Hexagonal tiling

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} (as a truncated triangular tiling). English mathematician John Conway called it a hextille. The internal angle of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. Hexagonal tiling is the densest way to arrange circles in two dimensions. 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. However, the less regular Weaire–Phelan structure is slightly better. This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised, known as carbon nanotubes. They have many potential applications, due to their high tensile strength and electrical properties. Silicene is similar. Chicken wire consists of a hexagonal lattice (often not regular) of wires. File:Kissing-2d.svg|The densest [[circle packing]] is arranged like the hexagons in this tiling File:Chicken Wire close-up.jpg|[[Chicken wire]] fencing File:Graphene xyz.jpg|[[Graphene]] File:Carbon nanotube zigzag povray.PNG|A [[carbon nanotube]] can be seen as a hexagon tiling on a [[Cylinder (geometry)|cylindrical]] surface File:Tile (AM 1955.117-1).jpg|alt=Hexagonal tile with blue bird and flowers|Hexagonal Persian tile c.1955 The hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest sphere packings in three dimensions.