# Truncated 5-orthoplexes

| 5-orthoplex | Truncated 5-orthoplex | Bitruncated 5-orthoplex |
| 5-cube | Truncated 5-cube | Bitruncated 5-cube |
| Orthogonal projections in B$_5$ Coxeter plane |||
In five-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.
There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube.

## Truncated 5-orthoplex

Truncated 5-orthoplex :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | t{3,3,3,4} t{3,3$^{1,1}$} | t{3,3,3,4} t{3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | | |
| 4-faces | 42 | 10 32 |
| Cells | 240 | 160 80 |
| Faces | 400 | 320 80 |
| Edges | 280 | 240 40 |
| Vertices | 80 | 80 |
| Vertex figure | ( )v{3,4} | ( )v{3,4} |
| Coxeter groups | B$_5$, [3,3,3,4], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 | B$_5$, [3,3,3,4], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 |
| Properties | convex | convex |

### Alternate names

* Truncated pentacross
* Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
    (±2,±1,0,0,0)

### Images

The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex. All edges are shortened, and two new vertices are added on each original edge.
orthographic projections :
| Coxeter plane | B$_5$ | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ |
| Graph | | | |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B$_2$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [4] | [4] |

## Bitruncated 5-orthoplex

Bitruncated 5-orthoplex :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | 2t{3,3,3,4} 2t{3,3$^{1,1}$} | 2t{3,3,3,4} 2t{3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | | |
| 4-faces | 42 | 10 32 |
| Cells | 280 | 40 160 80 |
| Faces | 720 | 320 320 80 |
| Edges | 720 | 480 240 |
| Vertices | 240 | 240 |
| Vertex figure | { }v{4} | { }v{4} |
| Coxeter groups | B$_5$, [3,3,3,4], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 | B$_5$, [3,3,3,4], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 |
| Properties | convex | convex |
The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb.

### Alternate names

* Bitruncated pentacross
* Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of
    (±2,±2,±1,0,0)

### Images

The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex.
orthographic projections :
| Coxeter plane | B$_5$ | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ |
| Graph | | | |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B$_2$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [4] | [4] |

## Related polytopes

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
B5 polytopes :
| β$_5$ | t$_1$β$_5$ | t$_2$γ$_5$ | t$_1$γ$_5$ | γ$_5$ | t$_{0,1}$β$_5$ | t$_{0,2}$β$_5$ | t$_{1,2}$β$_5$ |
| t$_{0,3}$β$_5$ | t$_{1,3}$γ$_5$ | t$_{1,2}$γ$_5$ | t$_{0,4}$γ$_5$ | t$_{0,3}$γ$_5$ | t$_{0,2}$γ$_5$ | t$_{0,1}$γ$_5$ | t$_{0,1,2}$β$_5$ |
| t$_{0,1,3}$β$_5$ | t$_{0,2,3}$β$_5$ | t$_{1,2,3}$γ$_5$ | t$_{0,1,4}$β$_5$ | t$_{0,2,4}$γ$_5$ | t$_{0,2,3}$γ$_5$ | t$_{0,1,4}$γ$_5$ | t$_{0,1,3}$γ$_5$ |
| t$_{0,1,2}$γ$_5$ | t$_{0,1,2,3}$β$_5$ | t$_{0,1,2,4}$β$_5$ | t$_{0,1,3,4}$γ$_5$ | t$_{0,1,2,4}$γ$_5$ | t$_{0,1,2,3}$γ$_5$ | t$_{0,1,2,3,4}$γ$_5$ |