Cantellated 7-simplexes
7-simplex |
Cantellated 7-simplex |
Bicantellated 7-simplex |
Tricantellated 7-simplex |
Birectified 7-simplex |
Cantitruncated 7-simplex |
Bicantitruncated 7-simplex |
Tricantitruncated 7-simplex |
Orthogonal projections in A7 Coxeter plane |
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In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.
There are unique 6 degrees of cantellation for the 7-simplex, including truncations.
Cantellated 7-simplex
Cantellated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | rr{3,3,3,3,3,3} or |
Coxeter-Dynkin diagram | or |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1008 |
Vertices | 168 |
Vertex figure | 5-simplex prism |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex |
Alternate names
- Small rhombated octaexon (acronym: saro) (Jonathan Bowers)[1]
Coordinates
The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Bicantellated 7-simplex
Bicantellated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | r2r{3,3,3,3,3,3} or |
Coxeter-Dynkin diagrams | or |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 2520 |
Vertices | 420 |
Vertex figure | |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex |
Alternate names
- Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)[2]
Coordinates
The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Tricantellated 7-simplex
Tricantellated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | r3r{3,3,3,3,3,3} or |
Coxeter-Dynkin diagrams | or |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 3360 |
Vertices | 560 |
Vertex figure | |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex |
Alternate names
- Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)[3]
Coordinates
The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Cantitruncated 7-simplex
Cantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | tr{3,3,3,3,3,3} or |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1176 |
Vertices | 336 |
Vertex figure | |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex |
Alternate names
- Great rhombated octaexon (acronym: garo) (Jonathan Bowers)[4]
Coordinates
The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Bicantitruncated 7-simplex
Bicantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t2r{3,3,3,3,3,3} or |
Coxeter-Dynkin diagrams | or |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 2940 |
Vertices | 840 |
Vertex figure | |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex |
Alternate names
- Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)[5]
Coordinates
The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Tricantitruncated 7-simplex
Tricantitruncated 7-simplex | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t3r{3,3,3,3,3,3} or |
Coxeter-Dynkin diagrams | or |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 3920 |
Vertices | 1120 |
Vertex figure | |
Coxeter groups | A7, [3,3,3,3,3,3] |
Properties | convex |
Alternate names
- Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)[6]
Coordinates
The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.
Images
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Related polytopes
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
See also
Notes
References
- 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. "7D uniform polytopes (polyexa)". x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / E9 / E10 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 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 |