# Stericated 7-simplexes

| 7-simplex | Stericated 7-simplex | Bistericated 7-simplex |
| Steritruncated 7-simplex | Bisteritruncated 7-simplex | Stericantellated 7-simplex |
| Bistericantellated 7-simplex | Stericantitruncated 7-simplex | Bistericantitruncated 7-simplex |
| Steriruncinated 7-simplex | Steriruncitruncated 7-simplex | Steriruncicantellated 7-simplex |
| Bisteriruncitruncated 7-simplex | Steriruncicantitruncated 7-simplex | Bisteriruncicantitruncated 7-simplex |
In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex.
There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations.

## Stericated 7-simplex

Stericated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2240 |
| Vertices | 280 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Small cellated octaexon (acronym: sco) (Jonathan Bowers)

### Coordinates

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

## Bistericated 7-simplex

bistericated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{1,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3360 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80320 |
| Properties | convex |

### Alternate names

* Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)

### Coordinates

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

## Steritruncated 7-simplex

steritruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7280 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Cellitruncated octaexon (acronym: cato) (Jonathan Bowers)

### Coordinates

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

## Bisteritruncated 7-simplex

bisteritruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{1,2,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 9240 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Bicellitruncated octaexon (acronym: bacto) (Jonathan Bowers)

### Coordinates

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

## Stericantellated 7-simplex

Stericantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10080 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Cellirhombated octaexon (acronym: caro) (Jonathan Bowers)

### Coordinates

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

## Bistericantellated 7-simplex

Bistericantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{1,3,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15120 |
| Vertices | 2520 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80320 |
| Properties | convex |

### Alternate names

* Bicellirhombihexadecaexon (acronym: bacroh) (Jonathan Bowers)

### Coordinates

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

## Stericantitruncated 7-simplex

stericantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16800 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Celligreatorhombated octaexon (acronym: cagro) (Jonathan Bowers)

### Coordinates

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

## Bistericantitruncated 7-simplex

bistericantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{1,2,3,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 22680 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Bicelligreatorhombated octaexon (acronym: bacogro) (Jonathan Bowers)

### Coordinates

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

## Steriruncinated 7-simplex

Steriruncinated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5040 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Celliprismated octaexon (acronym: cepo) (Jonathan Bowers)

### Coordinates

The vertices of the steriruncinated 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 steriruncinated 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] |

## Steriruncitruncated 7-simplex

steriruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Celliprismatotruncated octaexon (acronym: capto) (Jonathan Bowers)

### Coordinates

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

## Steriruncicantellated 7-simplex

steriruncicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Celliprismatorhombated octaexon (acronym: capro) (Jonathan Bowers)

### Coordinates

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

## Bisteriruncitruncated 7-simplex

bisteriruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{1,2,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80320 |
| Properties | convex |

### Alternate names

* Bicelliprismatotruncated hexadecaexon (acronym: bicpath) (Jonathan Bowers)

### Coordinates

The vertices of the bisteriruncitruncated 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 bisteriruncitruncated 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]] |

## Steriruncicantitruncated 7-simplex

steriruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23520 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Great cellated octaexon (acronym: gecco) (Jonathan Bowers)

### Coordinates

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

## Bisteriruncicantitruncated 7-simplex

bisteriruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{1,2,3,4,5}${3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 35280 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80320 |
| Properties | convex |

### Alternate names

* Great bicellated hexadecaexon (gabach) (Jonathan Bowers)

### Coordinates

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

This polytope is one 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$_{0,1,2,3,4,5,6}$ |