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. Construction. There are two Coxeter  groups associated with the "rectified hexacross", one with the C or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D or  Coxeter group. Cartesian coordinates. Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length formula_1 are all permutations of: Root vectors. The 60 vertices represent the root vectors of the simple Lie group D. 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 and C simple Lie groups. The 60 roots of D can be geometrically folded into H (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. Cartesian coordinates. Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length formula_1 are all permutations of: 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 B Coxeter plane, including the regular 6-cube or 6-orthoplex.