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 | ||
|---|---|---|
| Type | uniform 8-polytope | |
| Schläfli symbol | t0,7 | |
| 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 | |
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)
Its 72 vertices represent the root vectors of the simple Lie group A8.
| Omnitruncated 8-simplex | ||
|---|---|---|
| Type | uniform 8-polytope | |
| Schläfli symbol | t0,1,2,3,4,5,6,7 | |
| Coxeter-Dynkin diagrams | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 1451520 | |
| Vertices | 362880 | |
| Vertex figure | irr. 7-simplex | |
| Coxeter group | A8, [[37]], order 725760 | |
| Properties | convex | |
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
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 A8 symmetry.