Honeycomb honeycomb conjecture, this was given by Jan Brożek and proved much later by Thomas Hales. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. A second reason, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency. The closed ends of the honeycomb cells are
Bitruncated cubic honeycomb the Coxeter–Dynkin diagrams: This honeycomb can be alternated, creating pyritohedral icosahedra from the truncated octahedra with disphenoid tetrahedral cells created in the gaps. There are three constructions from three related Coxeter-Dynkin diagrams: , , and . These have symmetry [4,3,4], [4,(3)] and [3] respectively. The first and last symmetry can be doubled as <nowiki></nowiki>4,3,4 and [[3]]. The dual honeycomb is made of cells called ten-of-diamonds decahedra. This honeycomb is represented in the boron atoms of the α-rhombihedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice. Bitruncated cubic honeycomb The bitruncated cubic honeycomb is