# Cantellated 6-cubes

| 6-cube | 6-cube | 6-cube | 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Bicantellated 6-cube | Bicantellated 6-cube | Bicantellated 6-cube | Bicantellated 6-cube |
| 6-orthoplex | 6-orthoplex | 6-orthoplex | 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Bicantellated 6-orthoplex | Bicantellated 6-orthoplex | Bicantellated 6-orthoplex | Bicantellated 6-orthoplex |
| Cantitruncated 6-cube | Cantitruncated 6-cube | Cantitruncated 6-cube | Bicantitruncated 6-cube | Bicantitruncated 6-cube | Bicantitruncated 6-cube | Bicantitruncated 6-orthoplex | Bicantitruncated 6-orthoplex | Bicantitruncated 6-orthoplex | Cantitruncated 6-orthoplex | Cantitruncated 6-orthoplex | Cantitruncated 6-orthoplex |
| Orthogonal projections in B$_6$ Coxeter plane ||||||||||||
In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.
There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex.

## Cantellated 6-cube

Cantellated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | rr{4,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4800 |
| Vertices | 960 |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Cantellated hexeract
* Small rhombated hexeract (acronym: srox) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Bicantellated 6-cube

Cantellated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | 2rr{4,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3\\3,4\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Bicantellated hexeract
* Small birhombated hexeract (acronym: saborx) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Cantitruncated 6-cube

Cantellated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | tr{4,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Cantitruncated hexeract
* Great rhombihexeract (acronym: grox) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
It is fourth in a series of cantitruncated hypercubes:
Petrie polygon projections :
| Truncated cuboctahedron | Cantitruncated tesseract | Cantitruncated 5-cube | Cantitruncated 6-cube | Cantitruncated 7-cube | Cantitruncated 8-cube |

## Bicantitruncated 6-cube

Cantellated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | 2tr{4,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3\\3,4\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Bicantitruncated hexeract
* Great birhombihexeract (acronym: gaborx) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B$_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes :
| β$_6$ | t$_1$β$_6$ | t$_2$β$_6$ | t$_2$γ$_6$ | t$_1$γ$_6$ | γ$_6$ | t$_{0,1}$β$_6$ | t$_{0,2}$β$_6$ |
| t$_{1,2}$β$_6$ | t$_{0,3}$β$_6$ | t$_{1,3}$β$_6$ | t$_{2,3}$γ$_6$ | t$_{0,4}$β$_6$ | t$_{1,4}$γ$_6$ | t$_{1,3}$γ$_6$ | t$_{1,2}$γ$_6$ |
| t$_{0,5}$γ$_6$ | t$_{0,4}$γ$_6$ | t$_{0,3}$γ$_6$ | t$_{0,2}$γ$_6$ | t$_{0,1}$γ$_6$ | t$_{0,1,2}$β$_6$ | t$_{0,1,3}$β$_6$ | t$_{0,2,3}$β$_6$ |
| t$_{1,2,3}$β$_6$ | t$_{0,1,4}$β$_6$ | t$_{0,2,4}$β$_6$ | t$_{1,2,4}$β$_6$ | t$_{0,3,4}$β$_6$ | t$_{1,2,4}$γ$_6$ | t$_{1,2,3}$γ$_6$ | t$_{0,1,5}$β$_6$ |
| t$_{0,2,5}$β$_6$ | t$_{0,3,4}$γ$_6$ | t$_{0,2,5}$γ$_6$ | t$_{0,2,4}$γ$_6$ | t$_{0,2,3}$γ$_6$ | t$_{0,1,5}$γ$_6$ | t$_{0,1,4}$γ$_6$ | t$_{0,1,3}$γ$_6$ |
| t$_{0,1,2}$γ$_6$ | t$_{0,1,2,3}$β$_6$ | t$_{0,1,2,4}$β$_6$ | t$_{0,1,3,4}$β$_6$ | t$_{0,2,3,4}$β$_6$ | t$_{1,2,3,4}$γ$_6$ | t$_{0,1,2,5}$β$_6$ | t$_{0,1,3,5}$β$_6$ |
| t$_{0,2,3,5}$γ$_6$ | t$_{0,2,3,4}$γ$_6$ | t$_{0,1,4,5}$γ$_6$ | t$_{0,1,3,5}$γ$_6$ | t$_{0,1,3,4}$γ$_6$ | t$_{0,1,2,5}$γ$_6$ | t$_{0,1,2,4}$γ$_6$ | t$_{0,1,2,3}$γ$_6$ |
| t$_{0,1,2,3,4}$β$_6$ | t$_{0,1,2,3,5}$β$_6$ | t$_{0,1,2,4,5}$β$_6$ | t$_{0,1,2,4,5}$γ$_6$ | t$_{0,1,2,3,5}$γ$_6$ | t$_{0,1,2,3,4}$γ$_6$ | t$_{0,1,2,3,4,5}$γ$_6$ |