List of aperiodic sets of tiles

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A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units will be generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this. Fund un prim cell.svg
Click "show" for description.
A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units will be generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this.

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). [1] 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. [2] 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. [3] 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".)

Contents

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.

Explanations

AbbreviationMeaningExplanation
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
MLDMutually locally derivabletwo 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)

List

ImageNameNumber of tilesSpacePublication DateRefs.Comments
Trilobite and cross.png
Trilobite and cross tiles2E21999 [4] Tilings MLD from the chair tilings
Penrose P1.svg
Penrose P1 tiles 6E21974 [5] [6] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Kite Dart.svg
Penrose P2 tiles 2E21977 [7] [8] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P3 arcs.svg
Penrose P3 tiles 2E21978 [9] [10] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
Binary tiling arcs.svg
Binary tiles 2E21988 [11] [12] 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.svg
Robinson tiles 6E21971 [13] [14] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No imageAmmann A1 tiles6E21977 [15] [16] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2.svg
Ammann A2 tiles2E21986 [17] [18]
Ammann A3.svg
Ammann A3 tiles3E21986 [17] [18]
Ammann A4.svg
Ammann A4 tiles 2E21986 [17] [18] [19] Tilings MLD with Ammann A5.
Ammann A5.svg
Ammann A5 tiles 2E21982 [20] [21] [22] Tilings MLD with Ammann A4.
No imagePenrose hexagon-triangle tiles2E21997 [23] [23] [24]
Goldren Triangle 200px.png
Golden triangle tiles10E22001 [25] [26] date is for discovery of matching rules. Dual to Ammann A2
Socolar.svg
Socolar tiles3E21989 [27] [28] [29] Tilings MLD from the tilings by the Shield tiles
Shield.svg
Shield tiles4E21988 [30] [31] [32] Tilings MLD from the tilings by the Socolar tiles
Square triangle tiles.svg
Square triangle tiles5E21986 [33] [34]
Starfish ivyleaf hex.svg
Starfish, ivy leaf and hex tiles3E2 [35] [36] [37] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
Robinson triangle decompositions.svg
Robinson triangle 4E2 [17] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles.svg
Danzer triangles6E21996 [38] [39]
Pinwheel 1.svg
Pinwheel tiles E21994 [40] [41] [42] [43] Date is for publication of matching rules.
Socolar-Taylor tile.svg
Socolar–Taylor tile 1E22010 [44] [45] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426E21966 [46]
No image Wang tiles 104E22008 [47]
No imageWang tiles52E21971 [13] [48] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
Wang 32 tiles.svg
Wang tiles32E21986 [49] Locally derivable from the Penrose tiles.
No imageWang tiles24E21986 [49] Locally derivable from the A2 tiling
Wang 16 tiles.svg
Wang tiles16E21986 [17] [50] Derived from tiling A2 and its Ammann bars
Wang 14 tiles.svg
Wang tiles14E21996 [51] [52]
Wang 13 tiles.svg
Wang tiles13E21996 [53] [54]
Wang 11 tiles.svg
Wang tiles11E22015 [55]
No imageDecagonal Sponge tile1E22002 [56] [57] Porous tile consisting of non-overlapping point sets
No imageGoodman-Strauss strongly aperiodic tiles85H22005 [58]
No imageGoodman-Strauss strongly aperiodic tiles26H22005 [59]
Goodman-Strauss hyperbolic tile.svg
Böröczky hyperbolic tile 1Hn1974 [60] [61] [59] [62] Only weakly aperiodic
No image Schmitt tile 1E31988 [63] Screw-periodic
SCD tile.svg
Schmitt–Conway–Danzer tile 1E3 [63] Screw-periodic and convex
Socolar Taylor 3D.svg
Socolar–Taylor tile 1E32010 [44] [45] Periodic in third dimension
No imagePenrose rhombohedra2E31981 [64] [65] [66] [67] [68] [69] [70] [71]
Nets for icosahedral aperiodic tile set.svg
Mackay–Amman rhombohedra4E31981 [35] Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No imageWang cubes21E31996 [72]
No imageWang cubes18E31999 [73]
No imageDanzer tetrahedra4E31989 [74] [75]
I and L tiles.png
I and L tiles2En for all n ≥ 31999 [76]

Related Research Articles

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Prototile Basic shape(s) used in a tessellation

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 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.

Aperiodic tiling Specific form of plane tiling in mathematics

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.

Gyrobifastigium 26th Johnson solid (8 faces)

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Robert Ammann American mathematician

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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.

Quaquaversal tiling

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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. 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.

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 Aperiodic 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.

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Aperiodic set of prototiles

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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.

Danzer set Set of points touching all convex bodies of unit volume

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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).

Chair tiling Nonperiodic substitution tiling

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.

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