| 8-simplex | Heptellated 8-simplex | Heptihexipentisteriruncicantitruncated 8-simplex(Omnitruncated 8-simplex) |
| Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation) | ||
|---|---|---|
In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.
There are 35 unique heptellations for the 8-simplex, including all permutations of truncations, cantellations, runcinations, sterications, pentellations, and hexications. The simplest heptellated 8-simplex is also called an expanded 8-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 8-simplex. The highest form, the heptihexipentisteriruncicantitruncated 8-simplex is more simply called a omnitruncated 8-simplex with all of the nodes ringed.
Heptellated 8-simplex
| Heptellated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t0,7{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 504 |
| Vertices | 72 |
| Vertex figure | 6-simplex antiprism |
| Coxeter group | A8×2, [[37]], order 725760 |
| Properties | convex |
Alternate names
- Expanded 8-simplex
- Small exated enneazetton (soxeb) (Jonathan Bowers)1
Coordinates
The vertices of the heptellated 8-simplex can bepositioned in 8-space as permutations of (0,1,1,1,1,1,1,1,2). This construction is based on facets of the heptellated 9-orthoplex.
A second construction in 9-space, from the center of a rectified 9-orthoplex is given by coordinate permutations of:
(1,-1,0,0,0,0,0,0,0)Root vectors
Its 72 vertices represent the root vectors of the simple Lie group A8.
Images
orthographic projections| 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] |
Omnitruncated 8-simplex
| Omnitruncated 8-simplex | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t0,1,2,3,4,5,6,7{37} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1451520 |
| Vertices | 362880 |
| Vertex figure | irr. 7-simplex |
| Coxeter group | A8, [[37]], order 725760 |
| Properties | convex |
The symmetry order of an omnitruncated 8-simplex is 725760. The symmetry of a family of a uniform polytopes is equal to the number of vertices of the omnitruncation, being 362880 (9 factorial) in the case of the omnitruncated 8-simplex; but when the CD symbol is palindromic, the symmetry order is doubled, 725760 here, because the element corresponding to any element of the underlying 8-simplex can be exchanged with one of those corresponding to an element of its dual.
Alternate names
- Heptihexipentisteriruncicantitruncated 8-simplex
- Great exated enneazetton (goxeb) (Jonathan Bowers)2
Coordinates
The Cartesian coordinates of the vertices of the omnitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,1,2,3,4,5,6,7,8). This construction is based on facets of the heptihexipentisteriruncicantitruncated 9-orthoplex, t0,1,2,3,4,5,6,7{37,4}
Images
orthographic projections| 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] |
Permutohedron and related tessellation
The omnitruncated 8-simplex is the permutohedron of order 9. The omnitruncated 8-simplex is a zonotope, the Minkowski sum of nine line segments parallel to the nine lines through the origin and the nine vertices of the 8-simplex.
Like all uniform omnitruncated n-simplices, the omnitruncated 8-simplex can tessellate space by itself, in this case 8-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .
Related polytopes
The two presented polytopes are selected from 135 uniform 8-polytopes with A8 symmetry, shown in the table below.
| A8 polytopes | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| t0 | t1 | t2 | t3 | t01 | t02 | t12 | t03 | t13 | t23 | t04 | t14 | t24 | t34 | t05 |
| t15 | t25 | t06 | t16 | t07 | t012 | t013 | t023 | t123 | t014 | t024 | t124 | t034 | t134 | t234 |
| t015 | t025 | t125 | t035 | t135 | t235 | t045 | t145 | t016 | t026 | t126 | t036 | t136 | t046 | t056 |
| t017 | t027 | t037 | t0123 | t0124 | t0134 | t0234 | t1234 | t0125 | t0135 | t0235 | t1235 | t0145 | t0245 | t1245 |
| t0345 | t1345 | t2345 | t0126 | t0136 | t0236 | t1236 | t0146 | t0246 | t1246 | t0346 | t1346 | t0156 | t0256 | t1256 |
| t0356 | t0456 | t0127 | t0137 | t0237 | t0147 | t0247 | t0347 | t0157 | t0257 | t0167 | t01234 | t01235 | t01245 | t01345 |
| t02345 | t12345 | t01236 | t01246 | t01346 | t02346 | t12346 | t01256 | t01356 | t02356 | t12356 | t01456 | t02456 | t03456 | t01237 |
| t01247 | t01347 | t02347 | t01257 | t01357 | t02357 | t01457 | t01267 | t01367 | t012345 | t012346 | t012356 | t012456 | t013456 | t023456 |
| t123456 | t012347 | t012357 | t012457 | t013457 | t023457 | t012367 | t012467 | t013467 | t012567 | t0123456 | t0123457 | t0123467 | t0123567 | t01234567 |
Notes
- 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)". x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb