# Pentellated 7-orthoplexes

Orthogonal projections in B$_6$ Coxeter plane :
| 7-orthoplex | Pentellated 7-orthoplex | Pentitruncated 7-orthoplex | Penticantellated 7-orthoplex |
| Penticantitruncated 7-orthoplex | Pentiruncinated 7-orthoplex | Pentiruncitruncated 7-orthoplex | Pentiruncicantellated 7-orthoplex |
| Pentiruncicantitruncated 7-orthoplex | Pentistericated 7-orthoplex | Pentisteritruncated 7-orthoplex | Pentistericantellated 7-orthoplex |
| Pentistericantitruncated 7-orthoplex | Pentisteriruncinated 7-orthoplex | Pentisteriruncitruncated 7-orthoplex | Pentisteriruncicantellated 7-orthoplex |
| Pentisteriruncicantitruncated 7-orthoplex |
In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-orthoplex.
There are 32 unique pentellations of the 7-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-cube.
These polytopes are a part of a set of 127 uniform 7-polytopes with B$_7$ symmetry.

## Pentellated 7-orthoplex

Pentellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 2688 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Small terated hecatonicosoctaexon (acronym: Staz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,1,1,1,1,2)√2

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentitruncated 7-orthoplex

pentitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 87360 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teritruncated hecatonicosoctaexon (acronym: Tetaz) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Coordinates

Coordinates are permutations of (0,1,1,1,1,2,3).

## Penticantellated 7-orthoplex

Penticantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 188160 |
| Vertices | 26880 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Terirhombated hecatonicosoctaexon (acronym: Teroz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,1,1,2,2,3)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Penticantitruncated 7-orthoplex

penticantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 295680 |
| Vertices | 53760 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Terigreatorhombated hecatonicosoctaexon (acronym: Tograz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,1,1,2,3,4)√2.
orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentiruncinated 7-orthoplex

pentiruncinated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 174720 |
| Vertices | 26880 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teriprismated hecatonicosoctaexon (acronym: Topaz) (Jonathan Bowers)

### Coordinates

The coordinates are permutations of (0,1,1,2,2,2,3)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentiruncitruncated 7-orthoplex

pentiruncitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 443520 |
| Vertices | 80640 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teriprismatotruncated hecatonicosoctaexon (acronym: Toptaz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,1,2,2,3,4)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentiruncicantellated 7-orthoplex

pentiruncicantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 403200 |
| Vertices | 80640 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teriprismatorhombated hecatonicosoctaexon (acronym: Toparz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,1,2,3,3,4)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentiruncicantitruncated 7-orthoplex

pentiruncicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 725760 |
| Vertices | 161280 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Terigreatoprismated hecatonicosoctaexon (acronym: Tegopaz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,1,2,3,4,5)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | too complex | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentistericated 7-orthoplex

pentistericated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 67200 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericellated hecatonicosoctaexon (acronym: Tocaz) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Coordinates

Coordinates are permutations of (0,1,2,2,2,2,3)√2.

## Pentisteritruncated 7-orthoplex

pentisteritruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 241920 |
| Vertices | 53760 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericellitruncated hecatonicosoctaexon (acronym: Tacotaz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,2,2,3,4)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentistericantellated 7-orthoplex

pentistericantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 403200 |
| Vertices | 80640 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericellirhombated hecatonicosoctaexon (acronym: Tocarz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,2,3,3,4)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentistericantitruncated 7-orthoplex

pentistericantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 645120 |
| Vertices | 161280 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericelligreatorhombated hecatonicosoctaexon (acronym: Tecagraz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,2,3,4,5)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | too complex | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentisteriruncinated 7-orthoplex

Pentisteriruncinated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 241920 |
| Vertices | 53760 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Bipenticantitruncated 7-orthoplex as t$_{1,2,3,6}${3$^5$,4}
* Tericelliprismated hecatonicosoctaexon (acronym: Tecpaz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,3,3,3,4)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentisteriruncitruncated 7-orthoplex

pentisteriruncitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 645120 |
| Vertices | 161280 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericelliprismatotruncated hecatonicosoctaexon (acronym: Tecpotaz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,3,3,4,5)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | too complex | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentisteriruncicantellated 7-orthoplex

pentisteriruncicantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 645120 |
| Vertices | 161280 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Bipentiruncicantitruncated 7-orthoplex as t$_{1,2,3,4,6}${3$^5$,4}
* Tericelliprismatorhombated hecatonicosoctaexon (acronym: Tacparez) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,3,4,4,5)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | too complex | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Pentisteriruncicantitruncated 7-orthoplex

pentisteriruncicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,5}${3$^5$,4} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1128960 |
| Vertices | 322560 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Great terated hecatonicosoctaexon (acronym: Gotaz) (Jonathan Bowers)

### Coordinates

Coordinates are permutations of (0,1,2,3,4,5,6)√2.

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | too complex | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |