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
Order-5 cubic honeycomb and pentagonal prism cells, with an octahedron vertex figure. It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces: It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t{4,3,4}: The runcitruncated order-5 cubic honeycomb or runcicantellated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with a quad-pyramid vertex figure. It is similar to the Euclidean (order-4) runcitruncated cubic honeycomb, t{4,3,4}: The omnitruncated order-5 cubic honeycomb or omnitruncated order-4 dodecahedral honeycomb has Coxeter diagram . It is similar to the Euclidean (order-4) omnitruncated cubic honeycomb, t{4,3,4}: In 3-dimensional hyperbolic geometry, the alternated