List of aperiodic sets of tiles

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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".)

Contents

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

AbbreviationMeaningExplanation
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
MLDMutually locally derivabletwo 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

Trilobite and cross tiles2E21999 [4] Tilings MLD from the chair tilings
Penrose P1 tiles 6E21974 [5] [6] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P2 tiles 2E21977 [7] [8] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P3 tiles 2E21978 [9] [10] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
Binary tiles 2E21988 [11] [12] 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 6E21971 [13] [14] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No imageAmmann A1 tiles6E21977 [15] [16] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2 tiles2E21986 [17] [18]
Ammann A3 tiles3E21986 [17] [18]
Ammann A4 tiles 2E21986 [17] [18] [19] Tilings MLD with Ammann A5.
Ammann A5 tiles 2E21982 [20] [21] [22] Tilings MLD with Ammann A4.
No imagePenrose hexagon-triangle tiles2E21997 [23] [23] [24]
Golden triangle tiles10E22001 [25] [26] date is for discovery of matching rules. Dual to Ammann A2
Socolar tiles3E21989 [27] [28] [29] Tilings MLD from the tilings by the Shield tiles
Shield tiles4E21988 [30] [31] [32] Tilings MLD from the tilings by the Socolar tiles
Square triangle tiles5E21986 [33] [34]
Starfish, ivy leaf and hex tiles3E2 [35] [36] [37] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
Robinson triangle 4E2 [17] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles6E21996 [38] [39]
Pinwheel tiles E21994 [40] [41] [42] [43] Date is for publication of matching rules.
Socolar–Taylor tile 1E22010 [44] [45] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426E21966 [46]
No image Wang tiles 104E22008 [47]
No imageWang tiles52E21971 [13] [48] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
Wang tiles32E21986 [49] Locally derivable from the Penrose tiles.
No imageWang tiles24E21986 [49] Locally derivable from the A2 tiling
Wang tiles16E21986 [17] [50] Derived from tiling A2 and its Ammann bars
Wang tiles14E21996 [51] [52]
Wang tiles13E21996 [53] [54]
Wang tiles11E22015 [55]
No imageDecagonal Sponge tile1E22002 [56] [57] Porous tile consisting of non-overlapping point sets
No imageGoodman-Strauss strongly aperiodic tiles85H22005 [58]
No imageGoodman-Strauss strongly aperiodic tiles26H22005 [59]
Böröczky hyperbolic tile 1Hn1974 [60] [61] [59] [62] Only weakly aperiodic
No image Schmitt tile 1E31988 [63] Screw-periodic
Schmitt–Conway–Danzer tile 1E3 [63] Screw-periodic and convex
Socolar–Taylor tile 1E32010 [44] [45] Periodic in third dimension
No imagePenrose rhombohedra2E31981 [64] [65] [66] [67] [68] [69] [70] [71]
Mackay–Amman rhombohedra4E31981 [35] Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No imageWang cubes21E31996 [72]
No imageWang cubes18E31999 [73]
No imageDanzer tetrahedra4E31989 [74] [75]
I and L tiles2En for all n ≥ 31999 [76]

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