Weaire-Phelan structure

The Weaire-Phelan structure is a convex honeycomb in 3D Euclidean space, which consists of 2N pyritohedra and 6N partially-truncated hexagonal antitegums, meeting at two vertex types. However, it cannot be made CRF.

As of this writing, it is the most efficient known tiling of 3D space.

Coordinates
The centers of the structure's cells are given by

$$(4i,4j,4k)$$ and $$(2+4i,2+4j,2+4k)$$

for the pyritohedra, and all even permutations of

$$(2+2i,1+2j,2k)$$

for the partially-truncated hexagonal antitegums, in which $$\{i,j,k\}\in\mathbb{Z}$$.

If $$\{i,j,k\}\in\mathbb{Z}$$, the structure's vertices are given by

$$\biggl(4i\pm\frac{2\sqrt[3]{2}}{3},4j\pm\frac{2\sqrt[3]{2}}{3},4k\pm\frac{2\sqrt[3]{2}}{3}\biggr)$$ and all even permuations of

$$\biggl(4i,4j\pm\frac{\sqrt[3]{2}}{2},4k\pm\sqrt[3]{2}\biggr)$$ from one pyritohedral orientation, as well as

$$\biggl(4i\pm\frac{6+2\sqrt[3]{2}}{3},4j\pm\frac{6+2\sqrt[3]{2}}{3},4k\pm\frac{6+2\sqrt[3]{2}}{3},\biggr)$$ and all even permutations of

$$\biggl(4i,4j\pm\sqrt[3]{2},4k\pm\frac{\sqrt[3]{2}}{2}\biggr)$$ for the alternate pyritohedral orientation.

A few vertices are missed by the pyritohedra, which are given by all even permutations of

$$(2+4i,1+2j,4k)$$.