Truncated 8-orthoplexes
# Truncated 8-orthoplexes

| 8-orthoplex | Truncated 8-orthoplex | Bitruncated 8-orthoplex |
| - | - | - |
| Tritruncated 8-orthoplex | Quadritruncated 8-cube | Tritruncated 8-cube |
| Bitruncated 8-cube | Truncated 8-cube | 8-cube |
| Orthogonal projections in $B_8$ Coxeter plane |||

In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.

There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.

## Truncated 8-orthoplex

Truncated 8-orthoplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $t_{0,1}{3,3,3,3,3,3,4}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1456 |
| Vertices | 224 |
| Vertex figure | ( )v{3,3,3,4} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
| Properties | convex |

### Alternate names

* Truncated octacross (acronym tek) (Jonthan Bowers)

### Construction

There are two Coxeter groups associated with the truncated 8-orthoplex, one with the $C_8$ or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the $D_8$ or $[3^{5,1,1}]$ Coxeter group.

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

    (±2,±1,0,0,0,0,0,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Bitruncated 8-orthoplex

Bitruncated 8-orthoplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $t_{1,2}{3,3,3,3,3,3,4}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | { }v{3,3,3,4} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
| Properties | convex |

### Alternate names

* Bitruncated octacross (acronym batek) (Jonthan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

    (±2,±2,±1,0,0,0,0,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Tritruncated 8-orthoplex

Tritruncated 8-orthoplex :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | $t_{2,3}{3,3,3,3,3,3,4}$ |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {3}v{3,3,4} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
| Properties | convex |

### Alternate names

* Tritruncated octacross (acronym tatek) (Jonthan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

    (±2,±2,±2,±1,0,0,0,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
