Truncated 7-orthoplexes
# 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] |
