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
Dodecahedral-icosahedral honeycomb rhombic disphenoid vertex figure. It has a Coxeter diagram . Dodecahedral-icosahedral honeycomb In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and icosidodecahedron cells, in a rhombicosidodecahedron vertex figure. Wide-angle perspective views: There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group : , , , , . The rectified dodecahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosidodecahedron, and rhombicosidodecahedron cells, in a cuboid vertex figure. It has a Coxeter diagram . The cyclotruncated dodecahedral-icosahedral honeycomb is a compact