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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.
Uniform colorings. There are a total of 32 uniform colorings of the 11 uniform tilings: Triangular tiling – 9 uniform colorings, 4 wythoffian, 5 nonwythoffian. Square tiling – 9 colorings: 7 wythoffian, 2 nonwythoffian. Hexagonal tiling – 3 colorings, all wythoffian. Trihexagonal tiling – 2 colorings, both wythoffian.
The trihexagonal tiling has Schläfli symbol of r{6,3}, or Coxeter diagram, , symbolizing the fact that it is a rectified hexagonal tiling, {6,3}. Its symmetries can be described by the wallpaper group p6mm, (*632), [15] and the tiling can be derived as a Wythoff construction within the reflectional fundamental domains of this group.
Euclidean tilings by convex regular polygons. A regular tiling has one type of regular face. A semiregular or uniform tiling has one type of vertex, but two or more types of faces. A k -uniform tiling has k types of vertices, and two or more types of regular faces. A non-edge-to-edge tiling can have different-sized regular faces.
A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with three hexagonal faces around each vertex. A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D 3 symmetry.
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 applied to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations.
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2). Uniform solutions are constructed by a single generator point with 7 positions within the ...
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle.This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a tiling by ...
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