Honeycombs - their variety, topology and symmetry

The double-layer honeycomb with hexagonal cells, three rhombic faces between the two layers and p3m1 layer space-group symmetry, used universally by honeybees, is often considered to be the most efficient (from the point of view of wax economy) and the only honeycomb manufactured by bees. However, another variant of a symmetric and periodic double-layer hexagonal honeycomb with two hexagons and two rhombi between the two layers and slightly better wax economy was discovered theoretically in 1964 by Fejes Tóth and found in nature some years later. The present work shows that there is yet another possibility, with the interface formed by one hexagon and two quadrangles, in addition to the trivial case with flat hexagonal cell bottoms and very poor wax economy. Moreover, we demonstrate that the geometry of the Fejes Tóth honeycomb can be optimized for even better wax economy. All the theoretical honeycomb types are derived using the principle of Dirichlet-domain construction and shown to have more and less symmetric variants. Wax economy is calculated for each case, confirming that indeed the modified Fejes Tóth honeycomb is the most efficient, while the trivial flat-bottom case is the least.