Weaire-Phelan structure

The Weaire-Phelan structure is a convex honeycomb that consists of pyritohedra and tetragonal disphenoidal symmetric order-6 truncated hexagonal trapezohedra, meeting at two vertex types. It cannot be made CRF.

It is conjectured to be the most efficient honeycomb of three-dimensional space using cells of equal volume, disproving the earlier hypothesis that the bitruncated cubic honeycomb is the most efficient.

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 order-6 truncated hexagonal trapezohedra, 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

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