In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.
The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.
There are two Coxeter groups associated with the rectified hexacross, one with the C<sub>6</sub> or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D<sub>6</sub> or [3<sup>3,1,1</sup>] Coxeter group.
Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:
The 60 vertices represent the root vectors of the simple Lie group D<sub>6</sub>. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B<sub>6</sub> and C<sub>6</sub> simple Lie groups.
The 60 roots of D<sub>6</sub> can be geometrically folded into H<sub>3</sub> (Icosahedral symmetry), as to , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:
The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.
Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:
It can also be projected into 3D-dimensions as â , a dodecahedron envelope.
These polytopes are in a family of 63 uniform 6-polytopes generated from the B<sub>6</sub> Coxeter plane, including the regular 6-cube and 6-orthoplex.