Rectified 9-orthoplexes

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 BC9 symmetry.

Rectified 9-orthoplex
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 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 $$\sqrt{2}$$ 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 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.

Alternate names

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

Alternate names

 * trirectified enneacross (Acronym tarv) (Jonathan Bowers)