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 or (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.

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 or (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. If a circle is inscribed in each hexagon, the resulting figure is the densest way to arrange circles in two dimensions; its packing density is \frac{\pi}{2\sqrt{3}} \approx 0.907 . The honeycomb theorem 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. The hexagonal tiling is commonly found in nature, such as the sheet of graphene 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 has a similar structure as graphene. Chicken wire consists of a hexagonal lattice of wires, although the shape is not regular. File: File:| File: File:Carbon nanotube zigzag povray.PNG|A carbon nanotube can be seen as a hexagon tiling on a cylindrical surface File:Tile (AM 1955.117-1).jpg|alt=Hexagonal tile with blue bird and flowers|Hexagonal Persian tile File:Hex pavers sliding to Hudson W60 jeh.jpg|Hexagonal trylinka pavement crumbling in New York 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. === Chamfered hexagonal tiling=== A chamfered hexagonal tiling replaces edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling. at the limit]] === Symmetry mutations=== 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 n 3 , 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. There are 2 regular complex apeirogons, sharing the vertices of the hexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal. The first is made of 2-edges, three around every vertex, the second has hexagonal edges, three around every vertex. A third complex apeirogon, sharing the same vertices, is quasiregular, which alternates 2-edges and 6-edges. ==Aside Euclidean plane== Goldberg polyhedron is a partial solution for a sphere Solutions for hyperbolic planes ==See also== Hexagonal lattice Hexagonal prismatic honeycomb Tilings of regular polygons List of uniform tilings List of regular polytopes Hexagonal tiling honeycomb Hex map board game design John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]

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