List of aperiodic sets of tiles


In geometry, a tiling is a partition of the plane into closed sets, without gaps or overlaps. 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. 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 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

List

ImageNameNumber of tilesSpacePublication DateRefs.Comments
Trilobite and cross tiles2E21999Tilings MLD from the chair tilings
Penrose P1 tiles6E21974Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P2 tiles2E21977Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P3 tiles2E21978Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
Binary tiles2E21988Although 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 tiles6E21971Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No imageAmmann A1 tiles6E21977Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2 tiles2E21986
Ammann A3 tiles3E21986
Ammann A4 tiles2E21986Tilings MLD with Ammann A5.
Ammann A5 tiles2E21982Tilings MLD with Ammann A4.
No imagePenrose hexagon-triangle tiles2E21997
No imageGolden triangle tiles10E22001date is for discovery of matching rules. Dual to Ammann A2
Socolar tiles3E21989Tilings MLD from the tilings by the Shield tiles
Shield tiles4E21988Tilings MLD from the tilings by the Socolar tiles
Square triangle tiles5E21986
Sphinx tiling91E2
Starfish, ivy leaf and hex tiles3E2Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
Robinson triangle4E2Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles6E21996
Pinwheel tilesE21994Date is for publication of matching rules.
Socolar–Taylor tile1E22010Not a connected set. Aperiodic hierarchical tiling.
No imageWang tiles20426E21966
No imageWang tiles104E22008
No imageWang tiles52E21971Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
Wang tiles32E21986Locally derivable from the Penrose tiles.
No imageWang tiles24E21986Locally derivable from the A2 tiling
Wang tiles16E21986Derived from tiling A2 and its Ammann bars
Wang tiles14E21996
Wang tiles13E21996
No imageDecagonal Sponge tile1E22002Porous tile consisting of non-overlapping point sets
No imageGoodman-Strauss strongly aperiodic tiles85H22005
No imageGoodman-Strauss strongly aperiodic tiles26H22005
Böröczky hyperbolic tile1Hn1974Only weakly aperiodic
No imageSchmitt tile1E31988Screw-periodic
Schmitt–Conway–Danzer tile1E3Screw-periodic and convex
Socolar–Taylor tile1E32010Periodic in third dimension
No imagePenrose rhombohedra2E31981
Mackay–Amman rhombohedra4E31981Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No imageWang cubes21E31996
No imageWang cubes18E31999
No imageDanzer tetrahedra4E31989
I and L tiles2En for all n ≥ 31999