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 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. 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 (preprint available) Tilings MLD from the chair tilings. 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". Penrose P3 tiles 2 E2 1978 Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex". 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. Ammann A1 tiles 6 E2 1977 , according to ; cf. Tiles enforce aperiodicity by forming an infinite hierarchal binary tree. Ammann A2 tiles 2 E2 1986 Ammann A3 tiles 3 E2 1986 Ammann A4 tiles 2 E2 1986 Tilings MLD with Ammann A5. Ammann A5 tiles 2 E2 1982 Tilings MLD with Ammann A4. No image Penrose hexagon-triangle tiles 3 E2 1997 Uses mirror images of tiles for tiling. No image Pegasus tiles 2 E2 2016 Variant of the Penrose hexagon-triangle tiles. Discovered in 2003 or earlier. Golden triangle tiles 10 E2 2001 Date is for discovery of matching rules. Dual to Ammann A2. Socolar tiles 3 E2 1989 Tilings MLD from the tilings by the Shield tiles. Shield tiles 4 E2 1988 Tilings MLD from the tilings by the Socolar tiles. Square triangle tiles 5 E2 1986 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". Danzer triangles 6 E2 1996 Pinwheel tiles 1 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. Wang tiles 32 E2 1986 Locally derivable from the Penrose tiles. No image Wang tiles 24 E2 1986 Locally derivable from the A2 tiling. Wang tiles 16 E2 1986 Derived from tiling A2 and its Ammann bars. Wang tiles 14 E2 1996 Wang tiles 13 E2 1996 Wang tiles 11 E2 2015 Smallest aperiodic set of Wang tiles. 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 Böröczky hyperbolic tile 1 Hn 1974 Only weakly aperiodic. No image Schmitt tile 1 E3 1988 Screw-periodic. Schmitt–Conway–Danzer tile 1 E3 1988 Screw-periodic and convex. Socolar–Taylor tile 1 E3 2010 Periodic in third dimension. No image Penrose rhombohedra 2 E3 1981 Mackay–Amman rhombohedra 4 E3 1981 Icosahedral symmetry. These are decorated Penrose rhombohedra 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 I and L tiles 2 En for all n ≥ 3 1999 (preprint available) Aperiodic monotile using dendrites (Mampusti/Whittaker) 1 E2 2021 Monotile using dendrites and a seed tile Aperiodic spiral monotile (Klaassen) 1 E2 2022 Monotile (one matching rule) using a seed tile Hilbert curve monotile (Klaassen) 1 E2 2022 Monotile creating a Hilbert curve using a seed tile Smith–Myers–Kaplan–Goodman-Strauss or "Hat" polytile 1 E2 2023 Mirrored monotiles, the first example of an "einstein". Smith–Myers–Kaplan–Goodman-Strauss or "Spectre" polytile 1 E2 2023 "Strictly chiral" aperiodic monotile, the first example of a real "einstein".TS12E22014Supertile made of 2 tiles
References
External links Stephens P. W., Goldman A. I. The Structure of Quasicrystals Levine D., Steinhardt P. J. Quasicrystals I Definition and structure Tilings Encyclopedia