# Rectified 9-orthoplexes

| 9-orthoplex | Rectified 9-orthoplex | Birectified 9-orthoplex | Trirectified 9-orthoplex | Quadrirectified 9-cube |
| Trirectified 9-cube | Birectified 9-cube | Rectified 9-cube | 9-cube |
| 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-orthoplex.
There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.
These polytopes are part of a family 511 uniform 9-polytopes with BC$_9$ symmetry.

## Rectified 9-orthoplex

Rectified 9-orthoplex :
| Type | uniform 9-polytope |
| Schläfli symbol | t$_1${3$^7$,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2016 |
| Vertices | 144 |
| Vertex figure | 7-orthoplex prism |
| Petrie polygon | octakaidecagon |
| Coxeter groups | C$_9$, [4,3$^7$] D$_9$, [3$^{6,1,1}$] |
| Properties | convex |
The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.
    or

### Alternate names

* rectified enneacross (Acronym riv) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C$_9$ or [4,3$^7$] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D$_9$ or [3$^{6,1,1}$] Coxeter group.

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:
    (±1,±1,0,0,0,0,0,0,0)

#### Root vectors

Its 144 vertices represent the root vectors of the simple Lie group D$_9$. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B$_9$ and C$_9$ simple Lie groups.

### Images

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

## Birectified 9-orthoplex
### Alternate names

* Rectified 9-demicube
* Birectified enneacross (Acronym brav) (Jonathan Bowers)

### Images

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

## Trirectified 9-orthoplex
### Alternate names

* trirectified enneacross (Acronym tarv) (Jonathan Bowers)

### Images

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