# Pentellated 6-cubes

| 6-cube | 6-orthoplex | Pentellated 6-cube |
| Pentitruncated 6-cube | Penticantellated 6-cube | Penticantitruncated 6-cube |
| Pentiruncitruncated 6-cube | Pentiruncicantellated 6-cube | Pentiruncicantitruncated 6-cube |
| Pentisteritruncated 6-cube | Pentistericantitruncated 6-cube | Omnitruncated 6-cube |
| Orthogonal projections in B$_6$ Coxeter plane |||
In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.
There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex.

## Pentellated 6-cube

Pentellated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{0,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1920 |
| Vertices | 384 |
| Vertex figure | 5-cell antiprism |
| Coxeter group | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Pentellated 6-orthoplex
* Expanded 6-cube, expanded 6-orthoplex
* Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (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] |

## Pentitruncated 6-cube

Pentitruncated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8640 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teritruncated hexeract (Acronym: tacog) (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] |

## Penticantellated 6-cube

Penticantellated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 21120 |
| Vertices | 3840 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Terirhombated hexeract (Acronym: topag) (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] |

## Penticantitruncated 6-cube

Penticantitruncated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30720 |
| Vertices | 7680 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Terigreatorhombated hexeract (Acronym: togrix) (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] |

## Pentiruncitruncated 6-cube

Pentiruncitruncated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,3,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 151840 |
| Vertices | 11520 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (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] |

## Pentiruncicantellated 6-cube

Pentiruncicantellated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,3,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 46080 |
| Vertices | 11520 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (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] |

## Pentiruncicantitruncated 6-cube

Pentiruncicantitruncated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 23040 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Terigreatoprismated hexeract (Acronym: tagpox) (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] |

## Pentisteritruncated 6-cube

Pentisteritruncated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,4,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30720 |
| Vertices | 7680 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (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] |

## Pentistericantitruncated 6-cube

Pentistericantitruncated 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,4,5}${4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 23040 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericelligreatorhombated hexeract (Acronym: tocagrax) (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] |

## Omnitruncated 6-cube

Omnitruncated 6-cube :
| Type | Uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,5}${3$^5$} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 728: 12 t$_{0,1,2,3,4}${3,3,3,4} 60×t$_{0,1,2,3}${3,3,4} × 160 {6}×t$_{0,1,2}${3,4} × 240 {8}×t$_{0,1,2}${3,3} × 192×t$_{0,1,2,3}${3$^3$} × 64 t$_{0,1,2,3,4}${3$^4$} |
| 4-faces | 14168 |
| Cells | 72960 |
| Faces | 151680 |
| Edges | 138240 |
| Vertices | 46080 |
| Vertex figure | irregular 5-simplex |
| Coxeter group | B$_6$, [4,3,3,3,3] |
| Properties | convex, isogonal |
The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.

### Alternate names

* Pentisteriruncicantitruncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
* Omnitruncated hexeract
* Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (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] |

### Full snub 6-cube

The full snub 6-cube or omnisnub 6-cube, defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3,3]$^+$, and constructed from 12 snub 5-cubes, 64 snub 5-simplexes, 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr{4,3} duoantiprisms, 240 4-s{3,4} duoantiprisms, and 23040 irregular 5-simplexes filling the gaps at the deleted vertices.

## 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$_6$ |