# List of aperiodic sets of tiles

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 diagram to the right (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

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[Note 1] [5] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P2 tiles 2 E2 1977[Note 2] [6] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P3 tiles 2 E2 1978[Note 3] [7] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
Binary tiles 2 E2 1988 [8][9] 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[Note 4] [10] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No image Ammann A1 tiles 6 E2 1977[11] [12] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2 tiles 2 E2 1986[Note 5] [13]
Ammann A3 tiles 3 E2 1986[Note 5] [13]
Ammann A4 tiles 2 E2 1986[Note 5] [13][14] Tilings MLD with Ammann A5.
Ammann A5 tiles 2 E2 1982[Note 6] [15][16] Tilings MLD with Ammann A4.
No image Penrose Hexagon-Triangle tiles 2 E2 1997[17] [17][18]
No image Golden Triangle tiles 10 E2 2001 [19] [20] date is for discovery of matching rules. Dual to Ammann A2
Socolar tiles 3 E2 1989[Note 7] [21][22] Tilings MLD from the tilings by the Shield tiles
Shield tiles 4 E2 1988[Note 8] [23][24] Tilings MLD from the tilings by the Socolar tiles
Square triangle tiles 5 E2 1986[25] [26]
Sphinx tiling 91 E2 [27]
Starfish, ivy leaf and hex tiles 3 E2 [28][29][30] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
Robinson triangle 4 E2 [12] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles 6 E2 1996[31] [32]
Pinwheel tiles E2 1994[33][34] [35][36] Date is for publication of matching rules.
Socolar–Taylor tile 1 E2 2010 [37][38] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426 E2 1966 [39]
No image Wang tiles 104 E2 2008 [40]
No image Wang tiles 52 E2 1971[Note 4] [41] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
Wang tiles 32 E2 1986 [42] Locally derivable from the Penrose tiles.
No image Wang tiles 24 E2 1986 [42] Locally derivable from the A2 tiling
Wang tiles 16 E2 1986 [43][44] Derived from tiling A2 and its Ammann bars
Wang tiles 14 E2 1996 [45][46]
Wang tiles 13 E2 1996 [47][48]
No image Decagonal Sponge tile 1 E2 2002 [49][50] Porous tile consisting of non-overlapping point sets
No image Goodman-Strauss strongly aperiodic tiles 85 H2 2005 [51]
No image Goodman-Strauss strongly aperiodic tiles 26 H2 2005 [52]
Böröczky hyperbolic tile 1 Hn 1974[53] [52] [54] Only weakly aperiodic
No image Schmitt tile 1 E3 1988 [55] Screw-periodic
Schmitt–Conway–Danzer tile 1 E3 [55] Screw-periodic and convex
Socolar-Taylor tile 1 E3 2010 [37][38] Periodic in third dimension
No image Penrose rhombohedra 2 E3 1981[56] [57][58][59][60][61][62][63]
Mackay-Amman rhombohedra 4 E3 1981 [64] Icosahedral symmetry. These are decorated Penrose rhomohedra with a matching rule that force aperiodicity.
No image Wang cubes 21 E3 1996 [65]
No image Wang cubes 18 E3 1999 [66]
No image Wang cubes 16 E3 [67]
No image Danzer tetrahedra 4 E3 1989[68] [69]
I and L tiles 2 En for all n ≥ 3 1999 [70]

## Notes

First published in

1.^ Penrose, R. (1974), "The role of Aesthetics in Pure and Applied Mathematical Research", Bull. Inst. Math. and its Appl. 10: 266-271
2.^ Gardner, M. (January 1977), "Extraordinary nonperiodic tiling that enriches the theory of tiles", Scientific American 236: 110-121
3.^ Penrose, R. (1978), "Pentaplexity", Eureka 39: 16-22
4.^ Robinson, R. (1971), "Undecidability and nonperiodicity of tilings in the plane", Inv. Math. 12: 177-209
5.^ Grünbaum, B.; Shephard, G. C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1194-X.
6.^ Beenker, F. P. M.(1982), "Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus", Eindhoven University of Technology, TH Report 82-WSK04
7.^ Socolar, J. E. S. (1989), "Simple octagonal and dodecagonal quasicrystals", Phys. Rev. A 39: 10519-51
8.^ Gähler, F., "Crystallography of dodecagonal quasicrystals", published in Janot, C.: Quasicrystalline materials : Proceedings of the I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988. Singapore : World Scientific, 1988, 272-284

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