Rectified cubic honeycomb

The rectified cubic honeycomb, or rich, is a convex uniform honeycomb. 2 octahedra and 4 cuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the rectification of the cubic honeycomb. It is also the rectification of the tetrahedral-octahedral honeycomb.

Vertex coordinates
The vertices of a rectified cubic honeycomb of edge length 1 are given by all permutations of:


 * $$\left(\sqrt2i,\,±\frac{\sqrt2}{2}+\sqrt2j,\,±\frac{\sqrt2}{2}+\sqrt2k\right),$$

where i, j, and k range over the integers.

Representations
A rectified cubic honeycomb has the following Coxeter diagrams:


 * (full symmetry)
 * (S4 symmetry, as rectified tetrahedral-octahedral honeycomb)
 * (S4 symmetry)
 * (P4 symmetry, as rectified cyclotetrahedral honeycomb)
 * (as alternated faceting)
 * (as alternated faceting)
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