![]() 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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In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.
There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicanti
Runcinated 8-simplex
[edit]Runcinated 8-simplex | |
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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
[edit]- Runcinated enneazetton
- Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)[1]
Coordinates
[edit]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
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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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] |
Biruncinated 8-simplex
[edit]Biruncinated 8-simplex | |
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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 |
Alternate names
[edit]- Biruncinated enneazetton
- Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)[2]
Coordinates
[edit]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
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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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] |
Triruncinated 8-simplex
[edit]Triruncinated 8-simplex | |
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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 |
Alternate names
[edit]- Triruncinated enneazetton
- Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)[3]
Coordinates
[edit]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
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Runcitruncated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Biruncitruncated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Triruncitruncated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Runcicantellated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Biruncicantellated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Runcicantitruncated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Biruncicantitruncated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Triruncicantitruncated 8-simplex
[edit]
Images
[edit]Ak Coxeter plane | A8 | A7 | A6 | A5 |
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Graph | ![]() |
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Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
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Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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)". x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb