Rectified 8-orthoplexes

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.

Rectified 8-orthoplex
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.

Related polytopes
The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
 * or

Alternate names

 * rectified octacross
 * rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.

Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,0,0,0,0,0,0)

Alternate names

 * birectified octacross
 * birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)

Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,±1,0,0,0,0,0)

Trirectified 8-orthoplex
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.

Alternate names

 * trirectified octacross
 * trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)

Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length $$\sqrt{2}$$ are all permutations of:
 * (±1,±1,±1,±1,0,0,0,0)