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.
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D<sub>8</sub>. 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 B<sub>8</sub> and C<sub>8</sub> simple Lie groups.
The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C<sub>8</sub> or [4,3<sup>6</sup>] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D<sub>8</sub> or [3<sup>5,1,1</sup>] Coxeter group.
Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length are all permutations of:
Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length are all permutations of:
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.
Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length are all permutations of: