# Rectified 9-simplexes

| 9-simplex | Rectified 9-simplex |
| Birectified 9-simplex | Trirectified 9-simplex | Quadrirectified 9-simplex |
| Orthogonal projections in A$_9$ Coxeter plane ||
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.
These polytopes are part of a family of 271 uniform 9-polytopes with A$_9$ symmetry.
There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.

## Rectified 9-simplex

Rectified 9-simplex :
| Type | uniform 9-polytope |
| Schläfli symbol | t$_1${3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 8-faces | 20 |
| 7-faces | 135 |
| 6-faces | 480 |
| 5-faces | 1050 |
| 4-faces | 1512 |
| Cells | 1470 |
| Faces | 960 |
| Edges | 360 |
| Vertices | 45 |
| Vertex figure | 8-simplex prism |
| Petrie polygon | decagon |
| Coxeter groups | A$_9$, [3,3,3,3,3,3,3,3] |
| Properties | convex |
The rectified 9-simplex is the vertex figure of the 10-demicube.

### Alternate names

* Rectified decayotton (reday) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.

### Images

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

## Birectified 9-simplex

Birectified 9-simplex :
| Type | uniform 9-polytope |
| Schläfli symbol | t$_2${3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1260 |
| Vertices | 120 |
| Vertex figure | {3}×{3,3,3,3,3} |
| Coxeter groups | A$_9$, [3,3,3,3,3,3,3,3] |
| Properties | convex |
This polytope is the vertex figure for the 1$_{62}$ honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 9-dimensional sphere packing.

### Alternate names

* Birectified decayotton (breday) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.

### Images

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

## Trirectified 9-simplex

Trirectified 9-simplex :
| Type | uniform 9-polytope |
| Schläfli symbol | t$_3${3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {3,3}×{3,3,3,3} |
| Coxeter groups | A$_9$, [3,3,3,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Trirectified decayotton (treday) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.

### Images

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

## Quadrirectified 9-simplex

Quadrirectified 9-simplex :
| Type | uniform 9-polytope |
| Schläfli symbol | t$_4${3,3,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | or |
| 8-faces | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {3,3,3}×{3,3,3} |
| Coxeter groups | A$_9$×2, [[3$^8$]] |
| Properties | convex |

### Alternate names

* Quadrirectified decayotton
* Icosayotton (icoy) (Jonathan Bowers)

### Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.

### Images

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