Truncated hexagonal tiling

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}. Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille). There are 3 regular and 8 semiregular tilings in the plane. There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.) The dodecagonal faces can be distorted into different geometries, such as: Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling). Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.) This tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry. Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares. The truncated 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). This is the lowest density packing that can be created from a uniform tiling. The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by face configuration V3.12.

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Uniform tiling

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain.

Trihexagonal tiling

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.

List of Euclidean uniform tilings

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.

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