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

• Stephens P. W., Goldman A. I. The Structure of Quasicrystals
• Levine D., Steinhardt P. J. Quasicrystals I Definition and structure
• Tilings Encyclopedia