# Truncated 7-orthoplexes

| 7-orthoplex | Truncated 7-orthoplex | Bitruncated 7-orthoplex | Tritruncated 7-orthoplex |
| 7-cube | Truncated 7-cube | Bitruncated 7-cube | Tritruncated 7-cube |
| Orthogonal projections in B$_7$ Coxeter plane ||||
In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.
There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.

## Truncated 7-orthoplex

Truncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t{3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | 3920 |
| Faces | 2520 |
| Edges | 924 |
| Vertices | 168 |
| Vertex figure | ( )v{3,3,4} |
| Coxeter groups | B$_7$, [3$^5$,4] D$_7$, [3$^{4,1,1}$] |
| Properties | convex |

### Alternate names

* Truncated heptacross
* Truncated hecatonicosoctaexon (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of
    (±2,±1,0,0,0,0,0)

### 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] |

### Construction

There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C$_7$ or [4,3$^5$] Coxeter group, and a lower symmetry with the D$_7$ or [3$^{4,1,1}$] Coxeter group.

## Bitruncated 7-orthoplex

Bitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | 2t{3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4200 |
| Vertices | 840 |
| Vertex figure | { }v{3,3,4} |
| Coxeter groups | B$_7$, [3$^5$,4] D$_7$, [3$^{4,1,1}$] |
| Properties | convex |

### Alternate names

* Bitruncated heptacross
* Bitruncated hecatonicosoctaexon (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
    (±2,±2,±1,0,0,0,0)

### 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] |

## Tritruncated 7-orthoplex

The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.
Tritruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | 3t{3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10080 |
| Vertices | 2240 |
| Vertex figure | {3}v{3,4} |
| Coxeter groups | B$_7$, [3$^5$,4] D$_7$, [3$^{4,1,1}$] |
| Properties | convex |

### Alternate names

* Tritruncated heptacross
* Tritruncated hecatonicosoctaexon (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
    (±2,±2,±2,±1,0,0,0)

### 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] |