Heptellated 8-simplexes
![]() 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Heptellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Heptihexipentisteriruncicantitruncated 8-simplex (Omnitruncated 8-simplex) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Orthogonal projections in A8 Coxeter plane (A7 for omnitruncation) |
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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
[edit]Heptellated 8-simplex | |
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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
[edit]- Expanded 8-simplex
- Small exated enneazetton (soxeb) (Jonathan Bowers)[1]
Coordinates
[edit]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
[edit]Its 72 vertices represent the root vectors of the simple Lie group A8.
Images
[edit]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
[edit]Omnitruncated 8-simplex | |
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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
[edit]- Heptihexipentisteriruncicantitruncated 8-simplex
- Great exated enneazetton (goxeb) (Jonathan Bowers)[2]
Coordinates
[edit]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
[edit]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
[edit]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
[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)". x3o3o3o3o3o3o3x - soxeb, x3x3x3x3x3x3x3x - goxeb