# Cantellated 5-cubes

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

## Cantellated 5-cube

Cantellated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | rr{4,3,3,3} = $r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$ | rr{4,3,3,3} = $r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | = | = |
| 4-faces | 122 | 10 80 32 |
| Cells | 680 | 40 320 160 160 |
| Faces | 1520 | 80 480 320 640 |
| Edges | 1280 | 320+960 |
| Vertices | 320 | 320 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, uniform | convex, uniform |

### Alternate names

* Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

### Coordinates

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

### 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] |

## Bicantellated 5-cube

Bicantellated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbols | 2rr{4,3,3,3} = $r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ r{3$^{2,1,1}$} = $r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$ | 2rr{4,3,3,3} = $r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ r{3$^{2,1,1}$} = $r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | = | = |
| 4-faces | 122 | 10 80 32 |
| Cells | 840 | 40 240 160 320 80 |
| Faces | 2160 | 240 320 960 320 320 |
| Edges | 1920 | 960+960 |
| Vertices | 480 |
| Vertex figure | | |
| Coxeter groups | B$_5$, [3,3,3,4] D$_5$, [3$^{2,1,1}$] | B$_5$, [3,3,3,4] D$_5$, [3$^{2,1,1}$] |
| Properties | convex, uniform | convex, uniform |
In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

### Alternate names

* Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
* Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:
    (0,1,1,2,2)

### 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] |

## Cantitruncated 5-cube

Cantitruncated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | tr{4,3,3,3} = $t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$ | tr{4,3,3,3} = $t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | = | = |
| 4-faces | 122 | 10 80 32 |
| Cells | 680 | 40 320 160 160 |
| Faces | 1520 | 80 480 320 640 |
| Edges | 1600 | 320+320+960 |
| Vertices | 640 | 640 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, uniform | convex, uniform |

### Alternate names

* Tricantitruncated 5-orthoplex / tricantitruncated pentacross
* Great rhombated penteract (girn) (Jonathan Bowers)

### Coordinates

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

### 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] |

### Related polytopes

It is third in a series of cantitruncated hypercubes:
Petrie polygon projections :
| Truncated cuboctahedron | Cantitruncated tesseract | Cantitruncated 5-cube | Cantitruncated 6-cube | Cantitruncated 7-cube | Cantitruncated 8-cube |

## Bicantitruncated 5-cube

Bicantitruncated 5-cube :
| Type | uniform 5-polytope | uniform 5-polytope |
| Schläfli symbol | 2tr{3,3,3,4} = $t\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ t{3$^{2,1,1}$} = $t\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$ | 2tr{3,3,3,4} = $t\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ t{3$^{2,1,1}$} = $t\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | = | = |
| 4-faces | 122 | 10 80 32 |
| Cells | 840 | 40 240 160 320 80 |
| Faces | 2160 | 240 320 960 320 320 |
| Edges | 2400 | 960+480+960 |
| Vertices | 960 | 960 |
| Vertex figure | | |
| Coxeter groups | B$_5$, [3,3,3,4] D$_5$, [3$^{2,1,1}$] | B$_5$, [3,3,3,4] D$_5$, [3$^{2,1,1}$] |
| Properties | convex, uniform | convex, uniform |

### Alternate names

* Bicantitruncated penteract
* Bicantitruncated pentacross
* Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of
    (±3,±3,±2,±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] |

## 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$_5$ |