Bitruncated cubic honeycomb

The bitruncated cubic honeycomb, or batch, is a convex noble uniform honeycomb. 4 truncated octahedra join at each vertex of this honeycomb. As the name suggests, it is the bitruncation of the cubic honeycomb, the medial stage in the series of truncations between a cubic honeycomb and its dual.

This honeycomb can be alternated into a bisnub cubic honeycomb, although it cannot be made uniform.

Before the discovery of the Weaire-Phelan structure, it was the most efficient known tiling of 3D Euclidean space.

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


 * $$\left(2\sqrt2i,\,±\frac{\sqrt2}{2}+2\sqrt2j,\,\sqert2+2\sqrt2k\right),$$

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

Representations
A bitruncated cubic honeycomb has the following Coxeter diagrams:


 * o4x3x4o (regular)
 * x3x3x *b4o (S4 symmetry)
 * x3x3x3x3*a (P4 symmetry, as omnitruncated tetrahedral honeycomb)
 * s4x3x4o (as alternated faceting)
 * x3x3x *b4s