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 t{4,5} with truncated square and pentagonal faces: It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, with octahedral cells at the truncated vertices. Same as Bitruncated order-4 dodecahedral honeycomb The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron cells, with a wedge vertex figure. It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}: The cantitruncated order-5 cubic honeycomb, , has rhombicuboctahedron and icosidodecahedron cells, with a mirrored sphenoid vertex figure. It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}: The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron,