![]() 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[edit]
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[edit]
- Small rhombated octaexon (acronym: saro) (Jonathan Bowers)[1]
Coordinates[edit]
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[edit]
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Bicantellated 7-simplex[edit]
Bicantellated 7-simplex | |
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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[edit]
- Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)[2]
Coordinates[edit]
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[edit]
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Tricantellated 7-simplex[edit]
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[edit]
- Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)[3]
Coordinates[edit]
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[edit]
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Cantitruncated 7-simplex[edit]
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[edit]
- Great rhombated octaexon (acronym: garo) (Jonathan Bowers)[4]
Coordinates[edit]
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[edit]
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
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Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Bicantitruncated 7-simplex[edit]
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[edit]
- Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)[5]
Coordinates[edit]
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[edit]
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [5] | [4] | [3] |
Tricantitruncated 7-simplex[edit]
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[edit]
- Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)[6]
Coordinates[edit]
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[edit]
Ak Coxeter plane | A7 | A6 | A5 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [[7]] | [6] |
Ak Coxeter plane | A4 | A3 | A2 |
Graph | ![]() |
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Dihedral symmetry | [[5]] | [4] | [[3]] |
Related polytopes[edit]
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
See also[edit]
Notes[edit]
References[edit]
- 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 [1]
- (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