 8-simplex   
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 Pentellated 8-simplex
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 Bipentitruncated 8-simplex
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| Orthogonal projections in A8 Coxeter plane | ||
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In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.
There are two unique pentellations of the 8-simplex. Including truncations, cantellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A8 symmetry. A8, [37] has order 9 factorial symmetry, or 362880. The bipentalled form is symmetrically ringed, doubling the symmetry order to 725760, and is represented the double-bracketed group [[37]]. The A8 Coxeter plane projection shows order [9] symmetry for the pentellated 8-simplex, while the bipentellated 8-simple is doubled to [18] symmetry.
Pentellated 8-simplex edit
| Pentellated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t0,5{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
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| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5040 |
| Vertices | 504 |
| Vertex figure | |
| Coxeter group | A8, [37], order 362880 |
| Properties | convex |
Coordinates edit
The Cartesian coordinates of the vertices of the pentellated 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 pentellated 9-orthoplex.
Images edit
| Ak Coxeter plane | A8 | A7 | A6 | A5 |
|---|---|---|---|---|
| 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] |
Bipentellated 8-simplex edit
| Bipentellated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t1,6{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |
|
| 7-faces | t0,5{3,3,3,3,3,3} |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7560 |
| Vertices | 756 |
| Vertex figure | |
| Coxeter group | A8×2, [[37]], order 725760 |
| Properties | convex, facet-transitive |
Coordinates edit
The Cartesian coordinates of the vertices of the bipentellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,1,1,1,1,1,2,2). This construction is based on facets of the bipentellated 9-orthoplex.
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] |
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)". x3o3o3o3o3x3o3o, o3x3o3o3o3o3x3o