Runcinated_8-simplex
In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.
More information Orthogonal projections in A8 Coxeter plane ...
 8-simplex    |
 Runcinated 8-simplex |
 Biruncinated 8-simplex |
 Triruncinated 8-simplex |
 Runcitruncated 8-simplex |
 Biruncitruncated 8-simplex |
 Triruncitruncated 8-simplex |
 Runcicantellated 8-simplex |
 Biruncicantellated 8-simplex |
 Runcicantitruncated 8-simplex |
 Biruncicantitruncated 8-simplex |
 Triruncicantitruncated 8-simplex |
| Orthogonal projections in A8 Coxeter plane | |||
|---|---|---|---|
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There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicanti
| Runcinated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t0,3{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4536 |
| Vertices | 504 |
| Vertex figure | |
| Coxeter group | A8, [37], order 362880 |
| Properties | convex |
Alternate names
- Runcinated enneazetton
- Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)[1]
Coordinates
The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex.
Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | |
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 |
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| Dihedral symmetry | [9] | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph |  |
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 | |
| Dihedral symmetry | [5] | [4] | [3] |
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More information Biruncinated 8-simplex ...
| Biruncinated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t1,4{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11340 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A8, [37], order 362880 |
| Properties | convex |
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Alternate names
- Biruncinated enneazetton
- Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)[2]
Coordinates
The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 9-orthoplex.
Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [5] | [4] | [3] |
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More information Triruncinated 8-simplex ...
| Triruncinated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t2,5{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A8×2, [[37]], order 725760 |
| Properties | convex |
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Alternate names
- Triruncinated enneazetton
- Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)[3]
Coordinates
The Cartesian coordinates of the vertices of the triruncinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,2,2,2). This construction is based on facets of the triruncinated 9-orthoplex.
Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Close
Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Close
Images
More information Ak Coxeter plane, A8 ...
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| Graph | ||||
| Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 | |
| Graph | ||||
| Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
More information A8 polytopes ...
| A8 polytopes | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
t0 |
 t1 |
 t2 |
 t3 |
 t01 |
 t02 |
 t12 |
t03 |
 t13 |
 t23 |
 t04 |
t14 |
 t24 |
 t34 |
 t05 |
 t15 |
t25 |
 t06 |
 t16 |
 t07 |
 t012 |
t013 |
t023 |
 t123 |
 t014 |
 t024 |
t124 |
 t034 |
t134 |
 t234 |
 t015 |
 t025 |
 t125 |
 t035 |
 t135 |
t235 |
 t045 |
 t145 |
 t016 |
 t026 |
 t126 |
 t036 |
 t136 |
 t046 |
 t056 |
 t017 |
 t027 |
 t037 |
t0123 |
 t0124 |
 t0134 |
 t0234 |
t1234 |
 t0125 |
 t0135 |
 t0235 |
 t1235 |
 t0145 |
 t0245 |
 t1245 |
 t0345 |
 t1345 |
t2345 |
 t0126 |
 t0136 |
 t0236 |
 t1236 |
 t0146 |
 t0246 |
 t1246 |
 t0346 |
 t1346 |
 t0156 |
 t0256 |
 t1256 |
 t0356 |
 t0456 |
 t0127 |
 t0137 |
 t0237 |
 t0147 |
 t0247 |
 t0347 |
 t0157 |
 t0257 |
 t0167 |
 t01234 |
 t01235 |
 t01245 |
 t01345 |
 t02345 |
 t12345 |
 t01236 |
 t01246 |
 t01346 |
 t02346 |
 t12346 |
 t01256 |
 t01356 |
 t02356 |
 t12356 |
 t01456 |
 t02456 |
 t03456 |
 t01237 |
 t01247 |
 t01347 |
 t02347 |
 t01257 |
 t01357 |
 t02357 |
 t01457 |
 t01267 |
 t01367 |
 t012345 |
 t012346 |
 t012356 |
 t012456 |
 t013456 |
 t023456 |
 t123456 |
 t012347 |
 t012357 |
 t012457 |
 t013457 |
 t023457 |
 t012367 |
 t012467 |
 t013467 |
 t012567 |
 t0123456 |
 t0123457 |
 t0123467 |
 t0123567 |
 t01234567 |
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- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- 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
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Klitzing, Richard. "8D uniform polytopes (polyzetta)". x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb
More information Family, Regular polygon ...
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds | ||||||||||||
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