Truncated 7-simplexes
# Truncated 7-simplexes

| 7-simplex | Truncated 7-simplex |
| - | - |
| Bitruncated 7-simplex | Tritruncated 7-simplex |
| Orthogonal projections in $A_7$ Coxeter plane ||

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.

## Truncated 7-simplex

Truncated 7-simplex :

| Type | uniform 7-polytope |
| - | - |
| Schläfli symbol | t{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 16 |
| 5-faces | |
| 4-faces | |
| Cells | 350 |
| Faces | 336 |
| Edges | 196 |
| Vertices | 56 |
| Vertex figure | ( )v{3,3,3,3} |
| Coxeter groups | $A_7,$ [3,3,3,3,3,3] |
| Properties | convex, Vertex-transitive |

In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.

### Alternate names

* Truncated octaexon (Acronym: toc) (Jonathan Bowers)

### Coordinates

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

## Bitruncated 7-simplex

Bitruncated 7-simplex :

| Type | uniform 7-polytope |
| - | - |
| Schläfli symbol | 2t{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 588 |
| Vertices | 168 |
| Vertex figure | { }v{3,3,3} |
| Coxeter groups | $A_7,$ [3,3,3,3,3,3] |
| Properties | convex, Vertex-transitive |

### Alternate names

* Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)

### Coordinates

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

## Tritruncated 7-simplex

Tritruncated 7-simplex :

| Type | uniform 7-polytope |
| - | - |
| Schläfli symbol | 3t{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 980 |
| Vertices | 280 |
| Vertex figure | {3}v{3,3} |
| Coxeter groups | $A_7,$ [3,3,3,3,3,3] |
| Properties | convex, Vertex-transitive |

### Alternate names

* Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)

### Coordinates

The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 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 three polytopes are from a set of 71 uniform 7-polytopes with $A_7$ symmetry.

A7 polytopes :

| $t_0$ | $t_1$ | $t_2$ | $t_3$ | $t_{0,1}$ | $t_{0,2}$ | $t_{1,2}$ | $t_{0,3}$ |
| - | - | - | - | - | - | - | - |
| $t_{1,3}$ | $t_{2,3}$ | $t_{0,4}$ | $t_{1,4}$ | $t_{2,4}$ | $t_{0,5}$ | $t_{1,5}$ | $t_{0,6}$ |
| $t_{0,1,2}$ | $t_{0,1,3}$ | $t_{0,2,3}$ | $t_{1,2,3}$ | $t_{0,1,4}$ | $t_{0,2,4}$ | $t_{1,2,4}$ | $t_{0,3,4}$ |
| $t_{1,3,4}$ | $t_{2,3,4}$ | $t_{0,1,5}$ | $t_{0,2,5}$ | $t_{1,2,5}$ | $t_{0,3,5}$ | $t_{1,3,5}$ | $t_{0,4,5}$ |
| $t_{0,1,6}$ | $t_{0,2,6}$ | $t_{0,3,6}$ | $t_{0,1,2,3}$ | $t_{0,1,2,4}$ | $t_{0,1,3,4}$ | $t_{0,2,3,4}$ | $t_{1,2,3,4}$ |
| $t_{0,1,2,5}$ | $t_{0,1,3,5}$ | $t_{0,2,3,5}$ | $t_{1,2,3,5}$ | $t_{0,1,4,5}$ | $t_{0,2,4,5}$ | $t_{1,2,4,5}$ | $t_{0,3,4,5}$ |
| $t_{0,1,2,6}$ | $t_{0,1,3,6}$ | $t_{0,2,3,6}$ | $t_{0,1,4,6}$ | $t_{0,2,4,6}$ | $t_{0,1,5,6}$ | $t_{0,1,2,3,4}$ | $t_{0,1,2,3,5}$ |
| $t_{0,1,2,4,5}$ | $t_{0,1,3,4,5}$ | $t_{0,2,3,4,5}$ | $t_{1,2,3,4,5}$ | $t_{0,1,2,3,6}$ | $t_{0,1,2,4,6}$ | $t_{0,1,3,4,6}$ | $t_{0,2,3,4,6}$ |
| $t_{0,1,2,5,6}$ | $t_{0,1,3,5,6}$ | $t_{0,1,2,3,4,5}$ | $t_{0,1,2,3,4,6}$ | $t_{0,1,2,3,5,6}$ | $t_{0,1,2,4,5,6}$ | $t_{0,1,2,3,4,5,6}$ |
