8-simplex |
Heptellated 8-simplex |
Heptihexipentisteriruncicantitruncated 8-simplex (Omnitruncated 8-simplex) |
Orthogonal projections in A_{8} Coxeter plane (A_{7} 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 | |
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Type | uniform 8-polytope |
Schläfli symbol | t_{0,7}{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 | A_{8}×2, [[3^{7}]], order 725760 |
Properties | convex |
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:
Its 72 vertices represent the root vectors of the simple Lie group A_{8}.
A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
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Graph | ||||
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} | |
Graph | ||||
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Omnitruncated 8-simplex | |
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Type | uniform 8-polytope |
Schläfli symbol | t_{0,1,2,3,4,5,6,7}{3^{7}} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1451520 |
Vertices | 362880 |
Vertex figure | irr. 7-simplex |
Coxeter group | A_{8}, [[3^{7}]], 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.
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, t_{0,1,2,3,4,5,6,7}{3^{7},4}
A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} | |
Graph | ||||
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
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 .
This polytope is one of 135 uniform 8-polytopes with A_{8} symmetry.