# Truncated 5-cubes

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

## Truncated 5-cube

Truncated 5-cube :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | t{4,3,3,3} | t{4,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 42 | 10 32 |
| Cells | 200 | 40 160 |
| Faces | 400 | 80 320 |
| Edges | 400 | 80 320 |
| Vertices | 160 | 160 |
| Vertex figure | ( )v{3,3} | ( )v{3,3} |
| Coxeter group | B$_5$, [3,3,3,4], order 3840 | B$_5$, [3,3,3,4], order 3840 |
| Properties | convex | convex |

### Alternate names

* Truncated penteract (Acronym: tan) (Jonathan Bowers)

### Construction and coordinates

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at $1/({\sqrt {2}}+2)$ of the edge length. A regular 5-cell is formed at each truncated vertex.
The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:
    $\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)$

### Images

The truncated 5-cube is constructed by a truncation applied to the 5-cube. 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] |

### Related polytopes

The truncated 5-cube, is fourth in a sequence of truncated hypercubes:
Truncated hypercubes :
| Image | | | | | | | | ... |
| Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube | ... |
| Coxeter diagram | | | | | | | | ... |
| Vertex figure | ( )v( ) | ( )v{ } | ( )v{3} | ( )v{3,3} | ( )v{3,3,3} | ( )v{3,3,3,3} | ( )v{3,3,3,3,3} |

## Bitruncated 5-cube

Bitruncated 5-cube :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | 2t{4,3,3,3} | 2t{4,3,3,3} |
| Coxeter-Dynkin diagrams | | |
| 4-faces | 42 | 10 32 |
| Cells | 280 | 40 160 80 |
| Faces | 720 | 80 320 320 |
| Edges | 800 | 320 480 |
| Vertices | 320 | 320 |
| Vertex figure | { }v{3} | { }v{3} |
| Coxeter groups | B$_5$, [3,3,3,4], order 3840 | B$_5$, [3,3,3,4], order 3840 |
| Properties | convex | convex |

### Alternate names

* Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

### Construction and coordinates

The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at ${\sqrt {2}}$ of the edge length.
The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
    $\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)$

### Images

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

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:
Bitruncated hypercubes :
| Image | | | | | | | ... |
| Name | Bitruncated cube | Bitruncated tesseract | Bitruncated 5-cube | Bitruncated 6-cube | Bitruncated 7-cube | Bitruncated 8-cube | ... |
| Coxeter | | | | | | | ... |
| Vertex figure | ( )v{ } | { }v{ } | { }v{3} | { }v{3,3} | { }v{3,3,3} | { }v{3,3,3,3} |

## Related polytopes

This polytope is one of 31 uniform 5-polytope 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$_5$ |