B3×A1 | [31,1,1] × [4,3] × [ ] | |
22 | D4×H3×A1 | [31,1,1] × [5,3] × [ ] |
Three are based on the uniform polyhedra and uniform duoprism:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
23 | A3×I2(p)×I2(q)×A1 | [3,3] × [p] × [q] × [ ] | |
24 | B3×I2(p)×I2(q)×A1 | [4,3] × [p] × [q] × [ ] | |
25 | H3×I2(p)×I2(q)×A1 | [5,3] × [p] × [q] × [ ] |
There are 28 categorical uniform duoprismatic forms based on Cartesian products of lower dimensional uniform polytopes.
There are 4 based on the uniform 6-polytopes and regular polygons:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A6×I2(p) | [3,3,3,3,3] × [p] | |
2 | B6×I2(p) | [4,3,3,3,3] × [p] | |
3 | D6×I2(p) | [33,1,1] × [p] | |
4 | E6×I2(p) | [32,2,1] × [p] |
There are 9 based on the uniform 5-polytopes and uniform polyhedra:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
5 | A5×A2 | [3,3,3,3] × [3,3] | |
6 | A5×B2 | [3,3,3,3] × [4,3] | |
7 | A5×H2 | [3,3,3,3] × [5,3] | |
8 | B5×A2 | [4,3,3,3] × [3,3] | |
9 | B5×B2 | [4,3,3,3] × [4,3] | |
10 | B5×H2 | [4,3,3,3] × [5,3] | |
11 | D5×A2 | [32,1,1] × [3,3] | |
12 | D5×B2 | [32,1,1] × [4,3] | |
13 | D5×H2 | [32,1,1] × [5,3] |
Finally there are 20 based on two uniform 4-polytopes:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
14 | A4xA4 | [3,3,3] × [3,3,3] | |
15 | A4xB4 | [3,3,3] × [4,3,3] | |
16 | A4xD4 | [3,3,3] × [31,1,1] | |
17 | A4xF4 | [3,3,3] × [3,4,3] | |
18 | A4xH4 | [3,3,3] × [5,3,3] | |
19 | B4xB4 | [4,3,3] × [4,3,3] | |
20 | B4xD4 | [4,3,3] × [31,1,1] | |
21 | B4xF4 | [4,3,3] × [3,4,3] | |
22 | B4xH4 | [4,3,3] × [5,3,3] | |
23 | D4xD4 | [31,1,1] × [31,1,1] | |
24 | D4xF4 | [31,1,1] × [3,4,3] | |
25 | D4xH4 | [31,1,1] × [5,3,3] | |
26 | F4xF4 | [3,4,3] × [3,4,3] | |
27 | F4xH4 | [3,4,3] × [5,3,3] | |
28 | H4xH4 | [5,3,3] × [5,3,3] |
There are 11 categorical uniform triaprismatic forms based on Cartesian products of lower dimensional uniform polytopes, for example these regular products:
Six are based on products of the uniform 4-polytopes and uniform duoprisms:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | A4×I2(p)×I2(q) | [3,3,3] × [p] × [q] | |
2 | B4×I2(p)×I2(q) | [4,3,3] × [p] × [q] | |
3 | F4×I2(p)×I2(q) | [3,4,3] × [p] × [q] | |
4 | H4×I2(p)×I2(q) | [5,3,3] × [p] × [q] | |
5 | D4×I2(p)×I2(q) | [31,1,1] × [p] × [q] | |
6 | A3×A3×I2(p) | [3,3] × [3,3] × [p] |
Five are based on triprism products of two uniform polyhedra and regular polygons:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
7 | A3×B3×I2(p) | [3,3] × [4,3] × [p] | |
8 | A3×H3×I2(p) | [3,3] × [5,3] × [p] | |
9 | B3×B3×I2(p) | [4,3] × [4,3] × [p] | |
10 | B3×H3×I2(p) | [4,3] × [5,3] × [p] | |
11 | H3×A3×I2(p) | [5,3] × [5,3] × [p] |
There is one infinite family of uniform quadriprismatic figures based on Cartesian products of four regular polygons:
# | Coxeter group | Coxeter-Dynkin diagram | |
---|---|---|---|
1 | I2(p) x I2(q) x I2(r) x I2(s) | [p] x [q] x [r] x [s] |
Rank 9
[edit](9-space)
- A9: [3,3,3,3,3,3,3,3] (9-simplex)
- B9: [4,3,3,3,3,3,3,3] (9-orthoplex/9-hypercube)
- D9: [36,1,1] (9-orthoplex)
Rank 10
[edit](10-space)
Euclidean compact
[edit]Rank 2
[edit](1-space)
- I~1: [∞] (apeirogon)
Rank 3
[edit](2-space)
- A~2: (3 3 3) (triangular tiling)
- B~2: [4,4] (square tiling)
- H~2: [6,3] (triangular tiling/hexagonal tiling)
Rank 4
[edit](3-space)
- A~3: (3 3 3 3) (quarter cubic honeycomb)
- B~2: [4,3,4] (cubic honeycomb)
- D~3: [4,31,1] (Alternated cubic honeycomb)
(2-space prismatic)
- I~1xI~1: [∞]x[∞] = (square tiling)
Rank 5
[edit](4-space)
- A~4: (3 3 3 3 3)
- B~4: [4,3,3,4] (Tesseractic honeycomb)
- C~4: [4,3,31,1] (Demitesseractic honeycomb)
- D~4: [31,1,1,1] (Demitesseractic honeycomb)
- F~4: [3,4,3,3] (Icositetrachoric honeycomb/Demitesseractic honeycomb)
(3-space prismatic)
Rank 6
[edit](5-space)
(4-space prismatic)
- I~1xI~1xI2r: [∞] x [∞] x [r] = [4,4]x[r] - =
- B~3xI~1: [4,3,4]x[∞]
- D~3xI~1: [4,31,1]x[∞]
- A~3xI~1:
- A~2xA~2: [Δ]x[Δ]
- A~2xB~2: [Δ]x[4,4]
- A~2xH~2: [Δ]x[6,3]
- B~2xB~2: [4,4]x[4,4]
- B~2xH~2: [4,4]x[6,3]
- H~2xH~2: [6,3]x[6,3]
- A4xI~1: [3,3,3]x[∞] (5-cell column)
- B4xI~1: [4,3,3]x[∞] (tesseract/16-cell column)
- H4xI~1: [5,3,3]x[∞] (120-cell/600-cell column)
- F4xI~1: [3,4,3]x[∞] (24-cell column)
- D4xI~1: [31,1,1]x[∞] (24-cell column)
(3-space prismatic)
Rank 7
[edit](6-space)
(6-space prismatic)
- A5xI~1: [3,3,3] x [∞]
- B5xI~1: [4,3,3] x [∞]
- D5xI~1: [32,1,1] x [∞]
- A4xI~1xA1: [3,3,3] x [∞] x [ ]
- B4xI~1xA1: [4,3,3] x [∞] x [ ]
- F4xI~1xA1: [3,4,3] x [∞] x [ ]
- H4xI~1xA1: [5,3,3] x [∞] x [ ]
- D4xI~1xA1: [31,1,1] x [∞] x [ ]
- A3xI~1xI2q: [3,3] x [∞] x [q]
- B3xI~1xI2q: [4,3] x [∞] x [q]
- H3xI~1xI2q: [5,3] x [∞] x [q]
- I~1xI2qxI2rA1: [∞] x [q] x [r] x [ ]
(5-space prismatic)
- A3xI~1xI~1: [3,3] x [∞] x [∞]
- B3xI~1xI~1: [4,3] x [∞] x [∞]
- H3xI~1xI~1: [5,3] x [∞] x [∞]
- I~1xI~1xI2rxA1: [∞] x [∞] x [r] x [ ]
(4-space prismatic)
Rank 8
[edit](7-space)
Rank 9
[edit](8-space)
- A~9: (3 3 3 3 3 3 3 3)
- B~8: [4,3,3,3,3,3,3,4]
- C~8: [4,34,31,1]
- D~8: [31,1,34,31,1]
- E~8: [35,2,1] (E8 lattice)
Rank 10
[edit](9-space)
Euclidean noncompact
[edit]Rank 3
[edit](2-space)
- I~1xA1: [∞]x[ ] (apeirogonal prism)
Rank 4
[edit](3-space)
- B~2xA1: [4,4]x[ ] (cubic prismatic slab)
- H~2xA1: [6,3]x[ ] (triangular/hexagonal prismatic slab)
- A~2xA1: (3 3 3 3)x[ ] (triangular prismatic slab)
- I~1xA1xA1: [∞]x[ ]x[ ] = (4-∞ semi-infinite duoprism)
- I2(p)xI~1: [p]x[∞] (p-∞ semiinfinite duoprism)
Rank 5
[edit](4-space)
- I2pxI~1xA1: [p]x[∞]x[ ] -
- D~3xA1: [4,31,1]x[ ]
- A~3xA1: (3 3 3 3)x[ ]
- A~2xI2p: (3 3 3)x[p]
- B~2xI2p: [4,4]x[p]
- H~2xI2p: [6,3]x[p]
(3-space)
- I~1xI~1xA1: [∞]x[∞]x[ ] -
- B~2xI2p: [4,4]x[∞]
- H~2xI2p: [6,3]x[∞]
- A~2xI2pxA1: [Δ]x[∞]x[ ]
- B~2xA1: [4,3,4]x[ ]
Rank 6
[edit](5-space)
- A4xI~1: [3,3,3]x[∞] -
- B4xI~1: [4,3,3]x[∞] -
- F4xI~1: [3,4,3]x[∞] -
- H4xI~1: [5,3,3]x[∞] -
- D4xI~1: [31,1,1] x [∞] -
- A3xI~1xA1: [3,3] x [∞] x [ ] -
- B3xI~1xA1: [4,3] x [∞] x [ ] -
- H3xI~1xA1: [5,3] x [∞] x [ ] -
- I~1xI2qxI2r: [∞] x [q] x [r] -
- A~4xA1: (3 3 3 3)x[ ]
- B~4xA1: [4,3,3,4]x[ ]
- C~4xA1: [4,3,31,1]x[ ]
- D~4xA1: [31,1,1,1]x[ ]
- F~4xA1: [3,4,3,3]x[ ]
Rank 7
[edit]Rank 8
[edit]Rank 9
[edit]Rank 10
[edit]Hyperbolic compact
[edit]Rank 3
[edit](2-space)
Rank 4
[edit](3-space)
- [3,5,3] (Order-3 icosahedral honeycomb)
- [5,3,4] (Order-5 cubic honeycomb/Order-4 dodecahedral honeycomb)
- [5,3,5] (Order-5 dodecahedral honeycomb)
- [5,31,1]
- (4 3 3 3)
- (5 3 3 3)
- (4 3 3 3)
- (4 3 5 3)
- (5 3 5 3)
(2-space)
Rank 5
[edit](4-space)
- [5,3,3,3] (Order-5 pentachoronic tetracomb/Order-3 hecatonicosachoronic tetracomb)
- [5,3,3,4] (Order-5 tesseractic tetracomb/Order-4 hecatonicosachoronic tetracomb)
- [5,3,3,5] (Order-5 hecatonicosachoronic tetracomb)
- [5,3,31,1] (Order-5 demitesseractic tetracomb)
- (4 3 3 3 3)
Rank 6
[edit]Noncompact!
(5-space)
Rank 7
[edit]Rank 8
[edit]Rank 9
[edit]Rank 10
[edit]Hyperbolic noncompact
[edit]Rank 3
[edit](2-space)
Rank 4
[edit](3-space)
Rank 5
[edit](4-space)
Rank 6
[edit]Rank 7
[edit]Rank 8
[edit]Rank 9
[edit]Rank 10
[edit]Finite Coexter groups
[edit]Group symbol |
Alternate symbol |
Rank | Order | Related polytopes | Coxeter-Dynkin diagram |
---|---|---|---|---|---|
An | An | n | (n + 1)! | n-simplex | ... |
Bn = Cn | Cn | n | 2n n! | n-hypercube / n-cross-polytope | ... |
Dn | Bn | n | 2n−1 n! | demihypercube | ... |
I2(p) | D2p | 2 | 2p | p-gon | |
H3 | G3 | 3 | 120 | icosahedron / dodecahedron | |
F4 | F4 | 4 | 1152 | 24-cell | |
H4 | G4 | 4 | 14400 | 120-cell / 600-cell | |
E6 | E6 | 6 | 51840 | 122 polytope | |
E7 | E7 | 7 | 2903040 | 321 polytope | |
E8 | E8 | 8 | 696729600 | E8 polytope |
Finite Coxeter groups
[edit]Families of convex uniform polytopes are defined by Coxeter groups.
Note: (Alternate names as Simple Lie groups also given)
- An forms the simplex polytope family. (Same An)
- Bn is the family of demihypercubes, beginning at n=4 with the 16-cell, and n=5 with the demipenteract. (Also named Dn)
- Cn forms the hypercube polytope family. (Same Cn)
- 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)
Affine Coxeter groups (Euclidean)
[edit]Group symbol |
Alternate symbol |
Related uniform tessellation(s) | Coxeter-Dynkin diagram |
---|---|---|---|
A~n-1 | Pn | Simplex-rectified-simplex honeycomb A~2:Triangular tiling A~3:Tetrahedral-octahedral honeycomb |
... |
B~n-1 | Rn | Hypercubic honeycomb | ... |
C~n-1 | Sn | Demihypercubic honeycomb | ... |
D~n-1 | Qn | Demihypercubic honeycomb | ... |
I~1 | W2 | apeirogon | |
H~2 | G3 | Hexagonal tiling and Triangular tiling |
|
F~4 | V5 | Hexadecachoric tetracomb and Icositetrachoronic tetracomb or F4 lattice |
|
E~6 | T7 | E6 lattice | |
E~7 | T8 | E7 lattice | |
E~8 | T9 | E8 lattice |
n | A~2+ (3n) |
D~4+ [31,1,3n-5,31,1] |
B~2+ [4,3n-3,4] |
C~3+ [4,3n-4,31,1] |
E~6-8 [3a,b,c] |
F~4 [3,4,3,3] |
H~2 [6,3] |
I~1 |
---|---|---|---|---|---|---|---|---|
2 | I~1=[∞] | |||||||
3 | A~2=h[6,3] | B~2=[4,4] | F~2=[6,3] | |||||
4 | A~3=q[4,3,4] | B~3=[4,3,4] | C~3=h[4,3,4] | |||||
5 | A~4 | D~4=q[4,32,4] | B~4=[4,32,4] | C~4=h[4,32,4] | U5=[3,4,3,3] | |||
6 | A~5 | D~5=q[4,33,4] | B~5=[4,33,4] | C~5=h[4,33,4] | ||||
7 | A~6 | D~6=q[4,34,4] | B~6=[4,34,4] | C~6=h[4,34,4] | E~6=[32,2,2] | |||
8 | A~7 | D~7=q[4,35,4] | B~7=[4,35,4] | C~7=h[4,35,4] | E~7=[33,3,1] | |||
9 | A~8 | D~8=q[4,36,4] | B~8=[4,36,4] | C~8=h[4,36,4] | E~8=[35,2,1] | |||
10 | A~9 | D~9=q[4,37,4] | B~9=[4,37,4] | C~9=h[4,37,4] | ||||
11 | ... | ... | ... | ... |
See also
[edit]References
[edit]- THE GROWTH SERIES OF COMPACT HYPERBOLIC COXETER GROUPS WITH 4 AND 5 GENERATORS
- Spacelike Singularities and Hidden Symmetries of Gravity: 3 Hyperbolic Coxeter Groups
- Reflection Groups and Coxeter groups (Cambridge studies in advanced mathematics 29) , James E. Humphreys, Cambridge University press, 1990 ISBN 0521436133 (paperback), ISBN 052137510X (hardcopy)
- 2.4 Some positive definite graphs, 2.5 Some positive semidefinite graphs, 6.8 Coxeter Hyperbolic groups