# Cantellated 6-simplexes

| 6-simplex | Cantellated 6-simplex | Bicantellated 6-simplex |
| Birectified 6-simplex | Cantitruncated 6-simplex | Bicantitruncated 6-simplex |
| Orthogonal projections in A$_6$ Coxeter plane |||
In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.
There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

## Cantellated 6-simplex

Cantellated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | rr{3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | 35 |
| 4-faces | 210 |
| Cells | 560 |
| Faces | 805 |
| Edges | 525 |
| Vertices | 105 |
| Vertex figure | 5-cell prism |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)

### Coordinates

The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.

### Images

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

## Bicantellated 6-simplex

Bicantellated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | 2rr{3,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | 49 |
| 4-faces | 329 |
| Cells | 980 |
| Faces | 1540 |
| Edges | 1050 |
| Vertices | 210 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)

### Coordinates

The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.

### Images

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

## Cantitruncated 6-simplex

cantitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | tr{3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | 35 |
| 4-faces | 210 |
| Cells | 560 |
| Faces | 805 |
| Edges | 630 |
| Vertices | 210 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)

### Coordinates

The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.

### Images

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

## Bicantitruncated 6-simplex

bicantitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | 2tr{3,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | 49 |
| 4-faces | 329 |
| Cells | 980 |
| Faces | 1540 |
| Edges | 1260 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)

### Coordinates

The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.

### Images

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

## Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A$_6$ Coxeter plane orthographic projections.
A6 polytopes :
| t$_0$ | t$_1$ | t$_2$ | 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$_{0,5}$ | 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$_{0,1,5}$ | t$_{0,2,5}$ | 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$_{0,1,4,5}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ | t$_{0,1,2,4,5}$ | t$_{0,1,2,3,4,5}$ |