Hexagonal tiling

The honeycomb conjecture states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter.

The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is optimal.

This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds.

They have many potential applications, due to their high tensile strength and electrical properties.

In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures.

There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions.

The (h,k) represent the periodic repeat of one colored tile, counting hexagonal distances as h first, and k second.

The same counting is used in the Goldberg polyhedra, with a notation {p+,3}h,k, and can be applied to hyperbolic tilings for p > 6.

In the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling.

This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of a sequence that continues into the hyperbolic plane.

A chevron pattern has pmg (22*) symmetry, which is lowered to p1 (°) with 3 or 4 colored tiles.

There are 2 regular complex apeirogons, sharing the vertices of the hexagonal tiling.