Truncated 6-orthoplexes
# Truncated 6-orthoplexes

| 6-orthoplex | Truncated 6-orthoplex | Bitruncated 6-orthoplex | Tritruncated 6-cube |
| - | - | - | - |
| 6-cube | Truncated 6-cube | Bitruncated 6-cube | Tritruncated 6-cube |
| Orthogonal projections in $B_6$ Coxeter plane ||||

In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.

There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.

## Truncated 6-orthoplex

Truncated 6-orthoplex :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | t{3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 76 |
| 4-faces | 576 |
| Cells | 1200 |
| Faces | 1120 |
| Edges | 540 |
| Vertices | 120 |
| Vertex figure | ( )v{3,4} |
| Coxeter groups | $B_6,$ [3,3,3,3,4] $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

### Alternate names

* Truncated hexacross
* Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the truncated hexacross, one with the $C_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the $D_6$ or $[3^{3,1,1}]$ Coxeter group.

### Coordinates

Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of

    (±2,±1,0,0,0,0)

### Images

orthographic projections :

| Coxeter plane | $B_6$ | $B_5$ | $B_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | $B_3$ | $B_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | $A_5$ | $A_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Bitruncated 6-orthoplex

Bitruncated 6-orthoplex :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | 2t{3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | { }v{3,4} |
| Coxeter groups | $B_6,$ [3,3,3,3,4] $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

### Alternate names

* Bitruncated hexacross
* Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)

### Images

orthographic projections :

| Coxeter plane | $B_6$ | $B_5$ | $B_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | $B_3$ | $B_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | $A_5$ | $A_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

These polytopes are a part of a set of 63 uniform 6-polytopes generated from the $B_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.

B6 polytopes :

| $β_6$ | $t_1β_6$ | $t_2β_6$ | $t_2γ_6$ | $t_1γ_6$ | $γ_6$ | $t_{0,1}β_6$ | $t_{0,2}β_6$ |
| - | - | - | - | - | - | - | - |
| $t_{1,2}β_6$ | $t_{0,3}β_6$ | $t_{1,3}β_6$ | $t_{2,3}γ_6$ | $t_{0,4}β_6$ | $t_{1,4}γ_6$ | $t_{1,3}γ_6$ | $t_{1,2}γ_6$ |
| $t_{0,5}γ_6$ | $t_{0,4}γ_6$ | $t_{0,3}γ_6$ | $t_{0,2}γ_6$ | $t_{0,1}γ_6$ | $t_{0,1,2}β_6$ | $t_{0,1,3}β_6$ | $t_{0,2,3}β_6$ |
| $t_{1,2,3}β_6$ | $t_{0,1,4}β_6$ | $t_{0,2,4}β_6$ | $t_{1,2,4}β_6$ | $t_{0,3,4}β_6$ | $t_{1,2,4}γ_6$ | $t_{1,2,3}γ_6$ | $t_{0,1,5}β_6$ |
| $t_{0,2,5}β_6$ | $t_{0,3,4}γ_6$ | $t_{0,2,5}γ_6$ | $t_{0,2,4}γ_6$ | $t_{0,2,3}γ_6$ | $t_{0,1,5}γ_6$ | $t_{0,1,4}γ_6$ | $t_{0,1,3}γ_6$ |
| $t_{0,1,2}γ_6$ | $t_{0,1,2,3}β_6$ | $t_{0,1,2,4}β_6$ | $t_{0,1,3,4}β_6$ | $t_{0,2,3,4}β_6$ | $t_{1,2,3,4}γ_6$ | $t_{0,1,2,5}β_6$ | $t_{0,1,3,5}β_6$ |
| $t_{0,2,3,5}γ_6$ | $t_{0,2,3,4}γ_6$ | $t_{0,1,4,5}γ_6$ | $t_{0,1,3,5}γ_6$ | $t_{0,1,3,4}γ_6$ | $t_{0,1,2,5}γ_6$ | $t_{0,1,2,4}γ_6$ | $t_{0,1,2,3}γ_6$ |
| $t_{0,1,2,3,4}β_6$ | $t_{0,1,2,3,5}β_6$ | $t_{0,1,2,4,5}β_6$ | $t_{0,1,2,4,5}γ_6$ | $t_{0,1,2,3,5}γ_6$ | $t_{0,1,2,3,4}γ_6$ | $t_{0,1,2,3,4,5}γ_6$ |
