# Runcinated 6-simplexes

| 6-simplex | Runcinated 6-simplex | Biruncinated 6-simplex |
| Runcitruncated 6-simplex | Biruncitruncated 6-simplex | Runcicantellated 6-simplex |
| Runcicantitruncated 6-simplex | Biruncicantitruncated 6-simplex |
| Orthogonal projections in A$_6$ Coxeter plane |||
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.
There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.

## Runcinated 6-simplex

Runcinated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,3}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 455 |
| Cells | 1330 |
| Faces | 1610 |
| Edges | 840 |
| Vertices | 140 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)

### Coordinates

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

## Biruncinated 6-simplex

biruncinated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{1,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 84 |
| 4-faces | 714 |
| Cells | 2100 |
| Faces | 2520 |
| Edges | 1260 |
| Vertices | 210 |
| Vertex figure | |
| Coxeter group | A$_6$, [[3$^5$]], order 10080 |
| Properties | convex |

### Alternate names

* Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)

### Coordinates

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

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_6$ | A$_5$ | A$_4$ |
| Graph | | | |
| Symmetry | [[7]]$^{(*)}$=[14] | [6] | [[5]]$^{(*)}$=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Symmetry | [4] | [[3]]$^{(*)}$=[6] |
    Note: (*) Symmetry doubled for A$_k$ graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

## Runcitruncated 6-simplex

Runcitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,3}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 560 |
| Cells | 1820 |
| Faces | 2800 |
| Edges | 1890 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)

### Coordinates

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

## Biruncitruncated 6-simplex

biruncitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{1,2,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 84 |
| 4-faces | 714 |
| Cells | 2310 |
| Faces | 3570 |
| Edges | 2520 |
| Vertices | 630 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)

### Coordinates

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

## Runcicantellated 6-simplex

Runcicantellated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,3}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 455 |
| Cells | 1295 |
| Faces | 1960 |
| Edges | 1470 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)

### Coordinates

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

## Runcicantitruncated 6-simplex

Runcicantitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 560 |
| Cells | 1820 |
| Faces | 3010 |
| Edges | 2520 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Properties | convex |

### Alternate names

* Runcicantitruncated heptapeton
* Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)

### Coordinates

The vertices of the runcicantitruncated 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 runcicantitruncated 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] |

## Biruncicantitruncated 6-simplex

biruncicantitruncated 6-simplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{1,2,3,4}${3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 84 |
| 4-faces | 714 |
| Cells | 2520 |
| Faces | 4410 |
| Edges | 3780 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A$_6$, [[3$^5$]], order 10080 |
| Properties | convex |

### Alternate names

* Biruncicantitruncated heptapeton
* Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)

### Coordinates

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

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_6$ | A$_5$ | A$_4$ |
| Graph | | | |
| Symmetry | [[7]]$^{(*)}$=[14] | [6] | [[5]]$^{(*)}$=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Symmetry | [4] | [[3]]$^{(*)}$=[6] |
    Note: (*) Symmetry doubled for A$_k$ graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

## 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$_{0,1,2,3,4,5}$ |