Алтыбұрышты мозаика және оның геометриялық қасиеттері

Regular tiling of a two dimensional space

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. 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. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, with the face centered cubic being the more regular of the two. Pure copper, amongst other materials, forms a face centered cubic lattice.

This tiling is topologically related as a part of a sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity. This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of a sequence that continues into the hyperbolic plane. It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6. This tiling is also part of a sequence of truncated rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n gons at the truncated vertices, and nonregular hexagonal faces.

Like the uniform polyhedra there are eight uniform tilings that can be based on the regular hexagonal tiling (or the dual triangular tiling). Drawing the tiles colored red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 of which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

The hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number). The gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle in contact with a maximum of 6 circles.