# Runcinated 5-orthoplexes

| 5-orthoplex | Runcinated 5-orthoplex | Runcinated 5-cube |
| Runcitruncated 5-orthoplex | Runcicantellated 5-orthoplex | Runcicantitruncated 5-orthoplex |
| Runcitruncated 5-cube | Runcicantellated 5-cube | Runcicantitruncated 5-cube |
| Orthogonal projections in B$_5$ Coxeter plane |||
In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.
There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.

## Runcinated 5-orthoplex

Runcinated 5-orthoplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,3}${3,3,3,4} | t$_{0,3}${3,3,3,4} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 162 |
| Cells | 1200 |
| Faces | 2160 |
| Edges | 1440 | 1440 |
| Vertices | 320 | 320 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] D$_5$ [3$^{2,1,1}$] | B$_5$ [4,3,3,3] D$_5$ [3$^{2,1,1}$] |
| Properties | convex | convex |

### Alternate names

* Runcinated pentacross
* Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)

### Coordinates

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

### 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] |

## Runcitruncated 5-orthoplex

Runcitruncated 5-orthoplex :
| Type | uniform 5-polytope |
| Schläfli symbol | t$_{0,1,3}${3,3,3,4} t$_{0,1,3}${3,3$^{1,1}$} |
| Coxeter-Dynkin diagrams | |
| 4-faces | 162 |
| Cells | 1440 |
| Faces | 3680 |
| Edges | 3360 |
| Vertices | 960 |
| Vertex figure | |
| Coxeter groups | B$_5$, [3,3,3,4] D$_5$, [3$^{2,1,1}$] |
| Properties | convex |

### Alternate names

* Runcitruncated pentacross
* Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
    (±3,±2,±1,±1,0)

### 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] |

## Runcicantellated 5-orthoplex

Runcicantellated 5-orthoplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,2,3}${3,3,3,4} t$_{0,2,3}${3,3,3$^{1,1}$} | t$_{0,2,3}${3,3,3,4} t$_{0,2,3}${3,3,3$^{1,1}$} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 162 |
| Cells | 1200 |
| Faces | 2960 |
| Edges | 2880 |
| Vertices | 960 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] D$_5$ [3$^{2,1,1}$] | B$_5$ [4,3,3,3] D$_5$ [3$^{2,1,1}$] |
| Properties | convex | convex |

### Alternate names

* Runcicantellated pentacross
* Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)

### Coordinates

The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of:
    (0,1,2,2,3)

### 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] |

## Runcicantitruncated 5-orthoplex

Runcicantitruncated 5-orthoplex :
| Type | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2,3}${3,3,3,4} |
| Coxeter-Dynkin diagram | |
| 4-faces | 162 |
| Cells | 1440 |
| Faces | 4160 |
| Edges | 4800 |
| Vertices | 1920 |
| Vertex figure | Irregular 5-cell | Irregular 5-cell |
| Coxeter groups | B$_5$ [4,3,3,3] D$_5$ [3$^{2,1,1}$] | B$_5$ [4,3,3,3] D$_5$ [3$^{2,1,1}$] |
| Properties | convex, isogonal |

### Alternate names

* Runcicantitruncated pentacross
* Great prismated triacontiditeron (gippit) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of √2 are given by all permutations of coordinates and sign of:
    $\left(0,1,2,3,4\right)$

### 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] |

### Snub 5-demicube

The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [3$^{2,1,1}$]$^+$ or [4,(3,3,3)$^+$], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.

## 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$_5$ |