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Aperiodic tiling with "Tile(1,1)". The tiles are colored according to their rotational orientation modulo 60 degrees. ( Smith, Myers, Kaplan, and Goodman-Strauss) In plane geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way.
An aperiodic tiling using a single shape and its reflection, discovered by David Smith. An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non- periodic tilings.
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 .
Concretely, if A S has side lengths (1, 1, φ), then A L has side lengths (φ, φ, 1). B-tiles can be related to such A-tiles in two ways: If B S has the same size as A L then B L is an enlarged version φ A S of A S, with side lengths (φ, φ, φ 2 = 1 + φ) – this decomposes into an A L tile and A S tile joined along a common side of length 1.
Wang tiles: 16: E 2: 1986: Derived from tiling A2 and its Ammann bars. Wang tiles: 14: E 2: 1996: Wang tiles: 13: E 2: 1996: Wang tiles: 11: E 2: 2015: Smallest aperiodic set of Wang tiles. No image: Decagonal Sponge tile: 1: E 2: 2002: Porous tile consisting of non-overlapping point sets. No image: Goodman-Strauss strongly aperiodic tiles: 85 ...
Domino tiling. In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two ...
The 15th monohedral convex pentagonal type, discovered in 2015. In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon . A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle ...
Formally, the pinwheel tilings are the tilings whose tiles are isometric copies of , in which a tile may intersect another tile only either on a whole side or on half the length side, and such that the following property holds. Given any pinwheel tiling , there is a pinwheel tiling which, once each tile is divided in five following the Conway ...
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