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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.

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".)

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

Abbreviation Meaning Explanation
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)

List

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.
Image Name Number of tiles Space Publication Date Refs. Comments
Trilobite and cross tiles 2 E2 1999 [4] Tilings MLD from the chair tilings
Penrose P1 tiles 6 E2 1974[5] [6] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P2 tiles 2 E2 1977[7] [8] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P3 tiles 2 E2 1978[9] [10] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
Binary tiles 2 E2 1988 [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 6 E2 1971[13] [14] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No image Ammann A1 tiles 6 E2 1977[15] [16] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2 tiles 2 E2 1986[17] [18]
Ammann A3 tiles 3 E2 1986[17] [18]
Ammann A4 tiles 2 E2 1986[17] [18][19] Tilings MLD with Ammann A5.
Ammann A5 tiles 2 E2 1982[20] [21][22] Tilings MLD with Ammann A4.
No image Penrose hexagon-triangle tiles 2 E2 1997[23] [23][24]
Golden triangle tiles 10 E2 2001[25] [26] date is for discovery of matching rules. Dual to Ammann A2
Socolar tiles 3 E2 1989[27] [28][29] Tilings MLD from the tilings by the Shield tiles
Shield tiles 4 E2 1988[30] [31][32] Tilings MLD from the tilings by the Socolar tiles
Square triangle tiles 5 E2 1986[33] [34]
Starfish, ivy leaf and hex tiles 3 E2 [35][36][37] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
Robinson triangle 4 E2 [17] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles 6 E2 1996[38] [39]
Pinwheel tiles E2 1994[40][41] [42][43] Date is for publication of matching rules.
Socolar–Taylor tile 1 E2 2010 [44][45] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426 E2 1966 [46]
No image Wang tiles 104 E2 2008 [47]
No image Wang tiles 52 E2 1971[13] [48] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
Wang tiles 32 E2 1986 [49] Locally derivable from the Penrose tiles.
No image Wang tiles 24 E2 1986 [49] Locally derivable from the A2 tiling
Wang tiles 16 E2 1986 [17][50] Derived from tiling A2 and its Ammann bars
Wang tiles 14 E2 1996 [51][52]
Wang tiles 13 E2 1996 [53][54]
Wang tiles 11 E2 2015 [55]
No image Decagonal Sponge tile 1 E2 2002 [56][57] Porous tile consisting of non-overlapping point sets
No image Goodman-Strauss strongly aperiodic tiles 85 H2 2005 [58]
No image Goodman-Strauss strongly aperiodic tiles 26 H2 2005 [59]
Böröczky hyperbolic tile 1 Hn 1974[60][61] [59][62] Only weakly aperiodic
No image Schmitt tile 1 E3 1988 [63] Screw-periodic
Schmitt–Conway–Danzer tile 1 E3 [63] Screw-periodic and convex
Socolar–Taylor tile 1 E3 2010 [44][45] Periodic in third dimension
No image Penrose rhombohedra 2 E3 1981[64] [65][66][67][68][69][70][71]
Mackay–Amman rhombohedra 4 E3 1981 [35] Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No image Wang cubes 21 E3 1996 [72]
No image Wang cubes 18 E3 1999 [73]
No image Danzer tetrahedra 4 E3 1989[74] [75]
I and L tiles 2 En for all n ≥ 3 1999 [76]

References

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