Cantellated 8-simplexes
Cantellated 8-simplex |
Bicantellated 8-simplex |
Tricantellated 8-simplex | |
Cantitruncated 8-simplex |
Bicantitruncated 8-simplex |
Tricantitruncated 8-simplex | |
Orthogonal projections in A8 Coxeter plane |
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In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.
There are six unique cantellations for the 8-simplex, including permutations of truncation.
Cantellated 8-simplex
[edit]Cantellated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | rr{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1764 |
Vertices | 252 |
Vertex figure | 6-simplex prism |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
[edit]- Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)[1]
Coordinates
[edit]The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex.
Images
[edit]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] |
Bicantellated 8-simplex
[edit]Bicantellated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | r2r{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 5292 |
Vertices | 756 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
[edit]- Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)[2]
Coordinates
[edit]The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex.
Images
[edit]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] |
Tricantellated 8-simplex
[edit]tricantellated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | r3r{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 8820 |
Vertices | 1260 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
[edit]- Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)[3]
Coordinates
[edit]The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the tricantellated 9-orthoplex.
Images
[edit]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] |
Cantitruncated 8-simplex
[edit]Cantitruncated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | tr{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
[edit]- Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)[4]
Coordinates
[edit]The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 9-orthoplex.
Images
[edit]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] |
Bicantitruncated 8-simplex
[edit]Bicantitruncated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t2r{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
[edit]- Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)[5]
Coordinates
[edit]The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.
Images
[edit]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] |
Tricantitruncated 8-simplex
[edit]Tricantitruncated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t3r{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
- Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)[6]
Coordinates
[edit]The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.
Images
[edit]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] |
Related polytopes
[edit]This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
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. "8D uniform polytopes (polyzetta)". x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene