Rectified 6-orthoplexes

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.

Rectified 6-orthoplex
The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.
 * or

Alternate names

 * rectified hexacross
 * rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

Construction
There are two Coxeter groups associated with the rectified hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group.

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

Root vectors
The 60 vertices represent the root vectors of the simple Lie group D6. 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 B6 and C6 simple Lie groups.

The 60 roots of D6 can be geometrically folded into H3 (Icosahedral symmetry), as to, creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:

Birectified 6-orthoplex
The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.

Alternate names

 * birectified hexacross
 * birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

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

Images
It can also be projected into 3D-dimensions as →, a dodecahedron envelope.

Related polytopes
These polytopes are a part a family of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.