Truncated 7-cubes
# Truncated 7-cubes

| 7-cube | Truncated 7-cube | Bitruncated 7-cube | Tritruncated 7-cube |
| - | - | - | - |
| 7-orthoplex | Truncated 7-orthoplex | Bitruncated 7-orthoplex | Tritruncated 7-orthoplex |
| Orthogonal projections in $B_7$ Coxeter plane ||||

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

There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.

## Truncated 7-cube

Truncated 7-cube :

| Type | uniform 7-polytope |
| - | - |
| Schläfli symbol | $t{4,3^5}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3136 |
| Vertices | 896 |
| Vertex figure | Elongated 5-simplex pyramid |
| Coxeter groups | $B_7,$ $[3^5,4]$ |
| Properties | convex |

### Alternate names

* Truncated hepteract (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of

    (1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)

### Images

orthographic projections :

| Coxeter plane | $B_7$ / $A_6$ | $B_6$ / $D_7$ | $B_5$ / $D_6$ / $A_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | $B_4$ / $D_5$ | $B_3$ / $D_4$ / $A_2$ | $B_2$ / $D_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | $A_5$ | $A_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Related polytopes

The truncated 7-cube, is sixth in a sequence of truncated hypercubes:

Truncated hypercubes :

| Image | | | | | | | | ... |
| - | - | - | - | - | - | - | - | - |
| Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube | ... |
| Coxeter diagram | | | | | | | | ... |
| Vertex figure | ( )v( ) | ( )v{ } | ( )v{3} | ( )v{3,3} | ( )v{3,3,3} | ( )v{3,3,3,3} | ( )v{3,3,3,3,3} |

## Bitruncated 7-cube

Bitruncated 7-cube :

| Type | uniform 7-polytope |
| - | - |
| Schläfli symbol | $2t{4,3^5}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 9408 |
| Vertices | 2688 |
| Vertex figure | { }v{3,3,3} |
| Coxeter groups | $B_7,$ $[3^5,4]$ $D_7,$ $[3^{4,1,1}]$ |
| Properties | convex |

### Alternate names

* Bitruncated hepteract (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

    (±2,±2,±2,±2,±2,±1,0)

### Images

orthographic projections :

| Coxeter plane | $B_7$ / $A_6$ | $B_6$ / $D_7$ | $B_5$ / $D_6$ / $A_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | $B_4$ / $D_5$ | $B_3$ / $D_4$ / $A_2$ | $B_2$ / $D_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | $A_5$ | $A_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Related polytopes

The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes :

| Image | | | | | | | ... |
| - | - | - | - | - | - | - | - |
| Name | Bitruncated cube | Bitruncated tesseract | Bitruncated 5-cube | Bitruncated 6-cube | Bitruncated 7-cube | Bitruncated 8-cube | ... |
| Coxeter | | | | | | | ... |
| Vertex figure | ( )v{ } | { }v{ } | { }v{3} | { }v{3,3} | { }v{3,3,3} | { }v{3,3,3,3} |

## Tritruncated 7-cube

Tritruncated 7-cube :

| Type | uniform 7-polytope |
| - | - |
| Schläfli symbol | $3t{4,3^5}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3360 |
| Vertex figure | {4}v{3,3} |
| Coxeter groups | $B_7,$ $[3^5,4]$ $D_7,$ $[3^{4,1,1}]$ |
| Properties | convex |

### Alternate names

* Tritruncated hepteract (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

    (±2,±2,±2,±2,±1,0,0)

### Images

orthographic projections :

| Coxeter plane | $B_7$ / $A_6$ | $B_6$ / $D_7$ | $B_5$ / $D_6$ / $A_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | $B_4$ / $D_5$ | $B_3$ / $D_4$ / $A_2$ | $B_2$ / $D_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | $A_5$ | $A_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
