Rectified 9-orthoplexes
From Wikipedia, the free encyclopedia
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.
 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 A9 Coxeter plane | ||||
|---|---|---|---|---|
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 BC9 symmetry.
Rectified 9-orthoplex
| Rectified 9-orthoplex | |
|---|---|
| Type | uniform 9-polytope |
| Schläfli symbol | t1{37,4} |
| 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 | C9, [4,37] D9, [36,1,1] |
| Properties | convex |
The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.
or 
Alternate names
- Rectified enneacross (Acronym: riv) (Jonathan Bowers)[1]
Construction
There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or [4,37] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or [36,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length 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 D9. 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 B9 and C9 simple Lie groups.
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
| — | — | — | |||
| [8] | [6] | [4] | |||
Birectified 9-orthoplex
Alternate names
- Rectified 9-demicube
- Birectified enneacross (Acronym: brav) (Jonathan Bowers)[2]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
| — | — | — | |||
| [8] | [6] | [4] | |||
Trirectified 9-orthoplex
Alternate names
- Trirectified enneacross (Acronym: tarv) (Jonathan Bowers)[3]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
| — | — | — | |||
| [8] | [6] | [4] | |||