# Pentellated 7-simplexes

| 7-simplex | Pentellated 7-simplex | Pentitruncated 7-simplex | Penticantellated 7-simplex |
| Penticantitruncated 7-simplex | Pentiruncinated 7-simplex | Pentiruncitruncated 7-simplex | Pentiruncicantellated 7-simplex |
| Pentiruncicantitruncated 7-simplex | Pentistericated 7-simplex | Pentisteritruncated 7-simplex | Pentistericantellated 7-simplex |
| Pentistericantitruncated 7-simplex | Pentisteriruncinated 7-simplex | Pentisteriruncitruncated 7-simplex | Pentisteriruncicantellated 7-simplex |
| Pentisteriruncicantitruncated 7-simplex ||||
In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-simplex.
There are 16 unique pentellations of the 7-simplex with permutations of truncations, cantellations, runcinations, and sterications.

## Pentellated 7-simplex

Pentellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1260 |
| Vertices | 168 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small terated octaexon (acronym: seto) (Jonathan Bowers)

### Coordinates

The vertices of the pentellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,1,2). This construction is based on facets of the pentellated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentitruncated 7-simplex

pentitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5460 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teritruncated octaexon (acronym: teto) (Jonathan Bowers)

### Coordinates

The vertices of the pentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,3). This construction is based on facets of the pentitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Penticantellated 7-simplex

Penticantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11760 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Terirhombated octaexon (acronym: tero) (Jonathan Bowers)

### Coordinates

The vertices of the penticantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,2,3). This construction is based on facets of the penticantellated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Penticantitruncated 7-simplex

penticantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Terigreatorhombated octaexon (acronym: tegro) (Jonathan Bowers)

### Coordinates

The vertices of the penticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,4). This construction is based on facets of the penticantitruncated 8-orthoplex.
orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentiruncinated 7-simplex

pentiruncinated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10920 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teriprismated octaexon (acronym: tepo) (Jonathan Bowers)

### Coordinates

The vertices of the pentiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,2,3). This construction is based on facets of the pentiruncinated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentiruncitruncated 7-simplex

pentiruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 27720 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teriprismatotruncated octaexon (acronym: tapto) (Jonathan Bowers)

### Coordinates

The vertices of the pentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,4). This construction is based on facets of the pentiruncitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentiruncicantellated 7-simplex

pentiruncicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 25200 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Teriprismatorhombated octaexon (acronym: tapro) (Jonathan Bowers)

### Coordinates

The vertices of the pentiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,3,4). This construction is based on facets of the pentiruncicantellated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentiruncicantitruncated 7-simplex

pentiruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 45360 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Terigreatoprismated octaexon (acronym: tegapo) (Jonathan Bowers)

### Coordinates

The vertices of the pentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,5). This construction is based on facets of the pentiruncicantitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentistericated 7-simplex

pentistericated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4200 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericellated octaexon (acronym: teco) (Jonathan Bowers)

### Coordinates

The vertices of the pentistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the pentistericated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentisteritruncated 7-simplex

pentisteritruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericellitruncated octaexon (acronym: tecto) (Jonathan Bowers)

### Coordinates

The vertices of the pentisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the pentisteritruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentistericantellated 7-simplex

pentistericantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 25200 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericellirhombated octaexon (acronym: tecro) (Jonathan Bowers)

### Coordinates

The vertices of the pentistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,3,4). This construction is based on facets of the pentistericantellated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentistericantitruncated 7-simplex

pentistericantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericelligreatorhombated octaexon (acronym: tecagro) (Jonathan Bowers)

### Coordinates

The vertices of the pentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,5). This construction is based on facets of the pentistericantitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentisteriruncinated 7-simplex

Pentisteriruncinated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Bipenticantitruncated 7-simplex as t$_{1,2,3,6}${3,3,3,3,3,3}
* Tericelliprismated octaexon (acronym: tacpo) (Jonathan Bowers)

### Coordinates

The vertices of the pentisteriruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,3,4). This construction is based on facets of the pentisteriruncinated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Pentisteriruncitruncated 7-simplex

pentisteriruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Tericelliprismatotruncated octaexon (acronym: tacpeto) (Jonathan Bowers)

### Coordinates

The vertices of the pentisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,3,4,5). This construction is based on facets of the pentisteriruncitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Pentisteriruncicantellated 7-simplex

pentisteriruncicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Bipentiruncicantitruncated 7-simplex as t$_{1,2,3,4,6}${3,3,3,3,3,3}
* Tericelliprismatorhombated octaexon (acronym: tacpro) (Jonathan Bowers)

### Coordinates

The vertices of the pentisteriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,4,5). This construction is based on facets of the pentisteriruncicantellated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Pentisteriruncicantitruncated 7-simplex

pentisteriruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 70560 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter groups | A$_7$, [3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great terated octaexon (acronym: geto) (Jonathan Bowers)

### Coordinates

The vertices of the pentisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,6). This construction is based on facets of the pentisteriruncicantitruncated 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Related polytopes

These polytopes are a part of a set of 71 uniform 7-polytopes with A$_7$ symmetry.
A7 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_3$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{2,3}$ | t$_{0,4}$ | t$_{1,4}$ | t$_{2,4}$ | t$_{0,5}$ | t$_{1,5}$ | t$_{0,6}$ |
| t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ | t$_{0,2,4}$ | t$_{1,2,4}$ | t$_{0,3,4}$ |
| t$_{1,3,4}$ | t$_{2,3,4}$ | t$_{0,1,5}$ | t$_{0,2,5}$ | t$_{1,2,5}$ | t$_{0,3,5}$ | t$_{1,3,5}$ | t$_{0,4,5}$ |
| t$_{0,1,6}$ | t$_{0,2,6}$ | t$_{0,3,6}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,2,3,4}$ | t$_{1,2,3,4}$ |
| t$_{0,1,2,5}$ | t$_{0,1,3,5}$ | t$_{0,2,3,5}$ | t$_{1,2,3,5}$ | t$_{0,1,4,5}$ | t$_{0,2,4,5}$ | t$_{1,2,4,5}$ | t$_{0,3,4,5}$ |
| t$_{0,1,2,6}$ | t$_{0,1,3,6}$ | t$_{0,2,3,6}$ | t$_{0,1,4,6}$ | t$_{0,2,4,6}$ | t$_{0,1,5,6}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ |
| t$_{0,1,2,4,5}$ | t$_{0,1,3,4,5}$ | t$_{0,2,3,4,5}$ | t$_{1,2,3,4,5}$ | t$_{0,1,2,3,6}$ | t$_{0,1,2,4,6}$ | t$_{0,1,3,4,6}$ | t$_{0,2,3,4,6}$ |
| t$_{0,1,2,5,6}$ | t$_{0,1,3,5,6}$ | t$_{0,1,2,3,4,5}$ | t$_{0,1,2,3,4,6}$ | t$_{0,1,2,3,5,6}$ | t$_{0,1,2,4,5,6}$ | t$_{0,1,2,3,4,5,6}$ |