# Runcinated 5-cubes

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

## Runcinated 5-cube

Runcinated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,3}${4,3,3,3} | t$_{0,3}${4,3,3,3} |
| Coxeter diagram | | |
| 4-faces | 202 | 10 80 80 32 |
| Cells | 1240 | 40 240 320 160 320 160 |
| Faces | 2160 | 240 960 640 320 |
| Edges | 1440 | 480+960 |
| Vertices | 320 | 320 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex | convex |

### Alternate names

* Small prismated penteract (Acronym: span) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:
    $\left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)$

### 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] |

## Runcitruncated 5-cube

Runcitruncated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,3}${4,3,3,3} | t$_{0,1,3}${4,3,3,3} |
| Coxeter-Dynkin diagrams | | |
| 4-faces | 202 | 10 80 80 32 |
| Cells | 1560 | 40 240 320 320 160 320 160 |
| Faces | 3760 | 240 960 320 960 640 640 |
| Edges | 3360 | 480+960+1920 |
| Vertices | 960 | 960 |
| Vertex figure | | |
| Coxeter group | B$_5$, [3,3,3,4] | B$_5$, [3,3,3,4] |
| Properties | convex | convex |

### Alternate names

* Runcitruncated penteract
* Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

### Construction and coordinates

The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:
    $\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)$

### 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] |

## Runcicantellated 5-cube

Runcicantellated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,2,3}${4,3,3,3} | t$_{0,2,3}${4,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 202 | 10 80 80 32 |
| Cells | 1240 | 40 240 320 320 160 160 |
| Faces | 2960 | 240 480 960 320 640 320 |
| Edges | 2880 | 960+960+960 |
| Vertices | 960 | 960 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex | convex |

### Alternate names

* Runcicantellated penteract
* Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:
    $\left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)$

### 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] |

## Runcicantitruncated 5-cube

Runcicantitruncated 5-cube :
| Type | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2,3}${4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 202 |
| Cells | 1560 |
| Faces | 4240 |
| Edges | 4800 |
| Vertices | 1920 |
| Vertex figure | Irregular 5-cell | Irregular 5-cell |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, isogonal |

### Alternate names

* Runcicantitruncated penteract
* Biruncicantitruncated pentacross
* great prismated penteract (gippin) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
    $\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+3{\sqrt {2}}\right)$

### 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

These polytopes are a part of a set of 31 uniform polytera 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$_5$ |