Chair tiling

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The chair substitution (left) and a portion of a chair tiling (right) L substitution tiling.svg
The chair substitution (left) and a portion of a chair tiling (right)

In geometry, a chair tiling (or L tiling) is a nonperiodic substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing the L tiles into smaller copies and then rescaling them to their original size can be used to cover patches of the plane. [1] :581 Chair tilings do not possess translational symmetry, i.e., they are examples of nonperiodic tilings, but the chair tiles are not aperiodic tiles since they are not forced to tile nonperiodically by themselves. [2] :482 The trilobite and cross tiles are aperiodic tiles that enforce the chair tiling substitution structure [3] and these tiles have been modified to a simple aperiodic set of tiles using matching rules enforcing the same structure. [4] Barge et al. have computed the Čech cohomology of the chair tiling [5] and it has been shown that chair tilings can also be obtained via a cut-and-project scheme. [6]

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Prototile

In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.

Wang tile

Wang tiles, first proposed by mathematician, logician, and philosopher Hao Wang in 1961, are a class of formal systems. They are modelled visually by square tiles with a color on each side. A set of such tiles is selected, and copies of the tiles are arranged side by side with matching colors, without rotating or reflecting them.

Tessellation Tiling of a plane in mathematics

A tessellation or tiling of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

Aperiodic tiling Non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches

An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings.

Tromino

A tromino is a polyomino of order 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.

Gyrobifastigium

In geometry, the gyrobifastigium is the 26th Johnson solid (J26). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.

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.

Pentagonal tiling

In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.

Jarkko Kari

Jarkko J. Kari is a Finnish mathematician and computer scientist, known for his contributions to the theory of Wang tiles and cellular automata. Kari is currently a professor at the Department of Mathematics, University of Turku.

In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.

Ammann–Beenker tiling

In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker. Because all tilings obtained with the tiles are non-periodic, Ammann–Beenker tilings are considered aperiodic tilings. They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns.

Penrose tiling Non-periodic tiling of the plane

A Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and aperiodic means that shifting any tiling with these shapes by any finite distance, without rotation, cannot produce the same tiling. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.

Pythagorean tiling Tiling by squares of two sizes

A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are based on it, explaining its name. It is commonly used as a pattern for floor tiles. When used for this, it is also known as a hopscotch pattern or pinwheel pattern, but it should not be confused with the mathematical pinwheel tiling, an unrelated pattern.

Socolar–Taylor tile

The Socolar–Taylor tile is a single non-connected tile which is aperiodic on the Euclidean plane, meaning that it admits only non-periodic tilings of the plane, with rotations and reflections of the tile allowed. It is the first known example of a single aperiodic tile, or "einstein". The basic version of the tile is a simple hexagon, with printed designs to enforce a local matching rule, regarding how the tiles may be placed. It is currently unknown whether this rule may be geometrically implemented in two dimensions while keeping the tile a connected set.

Aperiodic set of prototiles

A set of prototiles is aperiodic if copies of the prototiles can be assembled to create tilings, such that all possible tessellation patterns are non-periodic. The aperiodicity referred to is a property of the particular set of prototiles; the various resulting tilings themselves are just non-periodic.

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. Such a shape is called an "einstein", a play on the German words ein Stein, meaning one tile. Depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, the problem is either open or solved. The einstein problem can be seen as a natural extension of the second part of Hilbert's eighteenth problem, which asks for a single polyhedron that tiles Euclidean 3-space, but such that no tessellation by this polyhedron is isohedral. Such anisohedral tiles were found by Karl Reinhardt in 1928, but these anisohedral tiles all tile space periodically.

Chaim Goodman-Strauss American mathematician

Chaim Goodman-Strauss is an American mathematician who works in convex geometry, especially aperiodic tiling. He is on the faculty of the University of Arkansas and is a co-author with John H. Conway of The Symmetries of Things, a comprehensive book surveying the mathematical theory of patterns.

Henri Moscovici is a Romanian-American mathematician, specializing in non-commutative geometry and global analysis.

Shahar Mozes is an Israeli mathematician.

References

  1. Robinson Jr., E. Arthur (1999-12-20). "On the table and the chair". Indagationes Mathematicae. 10 (4): 581–599. doi: 10.1016/S0019-3577(00)87911-2 .
  2. Goodman-Strauss, Chaim (1999), "Aperiodic Hierarchical Tilings" (PDF), in Sadoc, J. F.; Rivier, N. (eds.), Foams and Emulsions, Dordrecht: Springer, pp. 481–496, doi:10.1007/978-94-015-9157-7_28, ISBN   978-90-481-5180-6
  3. Goodman-Strauss, Chaim (1999). "A Small Aperiodic Set of Planar Tiles". European Journal of Combinatorics. 20 (5): 375–384. doi: 10.1006/eujc.1998.0281 .
  4. Goodman-Strauss, Chaim (2018). "Lots of aperiodic sets of tiles". Journal of Combinatorial Theory, Series A. 160: 409–445. arXiv: 1608.07165 . doi:10.1016/j.jcta.2018.07.002.
  5. Barge, Marcy; Diamond, Beverly; Hunton, John; Sadun, Lorenzo (2010). "Cohomology of substitution tiling spaces". Ergodic Theory and Dynamical Systems. 30 (6): 1607–1627. arXiv: 0811.2507 . doi:10.1017/S0143385709000777.
  6. Baake, Michael; Moody, Robert V.; Schlottmann, Martin (1998). "Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces". Journal of Physics A: Mathematical and General. 31 (27): 5755–5766. arXiv: math-ph/9901008 . Bibcode:1998JPhA...31.5755B. doi:10.1088/0305-4470/31/27/006.