# 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] |