# Rectified 10-cubes

| 10-orthoplex | Rectified 10-orthoplex | Birectified 10-orthoplex | Trirectified 10-orthoplex |
| Quadirectified 10-orthoplex | Quadrirectified 10-cube | Trirectified 10-cube | Birectified 10-cube |
| Rectified 10-cube | 10-cube |
| Orthogonal projections in BC$_{10}$ Coxeter plane ||||
In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube.
There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex.
These polytopes are part of a family 1023 uniform 10-polytopes with BC$_{10}$ symmetry.

## Rectified 10-cube

Rectified 10-orthoplex :
| Type | uniform 10-polytope |
| Schläfli symbol | t$_1${3$^8$,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 46080 |
| Vertices | 5120 |
| Vertex figure | 8-simplex prism |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |

### Alternate names

* Rectified dekeract (Acronym rade) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

### Images

Orthographic projections :
| B$_{10}$ | B$_9$ | B$_8$ |
| [20] | [18] | [16] |
| B$_7$ | B$_6$ | B$_5$ |
| [14] | [12] | [10] |
| B$_4$ | B$_3$ | B$_2$ |
| [8] | [6] | [4] |
| A$_9$ | A$_9$ | A$_5$ |
| [10] | [10] | [6] |
| A$_7$ | A$_7$ | A$_3$ |
| [8] | [8] | [4] |

## Birectified 10-cube

Birectified 10-orthoplex :
| Type | uniform 10-polytope |
| Coxeter symbol | 0$_{711}$ |
| Schläfli symbol | t$_2${3$^8$,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 184320 |
| Vertices | 11520 |
| Vertex figure | {4}x{3$_6$} |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |

### Alternate names

* Birectified dekeract (Acronym brade) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,±1,±1,±1,±1,±1,±1,0,0)

### Images

Orthographic projections :
| B$_{10}$ | B$_9$ | B$_8$ |
| [20] | [18] | [16] |
| B$_7$ | B$_6$ | B$_5$ |
| [14] | [12] | [10] |
| B$_4$ | B$_3$ | B$_2$ |
| [8] | [6] | [4] |
| A$_9$ | A$_9$ | A$_5$ |
| [10] | [10] | [6] |
| A$_7$ | A$_7$ | A$_3$ |
| [8] | [8] | [4] |

## Trirectified 10-cube

Trirectified 10-orthoplex :
| Type | uniform 10-polytope |
| Schläfli symbol | t$_3${3$^8$,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 322560 |
| Vertices | 15360 |
| Vertex figure | {4,3}x{3$^5$} |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |

### Alternate names

* Tririrectified dekeract (Acronym trade) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,±1,±1,±1,±1,±1,0,0,0)

### Images

Orthographic projections :
| B$_{10}$ | B$_9$ | B$_8$ |
| [20] | [18] | [16] |
| B$_7$ | B$_6$ | B$_5$ |
| [14] | [12] | [10] |
| B$_4$ | B$_3$ | B$_2$ |
| [8] | [6] | [4] |
| A$_9$ | A$_9$ | A$_5$ |
| [10] | [10] | [6] |
| A$_7$ | A$_7$ | A$_3$ |
| [8] | [8] | [4] |

## Quadrirectified 10-cube

Quadrirectified 10-orthoplex :
| Type | uniform 10-polytope |
| Schläfli symbol | t$_4${3$^8$,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 322560 |
| Vertices | 13440 |
| Vertex figure | {4,3,3}x{3$^4$} |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |

### Alternate names

* Quadrirectified dekeract
* Quadrirectified decacross (Acronym terade) (Jonathan Bowers)

### Cartesian coordinates

Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,±1,±1,±1,±1,0,0,0,0)

### Images

Orthographic projections :
| B$_{10}$ | B$_9$ | B$_8$ |
| [20] | [18] | [16] |
| B$_7$ | B$_6$ | B$_5$ |
| [14] | [12] | [10] |
| B$_4$ | B$_3$ | B$_2$ |
| [8] | [6] | [4] |
| A$_9$ | A$_9$ | A$_5$ |
| [10] | [10] | [6] |
| A$_7$ | A$_7$ | A$_3$ |
| [8] | [8] | [4] |