# Heptellated 8-simplexes

| 8-simplex | Heptellated 8-simplex | Heptihexipentisteriruncicantitruncated 8-simplex (Omnitruncated 8-simplex) |
| Orthogonal projections in A$_8$ Coxeter plane (A$_7$ 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 | 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 |

### Alternate names

* Expanded 8-simplex
* Small exated enneazetton (soxeb) (Jonathan Bowers)

### 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 A$_8$.

### Images

orthographic projections :
| 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] |

## Omnitruncated 8-simplex

Omnitruncated 8-simplex :
| 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.

### Alternate names

* Heptihexipentisteriruncicantitruncated 8-simplex
* Great exated enneazetton (goxeb) (Jonathan Bowers)

### 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, t$_{0,1,2,3,4,5,6,7}${3$^7$,4}

### Images

orthographic projections :
| 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] |

### 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

This polytope is one of 135 uniform 8-polytopes with A$_8$ symmetry.
A8 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_3$ | t$_{01}$ | t$_{02}$ | t$_{12}$ | t$_{03}$ | t$_{13}$ | t$_{23}$ | t$_{04}$ | t$_{14}$ | t$_{24}$ | t$_{34}$ | t$_{05}$ |
| t$_{15}$ | t$_{25}$ | t$_{06}$ | t$_{16}$ | t$_{07}$ | t$_{012}$ | t$_{013}$ | t$_{023}$ | t$_{123}$ | t$_{014}$ | t$_{024}$ | t$_{124}$ | t$_{034}$ | t$_{134}$ | t$_{234}$ |
| t$_{015}$ | t$_{025}$ | t$_{125}$ | t$_{035}$ | t$_{135}$ | t$_{235}$ | t$_{045}$ | t$_{145}$ | t$_{016}$ | t$_{026}$ | t$_{126}$ | t$_{036}$ | t$_{136}$ | t$_{046}$ | t$_{056}$ |
| t$_{017}$ | t$_{027}$ | t$_{037}$ | t$_{0123}$ | t$_{0124}$ | t$_{0134}$ | t$_{0234}$ | t$_{1234}$ | t$_{0125}$ | t$_{0135}$ | t$_{0235}$ | t$_{1235}$ | t$_{0145}$ | t$_{0245}$ | t$_{1245}$ |
| t$_{0345}$ | t$_{1345}$ | t$_{2345}$ | t$_{0126}$ | t$_{0136}$ | t$_{0236}$ | t$_{1236}$ | t$_{0146}$ | t$_{0246}$ | t$_{1246}$ | t$_{0346}$ | t$_{1346}$ | t$_{0156}$ | t$_{0256}$ | t$_{1256}$ |
| t$_{0356}$ | t$_{0456}$ | t$_{0127}$ | t$_{0137}$ | t$_{0237}$ | t$_{0147}$ | t$_{0247}$ | t$_{0347}$ | t$_{0157}$ | t$_{0257}$ | t$_{0167}$ | t$_{01234}$ | t$_{01235}$ | t$_{01245}$ | t$_{01345}$ |
| t$_{02345}$ | t$_{12345}$ | t$_{01236}$ | t$_{01246}$ | t$_{01346}$ | t$_{02346}$ | t$_{12346}$ | t$_{01256}$ | t$_{01356}$ | t$_{02356}$ | t$_{12356}$ | t$_{01456}$ | t$_{02456}$ | t$_{03456}$ | t$_{01237}$ |
| t$_{01247}$ | t$_{01347}$ | t$_{02347}$ | t$_{01257}$ | t$_{01357}$ | t$_{02357}$ | t$_{01457}$ | t$_{01267}$ | t$_{01367}$ | t$_{012345}$ | t$_{012346}$ | t$_{012356}$ | t$_{012456}$ | t$_{013456}$ | t$_{023456}$ |
| t$_{123456}$ | t$_{012347}$ | t$_{012357}$ | t$_{012457}$ | t$_{013457}$ | t$_{023457}$ | t$_{012367}$ | t$_{012467}$ | t$_{013467}$ | t$_{012567}$ | t$_{0123456}$ | t$_{0123457}$ | t$_{0123467}$ | t$_{0123567}$ | t$_{01234567}$ |