# Stericated 6-simplexes

| 6-simplex | Stericated 6-simplex | Steritruncated 6-simplex |
| Stericantellated 6-simplex | Stericantitruncated 6-simplex | Steriruncinated 6-simplex |
| Steriruncitruncated 6-simplex | Steriruncicantellated 6-simplex | Steriruncicantitruncated 6-simplex |
| Orthogonal projections in A$_6$ Coxeter plane |||
In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.
There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.

## Stericated 6-simplex

Stericated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 700 |
| Cells | 1470 |
| Faces | 1400 |
| Edges | 630 |
| Vertices | 105 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Steritruncated 6-simplex

Steritruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 945 |
| Cells | 2940 |
| Faces | 3780 |
| Edges | 2100 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Stericantellated 6-simplex

Stericantellated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1050 |
| Cells | 3465 |
| Faces | 5040 |
| Edges | 3150 |
| Vertices | 630 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Stericantitruncated 6-simplex

stericantitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1155 |
| Cells | 4410 |
| Faces | 7140 |
| Edges | 5040 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Steriruncinated 6-simplex

steriruncinated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,3,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 700 |
| Cells | 1995 |
| Faces | 2660 |
| Edges | 1680 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Steriruncitruncated 6-simplex

steriruncitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,3,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 945 |
| Cells | 3360 |
| Faces | 5670 |
| Edges | 4410 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Steriruncicantellated 6-simplex

steriruncicantellated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,3,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1050 |
| Cells | 3675 |
| Faces | 5880 |
| Edges | 4410 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Bistericantitruncated 6-simplex as t$_{1,2,3,5}${3,3,3,3,3}
* Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Steriruncicantitruncated 6-simplex

Steriuncicantitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 105 |
| 4-faces | 1155 |
| Cells | 4620 |
| Faces | 8610 |
| Edges | 7560 |
| Vertices | 2520 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)

### Coordinates

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

### Images

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

## Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A$_6$ Coxeter plane orthographic projections.
A6 polytopes :
| t$_0$ | t$_1$ | t$_2$ | 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$_{0,5}$ | 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$_{0,1,5}$ | t$_{0,2,5}$ | 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$_{0,1,4,5}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ | t$_{0,1,2,4,5}$ | t$_{0,1,2,3,4,5}$ |