# Rectified 10-orthoplexes

| 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 A$_{10}$ Coxeter plane ||||
In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex.
There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex.
These polytopes are part of a family 1023 uniform 10-polytopes with BC$_{10}$ symmetry.

## Rectified 10-orthoplex

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 | 2880 |
| Vertices | 180 |
| Vertex figure | 8-orthoplex prism |
| Petrie polygon | icosagon |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |
In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex.

## Rectified 10-orthoplex

The rectified 10-orthoplex is the vertex figure for the demidekeractic honeycomb.
    or

### Alternate names

* rectified decacross (Acronym rake) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C$_{10}$ or [4,3$^8$] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D$_{10}$ or [3$^{7,1,1}$] Coxeter group.

### Cartesian coordinates

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

### Root vectors

Its 180 vertices represent the root vectors of the simple Lie group D$_{10}$. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B$_{10}$.

### 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-orthoplex

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

### Alternate names

* Birectified decacross

### Cartesian coordinates

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

### 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] |

## Trirectified 10-orthoplex

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 | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |

### Alternate names

* Trirectified decacross (Acronym trake) (Jonathan Bowers)

### Cartesian coordinates

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

### 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] |

## Quadrirectified 10-orthoplex

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 | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | C$_{10}$, [4,3$^8$] D$_{10}$, [3$^{7,1,1}$] |
| Properties | convex |

### Alternate names

* Quadrirectified decacross (Acronym brake) (Jonthan Bowers)

### Cartesian coordinates

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

### 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] |