# Cantellated 8-simplexes

| Cantellated 8-simplex | Bicantellated 8-simplex | Tricantellated 8-simplex |
| Cantitruncated 8-simplex | Bicantitruncated 8-simplex | Tricantitruncated 8-simplex |
| Orthogonal projections in A$_8$ Coxeter plane |||
In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.
There are six unique cantellations for the 8-simplex, including permutations of truncation.

## Cantellated 8-simplex

Cantellated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | rr{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1764 |
| Vertices | 252 |
| Vertex figure | 6-simplex prism |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Alternate names

* Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)

### Coordinates

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

## Bicantellated 8-simplex

Bicantellated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | r2r{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5292 |
| Vertices | 756 |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Alternate names

* Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)

### Coordinates

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

## Tricantellated 8-simplex

tricantellated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | r3r{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8820 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

### Alternate names

* Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)

### Coordinates

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

## Cantitruncated 8-simplex

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

### Alternate names

* Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 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] |

## Bicantitruncated 8-simplex

Bicantitruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t2r{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |

## Alternate names

* Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 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] |

## Tricantitruncated 8-simplex

Tricantitruncated 8-simplex :
| Type | uniform 8-polytope |
| Schläfli symbol | t3r{3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |
* Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 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] |

## 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$_{01234567}$ |