# Stericated 5-cubes

| 5-cube | Stericated 5-cube | Steritruncated 5-cube |
| Stericantellated 5-cube | Steritruncated 5-orthoplex | Stericantitruncated 5-cube |
| Steriruncitruncated 5-cube | Stericantitruncated 5-orthoplex | Omnitruncated 5-cube |
| Orthogonal projections in B$_5$ Coxeter plane |||
In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.
There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the sterirunci​cantitruncated​ 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.

## Stericated 5-cube

Stericated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | 2r2r{4,3,3,3} | 2r2r{4,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 242 |
| Cells | 800 |
| Faces | 1040 |
| Edges | 640 |
| Vertices | 160 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex | convex |

### Alternate names

* Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
* Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
* Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)

### Coordinates

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

### Images

The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

### Dissections

The stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.
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] |

## Steritruncated 5-cube

Steritruncated 5-cube :
| Type | uniform 5-polytope |
| Schläfli symbol | t$_{0,1,4}${4,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 4-faces | 242 |
| Cells | 1600 |
| Faces | 2960 |
| Edges | 2240 |
| Vertices | 640 |
| Vertex figure | |
| Coxeter groups | B$_5$, [3,3,3,4] |
| Properties | convex |

### Alternate names

* Steritruncated penteract
* Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)

### Construction and coordinates

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

### 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] |

## Stericantellated 5-cube

Stericantellated 5-cube :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,2,4}${4,3,3,3} | t$_{0,2,4}${4,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 242 |
| Cells | 2080 |
| Faces | 4720 |
| Edges | 3840 |
| Vertices | 960 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex | convex |

### Alternate names

* Stericantellated penteract
* Stericantellated 5-orthoplex, stericantellated pentacross
* Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)

### Coordinates

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

### 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] |

## Stericantitruncated 5-cube

Stericantitruncated 5-cube :
| Type | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2,4}${4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 242 |
| Cells | 2400 |
| Faces | 6000 |
| Edges | 5760 |
| Vertices | 1920 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, isogonal |

### Alternate names

* Stericantitruncated penteract
* Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
* Celligreatorhombated penteract (cogrin) (Jonathan Bowers)

### Coordinates

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

### 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] |

## Steriruncitruncated 5-cube

Steriruncitruncated 5-cube :
| Type | Uniform 5-polytope |
| Schläfli symbol | 2t2r{4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 242 |
| Cells | 2160 |
| Faces | 5760 |
| Edges | 5760 |
| Vertices | 1920 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, isogonal |

### Alternate names

* Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
* Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)

### Coordinates

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

### 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] |

## Steritruncated 5-orthoplex

Steritruncated 5-orthoplex :
| Type | uniform 5-polytope |
| Schläfli symbol | t$_{0,1,4}${3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 4-faces | 242 |
| Cells | 1520 |
| Faces | 2880 |
| Edges | 2240 |
| Vertices | 640 |
| Vertex figure | |
| Coxeter group | B$_5$, [3,3,3,4] |
| Properties | convex |

### Alternate names

* Steritruncated pentacross
* Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of
    $\left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)$

### 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] |

## Stericantitruncated 5-orthoplex

Stericantitruncated 5-orthoplex :
| Type | Uniform 5-polytope |
| Schläfli symbol | t$_{0,2,3,4}${4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 242 |
| Cells | 2320 |
| Faces | 5920 |
| Edges | 5760 |
| Vertices | 1920 |
| Vertex figure | | |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, isogonal |

### Alternate names

* Stericantitruncated pentacross
* Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)

### Coordinates

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

### 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] |

## Omnitruncated 5-cube

Omnitruncated 5-cube :
| Type | Uniform 5-polytope |
| Schläfli symbol | tr2r{4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 242 |
| Cells | 2640 |
| Faces | 8160 |
| Edges | 9600 |
| Vertices | 3840 |
| Vertex figure | irr. {3,3,3} | irr. {3,3,3} |
| Coxeter group | B$_5$ [4,3,3,3] | B$_5$ [4,3,3,3] |
| Properties | convex, isogonal |

### Alternate names

* Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
* Omnitruncated penteract
* Omnitruncated triacontiditeron / omnitruncated pentacross
* Great cellated penteractitriacontiditeron (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of an omnitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
    $\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+4{\sqrt {2}}\right)$

### 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] |

### Full snub 5-cube

The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3]$^+$, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 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$_5$ |