# Runcinated 6-cubes

| 6-cube | Runcinated 6-cube | Biruncinated 6-cube | Runcinated 6-orthoplex | 6-orthoplex |
| Runcitruncated 6-cube | Biruncitruncated 6-cube | Runcicantellated 6-orthoplex | Runcicantellated 6-cube | Biruncitruncated 6-orthoplex |
| Runcitruncated 6-orthoplex | Runcicanti-truncated 6-cube | Biruncicanti-truncated 6-cube | Runcicanti-truncated 6-orthoplex |
| Orthogonal projections in B$_6$ Coxeter plane |||||
In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.
There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex.

## Runcinated 6-cube

Runcinated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{0,3}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7680 |
| Vertices | 1280 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small prismated hexeract (spox) (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] |

## Biruncinated 6-cube

Biruncinated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{1,4}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11520 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small biprismated hexeractihexacontatetrapeton (sobpoxog) (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] |

## Runcitruncated 6-cube

Runcitruncated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{0,1,3}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 17280 |
| Vertices | 3840 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Prismatotruncated hexeract (potax) (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] |

## Biruncitruncated 6-cube

Biruncitruncated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{1,2,4}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Biprismatotruncated hexeract (boprag) (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] |

## Runcicantellated 6-cube

Runcicantellated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{0,2,3}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3840 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Prismatorhombated hexeract (prox) (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] |

## Runcicantitruncated 6-cube

Runcicantitruncated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 7680 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great prismated hexeract (gippox) (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] |

## Biruncitruncated 6-cube

Biruncitruncated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{1,2,3,4}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter group | B$_6$ [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Biprismatotruncated hexeract (boprag) (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 from 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$_6$ |