Pentellated 8-simplexes
8-simplex |
Pentellated 8-simplex |
Bipentitruncated 8-simplex |
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, canetellations, runcinations, and sterications, there are 32 more pentellations. These polytopes are a part of a family 135 uniform 8-polytopes with A8 symmetry. A8, [38] 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 [[38]]. 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
Pentellated 8-simplex | |
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Type | uniform 8-polytope |
Schläfli symbol | t0,5{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
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
The Cartesian coordinates of the vertices of the pentellated 8-simplex can be most simply positioned in 8-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
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] |
Bipentellated 8-simplex
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
The Cartesian coordinates of the vertices of the bipentellated 8-simplex can be most simply positioned in 8-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
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ||||
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Related polytopes
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
Notes
References
- 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
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary