![]() 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bistericated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() Steri-truncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bisteri-truncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steri-cantellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bisteri-cantellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Stericanti-truncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bistericanti-truncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steri-runcinated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bisteri-runcinated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steriruncitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bisterirun-citruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Sterirunci-cantellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bisterirunci-cantellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Steriruncicanti-truncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Bisteriruncicanti-truncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Orthogonal projections in A8 Coxeter plane |
---|
In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.
Stericated 8-simplex
[edit]Stericated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t0,4{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 6300 |
Vertices | 630 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Coordinates
[edit]The Cartesian coordinates of the vertices of the stericated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 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] |
Bistericated 8-simplex
[edit]bistericated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t1,5{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 12600 |
Vertices | 1260 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Coordinates
[edit]The Cartesian coordinates of the vertices of the bistericated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 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] |
Steritruncated 8-simplex
[edit]Steritruncated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t0,1,4{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
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] |
Bisteritruncated 8-simplex
[edit]Bisteritruncated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t1,2,5{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
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] |
Stericantellated 8-simplex
[edit]
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] |
Bistericantellated 8-simplex
[edit]
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] |
Stericantitruncated 8-simplex
[edit]
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] |
Bistericantitruncated 8-simplex
[edit]
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] |
Steriruncinated 8-simplex
[edit]
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] |
Bisteriruncinated 8-simplex
[edit]
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] |
Steriruncitruncated 8-simplex
[edit]
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] |
Bisteriruncitruncated 8-simplex
[edit]
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] |
Steriruncicantellated 8-simplex
[edit]
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] |
Bisteriruncicantellated 8-simplex
[edit]
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] |
Steriruncicantitruncated 8-simplex
[edit]
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] |
Bisteriruncicantitruncated 8-simplex
[edit]
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)". x3o3o3o3x3o3o3o, o3x3o3o3o3x3o3o