In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles).A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single fundamental unit or primitive cell which repeats endlessly and regularly in two independent directions. An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".)
The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete.
|E2||Euclidean plane||normal flat plane|
|H2||hyperbolic plane||plane, where the parallel postulate does not hold|
|E3||Euclidean 3 space||space defined by three perpendicular coordinate axes|
|MLD||Mutually locally derivable||two tilings are said to be mutually locally derivable from each other, if one tiling can be obtained from the other by a simple local rule (such as deleting or inserting an edge)|
|Image||Name||Number of tiles||Space||Publication Date||Refs.||Comments|
|Trilobite and cross tiles||2||E2||1999||Tilings MLD from the chair tilings|
|Penrose P1 tiles||6||E2||1974||Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"|
|Penrose P2 tiles||2||E2||1977||Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"|
|Penrose P3 tiles||2||E2||1978||Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"|
|Binary tiles||2||E2||1988||Although similar in shape to the P3 tiles, the tilings are not MLD from each other. Developed in an attempt to model the atomic arrangement in binary alloys|
|Robinson tiles||6||E2||1971||Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices|
|No image||Ammann A1 tiles||6||E2||1977||Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.|
|Ammann A2 tiles||2||E2||1986|
|Ammann A3 tiles||3||E2||1986|
|Ammann A4 tiles||2||E2||1986||Tilings MLD with Ammann A5.|
|Ammann A5 tiles||2||E2||1982||Tilings MLD with Ammann A4.|
|No image||Penrose hexagon-triangle tiles||2||E2||1997|
|Golden triangle tiles||10||E2||2001||date is for discovery of matching rules. Dual to Ammann A2|
|Socolar tiles||3||E2||1989||Tilings MLD from the tilings by the Shield tiles|
|Shield tiles||4||E2||1988||Tilings MLD from the tilings by the Socolar tiles|
|Square triangle tiles||5||E2||1986|
|Starfish, ivy leaf and hex tiles||3||E2||Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles|
|Robinson triangle||4||E2||Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".|
|Pinwheel tiles||E2||1994||Date is for publication of matching rules.|
|Socolar–Taylor tile||1||E2||2010||Not a connected set. Aperiodic hierarchical tiling.|
|No image||Wang tiles||20426||E2||1966|
|No image||Wang tiles||104||E2||2008|
|No image||Wang tiles||52||E2||1971||Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices|
|Wang tiles||32||E2||1986||Locally derivable from the Penrose tiles.|
|No image||Wang tiles||24||E2||1986||Locally derivable from the A2 tiling|
|Wang tiles||16||E2||1986||Derived from tiling A2 and its Ammann bars|
|No image||Decagonal Sponge tile||1||E2||2002||Porous tile consisting of non-overlapping point sets|
|No image||Goodman-Strauss strongly aperiodic tiles||85||H2||2005|
|No image||Goodman-Strauss strongly aperiodic tiles||26||H2||2005|
|Böröczky hyperbolic tile||1||Hn||1974||Only weakly aperiodic|
|No image||Schmitt tile||1||E3||1988||Screw-periodic|
|Schmitt–Conway–Danzer tile||1||E3||Screw-periodic and convex|
|Socolar–Taylor tile||1||E3||2010||Periodic in third dimension|
|No image||Penrose rhombohedra||2||E3||1981|
|Mackay–Amman rhombohedra||4||E3||1981||Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.|
|No image||Wang cubes||21||E3||1996|
|No image||Wang cubes||18||E3||1999|
|No image||Danzer tetrahedra||4||E3||1989|
|I and L tiles||2||En for all n ≥ 3||1999|
A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical crystallographic restriction theorem, can possess only two-, three-, four-, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders—for instance, five-fold.
In the mathematical theory of tessellations, a prototile is one of the shapes of a tile in a tessellation.
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.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
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 is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings.
In geometry, the gyrobifastigium is the 26th Johnson solid. 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.
Robert Ammann was an amateur mathematician who made several significant and groundbreaking contributions to the theory of quasicrystals and aperiodic tilings.
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.
The quaquaversal tiling is a nonperiodic tiling of the euclidean 3-space introduced by John Conway and Charles Radin. The basic solid tiles are half prisms arranged in a pattern that relies essentially on their previous construct, the pinwheel tiling. The rotations relating these tiles belong to the group G(6,4) generated by two rotations of order 6 and 4 whose axes are perpendicular to each other. These rotations are dense in SO(3).
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. They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns.
In mathematics, a Meyer set or almost lattice is a set relatively dense X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who introduced and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for quasicrystals. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions.
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.
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.
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.
Charles Lewis Radin is an American mathematician, known for his work on aperiodic tilings and in particular for defining the pinwheel tiling and, with John Horton Conway, the quaquaversal tiling.
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.
In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved.
Quasicrystals and Geometry is a book on quasicrystals and aperiodic tiling by Marjorie Senechal, published in 1995 by Cambridge University Press (ISBN 0-521-37259-3).
In geometry, a chair 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. 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. The trilobite and cross tiles are aperiodic tiles that enforce the chair tiling substitution structure and these tiles have been modified to a simple aperiodic set of tiles using matching rules enforcing the same structure. Barge et al. have computed the Čech cohomology of the chair tiling and it has been shown that chair tilings can also be obtained via a cut-and-project scheme.