# Cantellated 7-simplexes

| 7-simplex | Cantellated 7-simplex | Bicantellated 7-simplex | Tricantellated 7-simplex |
| Birectified 7-simplex | Cantitruncated 7-simplex | Bicantitruncated 7-simplex | Tricantitruncated 7-simplex |
| Orthogonal projections in A$_7$ Coxeter plane ||||
In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.
There are unique 6 degrees of cantellation for the 7-simplex, including truncations.

## Cantellated 7-simplex

Cantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | rr{3,3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | or |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1008 |
| Vertices | 168 |
| Vertex figure | 5-simplex prism |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small rhombated octaexon (acronym: saro) (Jonathan Bowers)

### Coordinates

The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.

### Images

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

## Bicantellated 7-simplex

Bicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | r2r{3,3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2520 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)

### Coordinates

The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.

### Images

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

## Tricantellated 7-simplex

Tricantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | r3r{3,3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3360 |
| Vertices | 560 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)

### Coordinates

The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.

### Images

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

## Cantitruncated 7-simplex

Cantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | tr{3,3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1176 |
| Vertices | 336 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great rhombated octaexon (acronym: garo) (Jonathan Bowers)

### Coordinates

The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.

### Images

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

## Bicantitruncated 7-simplex

Bicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t2r{3,3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2940 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)

### Coordinates

The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.

### Images

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

## Tricantitruncated 7-simplex

Tricantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t3r{3,3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3920 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)

### Coordinates

The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [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 71 uniform 7-polytopes with A$_7$ symmetry.
A7 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_3$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{2,3}$ | t$_{0,4}$ | t$_{1,4}$ | t$_{2,4}$ | t$_{0,5}$ | t$_{1,5}$ | t$_{0,6}$ |
| t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ | t$_{0,2,4}$ | t$_{1,2,4}$ | t$_{0,3,4}$ |
| t$_{1,3,4}$ | t$_{2,3,4}$ | t$_{0,1,5}$ | t$_{0,2,5}$ | t$_{1,2,5}$ | t$_{0,3,5}$ | t$_{1,3,5}$ | t$_{0,4,5}$ |
| t$_{0,1,6}$ | t$_{0,2,6}$ | t$_{0,3,6}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,2,3,4}$ | t$_{1,2,3,4}$ |
| t$_{0,1,2,5}$ | t$_{0,1,3,5}$ | t$_{0,2,3,5}$ | t$_{1,2,3,5}$ | t$_{0,1,4,5}$ | t$_{0,2,4,5}$ | t$_{1,2,4,5}$ | t$_{0,3,4,5}$ |
| t$_{0,1,2,6}$ | t$_{0,1,3,6}$ | t$_{0,2,3,6}$ | t$_{0,1,4,6}$ | t$_{0,2,4,6}$ | t$_{0,1,5,6}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ |
| t$_{0,1,2,4,5}$ | t$_{0,1,3,4,5}$ | t$_{0,2,3,4,5}$ | t$_{1,2,3,4,5}$ | t$_{0,1,2,3,6}$ | t$_{0,1,2,4,6}$ | t$_{0,1,3,4,6}$ | t$_{0,2,3,4,6}$ |
| t$_{0,1,2,5,6}$ | t$_{0,1,3,5,6}$ | t$_{0,1,2,3,4,5}$ | t$_{0,1,2,3,4,6}$ | t$_{0,1,2,3,5,6}$ | t$_{0,1,2,4,5,6}$ | t$_{0,1,2,3,4,5,6}$ |