# 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] |