# Rectified 10-simplexes

| 10-simplex | Rectified 10-simplex | Birectified 10-simplex |
| Trirectified 10-simplex | Quadrirectified 10-simplex |
| Orthogonal projections in A$_9$ Coxeter plane |||
In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex.
These polytopes are part of a family of 527 uniform 10-polytopes with A$_{10}$ symmetry.
There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.

## Rectified 10-simplex

Rectified 10-simplex :
| Type | uniform polyxennon |
| Schläfli symbol | t$_1${3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 9-faces | 22 |
| 8-faces | 165 |
| 7-faces | 660 |
| 6-faces | 1650 |
| 5-faces | 2772 |
| 4-faces | 3234 |
| Cells | 2640 |
| Faces | 1485 |
| Edges | 495 |
| Vertices | 55 |
| Vertex figure | 9-simplex prism |
| Petrie polygon | decagon |
| Coxeter groups | A$_{10}$, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |
The rectified 10-simplex is the vertex figure of the 11-demicube.

### Alternate names

* Rectified hendecaxennon (Acronym ru) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_{10}$ | A$_9$ | A$_8$ |
| Graph | | | |
| Dihedral symmetry | [11] | [10] | [9] |
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Birectified 10-simplex

Birectified 10-simplex :
| Type | uniform 9-polytope |
| Schläfli symbol | t$_2${3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1980 |
| Vertices | 165 |
| Vertex figure | {3}x{3,3,3,3,3,3} |
| Coxeter groups | A$_{10}$, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Birectified hendecaxennon (Acronym bru) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_{10}$ | A$_9$ | A$_8$ |
| Graph | | | |
| Dihedral symmetry | [11] | [10] | [9] |
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Trirectified 10-simplex

Trirectified 10-simplex :
| Type | uniform polyxennon |
| Schläfli symbol | t$_3${3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4620 |
| Vertices | 330 |
| Vertex figure | {3,3}x{3,3,3,3,3} |
| Coxeter groups | A$_{10}$, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Trirectified hendecaxennon (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_{10}$ | A$_9$ | A$_8$ |
| Graph | | | |
| Dihedral symmetry | [11] | [10] | [9] |
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Quadrirectified 10-simplex

Quadrirectified 10-simplex :
| Type | uniform polyxennon |
| Schläfli symbol | t$_4${3,3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 6930 |
| Vertices | 462 |
| Vertex figure | {3,3,3}x{3,3,3,3} |
| Coxeter groups | A$_{10}$, [3,3,3,3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_{10}$ | A$_9$ | A$_8$ |
| Graph | | | |
| Dihedral symmetry | [11] | [10] | [9] |
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |