# Pentellated 6-orthoplexes

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

## Pentitruncated 6-orthoplex

Pentitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,5}${3,3,3,3,4} |
| 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 hexacontatetrapeton (Acronym: tacox) (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-orthoplex

Penticantellated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,5}${3,3,3,3,4} |
| 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 hexacontitetrapeton (Acronym: tapox) (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-orthoplex

Penticantitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,5}${3,3,3,3,4} |
| 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 hexacontitetrapeton (Acronym: togrig) (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-orthoplex

Pentiruncitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,3,5}${3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 51840 |
| Vertices | 11520 |
| Vertex figure | |
| Coxeter groups | B$_6$, [4,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teriprismatotruncated hexacontitetrapeton (Acronym: tocrax) (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-orthoplex

Pentiruncicantitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3,5}${3,3,3,3,4} |
| 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 hexacontitetrapeton (Acronym: tagpog) (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-orthoplex

Pentistericantitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,4,5}${3,3,3,3,4} |
| 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 hexacontitetrapeton (Acronym: tecagorg) (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$_6$ |