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.

Alternate names

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

Cartesian coordinates

for the vertices of a rectified 8-orthoplex, centered at the origin, edge length are all permutations of:

Images

Birectified 8-orthoplex

Alternate names

for the vertices of a birectified 8-orthoplex, centered at the origin, edge length are all permutations of:

Images

Trirectified 8-orthoplex

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

Alternate names

for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length are all permutations of:

Images