List of aperiodic sets of tiles

File:Fund un prim cell.svg

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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). 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 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. 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 expanding it with reliably sourced entries.

Image	Name	Number of tiles	Space	Publication Date	refs	Comments
100px	Trilobite and cross tiles	2	E2	1999		Tilings MLD from the chair tilings
100px	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"
100px	Penrose P3 tiles	2	E2	1978		Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
100px	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.
100px	Ammann A2 tiles	2	E2	1986
100px	Ammann A3 tiles	3	E2	1986
100px	Ammann A4 tiles	2	E2	1986		Tilings MLD with Ammann A5.
100px	Ammann A5 tiles	2	E2	1982		Tilings MLD with Ammann A4.
No image	Penrose Hexagon-Triangle tiles	2	E2	1997
No image	Golden Triangle tiles	10	E2	2001		date is for discovery of matching rules. Dual to Ammann A2
100px	Socolar tiles	3	E2	1989		Tilings MLD from the tilings by the Shield tiles
100px	Shield tiles	4	E2	1988		Tilings MLD from the tilings by the Socolar tiles
100px	Square triangle tiles	5	E2	1986
	Sphinx tiling	91	E2
100px	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".
100px	Danzer triangles	6	E2	1996
	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
100px	Wang tiles	32	E2	1986		Locally derivable from the Penrose tiles.
No image	Wang tiles	24	E2	1986		Locally derivable from the A2 tiling
100px	Wang tiles	16	E2	1986		Derived from tiling A2 and its Ammann bars
100px	Wang tiles	14	E2	1996
100px	Wang tiles	13	E2	1996
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
100px	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
100px	Socolar-Taylor tile	1	E3	2010		Periodic in third dimension
No image	Penrose rhombohedra	2	E3	1981
100px	Mackay-Amman rhombohedra	4	E3	1981		Icosahedral symmetry. These are decorated Penrose rhomohedra 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
100px	I and L tiles	2	En for all n ≥ 3	1999

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.^{^} .

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

External links

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