# Cantellated 5-orthoplexes

| 5-orthoplex | Cantellated 5-orthoplex | Bicantellated 5-cube | Cantellated 5-cube |
| 5-cube | Cantitruncated 5-orthoplex | Bicantitruncated 5-cube | Cantitruncated 5-cube |
| Orthogonal projections in B$_5$ Coxeter plane ||||
In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.
There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.

## Cantellated 5-orthoplex

Cantellated 5-orthoplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | rr{3,3,3,4} rr{3,3,3$^{1,1}$} | rr{3,3,3,4} rr{3,3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | | |
| 4-faces | 82 | 10 40 32 |
| Cells | 640 | 80 160 320 80 |
| Faces | 1520 | 640 320 480 80 |
| Edges | 1200 | 960 240 |
| Vertices | 240 | 240 |
| Vertex figure | Square pyramidal prism | Square pyramidal prism |
| Coxeter group | B$_5$, [4,3,3,3], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 | B$_5$, [4,3,3,3], order 3840 D$_5$, [3$^{2,1,1}$], order 1920 |
| Properties | convex | convex |

### Alternate names

* Cantellated 5-orthoplex
* Bicantellated 5-demicube
* Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)

### Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:
    (0,0,1,1,2)

### Images

The cantellated 5-orthoplex is constructed by a cantellation 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] |

## Cantitruncated 5-orthoplex

Cantitruncated 5-orthoplex :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | tr{3,3,3,4} tr{3,3$^{1,1}$} | tr{3,3,3,4} tr{3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | | |
| 4-faces | 82 | 10 40 32 |
| Cells | 640 | 80 160 320 80 |
| Faces | 1520 | 640 320 480 80 |
| Edges | 1440 | 960 240 240 |
| Vertices | 480 | 480 |
| Vertex figure | Square pyramidal pyramid | Square pyramidal pyramid |
| 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

* Cantitruncated pentacross
* Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of
    (±3,±2,±1,0,0)

### 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

These polytopes are from a set 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$_5$ |