# Truncated 8-simplexes

| 8-simplex | Truncated 8-simplex | Rectified 8-simplex |
| Quadritruncated 8-simplex | Tritruncated 8-simplex | Bitruncated 8-simplex |
| Orthogonal projections in A$_8$ Coxeter plane |||
In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.
There are four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated 8-simplex are located on the triangular faces of the 8-simplex. Vertices of the tritruncated 8-simplex are located inside the tetrahedral cells of the 8-simplex.

## Truncated 8-simplex

Truncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t{3$^7$} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 288 |
| Vertices | 72 |
| Vertex figure | ( )v{3,3,3,3,3} |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Alternate names

* Truncated enneazetton (Acronym: tene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the truncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 9-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_8$ | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Bitruncated 8-simplex

Bitruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | 2t{3$^7$} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1008 |
| Vertices | 252 |
| Vertex figure | { }v{3,3,3,3} |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Alternate names

* Bitruncated enneazetton (Acronym: batene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the bitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 9-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_8$ | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Tritruncated 8-simplex

tritruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | 3t{3$^7$} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2016 |
| Vertices | 504 |
| Vertex figure | {3}v{3,3,3} |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Alternate names

* Tritruncated enneazetton (Acronym: tatene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the tritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 9-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_8$ | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Quadritruncated 8-simplex

Quadritruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | 4t{3$^7$} |
| Coxeter-Dynkin diagrams | or |
| 6-faces | 18 3t{3,3,3,3,3,3} |
| 7-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2520 |
| Vertices | 630 |
| Vertex figure | {3,3}v{3,3} |
| Coxeter group | A$_8$, [[3$^7$]], order 725760 |
| Properties | convex, isotopic |
The quadritruncated 8-simplex an isotopic polytope, constructed from 18 tritruncated 7-simplex facets.

### Alternate names

* Octadecazetton (18-facetted 8-polytope) (Acronym: be) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the quadritruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,2,2,2). This construction is based on facets of the quadritruncated 9-orthoplex.

### 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] |

### Related polytopes

Isotopic uniform truncated simplices :
| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Name Coxeter | Hexagon = t{3} = {6} | Octahedron = r{3,3} = {3$^{1,1}$} = {3,4} $\left\{{\begin{array}{l}3\\3\end{array}}\right\}$ | Decachoron 2t{3$^3$} | Dodecateron 2r{3$^4$} = {3$^{2,2}$} $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | Tetradecapeton 3t{3$^5$} | Hexadecaexon 3r{3$^6$} = {3$^{3,3}$} $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ | Octadecazetton 4t{3$^7$} |
| Images | | | | | | | |
| Vertex figure | ( )∨( ) | { }×{ } | { }∨{ } | {3}×{3} | {3}∨{3} | {3,3}×{3,3} | {3,3}∨{3,3} |
| Facets | | {3} | t{3,3} | r{3,3,3} | 2t{3,3,3,3} | 2r{3,3,3,3,3} | 3t{3,3,3,3,3,3} |
| As intersecting dual simplexes | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ |

## 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$_{01234567}$ |