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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

AbbreviationMeaningExplanation
E2Euclidean planenormal flat plane
H2hyperbolic planeplane, where the parallel postulate does not hold
E3Euclidean 3 spacespace 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

ImageNameNumber of tilesSpacePublication DateRefs.Comments
Trilobite and cross tiles2E21999[4]Tilings MLD from the chair tilings.
Penrose P1 tiles6E21974[5][6]Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex".
Penrose P2 tiles2E21977[7][8]Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex".
Penrose P3 tiles2E21978[9][10]Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex".
Binary tiles2E21988[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 tiles6E21971[13][14]Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Ammann 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 tiles2E21986[17][18][19]Tilings MLD with Ammann A5.
Ammann A5 tiles2E21982[20][21][22]Tilings MLD with Ammann A4.
No imagePenrose hexagon-triangle tiles3E21997[23][23][24]Uses mirror images of tiles for tiling.
No imagePegasus tiles2E22016[25][25][26]Variant of the Penrose hexagon-triangle tiles. Discovered in 2003 or earlier.
Golden triangle tiles10E22001[27][28]Date is for discovery of matching rules. Dual to Ammann A2.
Socolar tiles3E21989[29][30][31]Tilings MLD from the tilings by the Shield tiles.
Shield tiles4E21988[32][33][34]Tilings MLD from the tilings by the Socolar tiles.
Square triangle tiles5E21986[35][36]
Starfish, ivy leaf and hex tiles3E2[37][38][39]Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles.
Robinson triangle4E2[17]Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles6E21996[40][41]
Pinwheel tilesE21994[42][43][44][45]Date is for publication of matching rules.
Socolar–Taylor tile1E22010[46][47]Not a connected set. Aperiodic hierarchical tiling.
No imageWang tiles20426E21966[48]
No imageWang tiles104E22008[49]
No imageWang tiles52E21971[13][50]Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Wang tiles32E21986[51]Locally derivable from the Penrose tiles.
No imageWang tiles24E21986[51]Locally derivable from the A2 tiling.
Wang tiles16E21986[17][52]Derived from tiling A2 and its Ammann bars.
Wang tiles14E21996[53][54]
Wang tiles13E21996[55][56]
Wang tiles11E22015[57]Smallest aperiodic set of Wang tiles.
No imageDecagonal Sponge tile1E22002[58][59]Porous tile consisting of non-overlapping point sets.
No imageGoodman-Strauss strongly aperiodic tiles85H22005[60]
No imageGoodman-Strauss strongly aperiodic tiles26H22005[61]
Böröczky hyperbolic tile1Hn1974[62][63][61][64]Only weakly aperiodic.
No imageSchmitt tile1E31988[65]Screw-periodic.
Schmitt–Conway–Danzer tile1E3[65]Screw-periodic and convex.
Socolar–Taylor tile1E32010[46][47]Periodic in third dimension.
No imagePenrose rhombohedra2E31981[66][67][68][69][70][71][72][73]
Mackay–Amman rhombohedra4E31981[37]Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No imageWang cubes21E31996[74]
No imageWang cubes18E31999[75]
No imageDanzer tetrahedra4E31989[76][77]
I and L tiles2En for all n ≥ 31999[78]
Aperiodic monotile construction diagram, based on Smith (2023)
Aperiodic monotile construction diagram, based on Smith (2023)
Smith–Myers–Kaplan–Goodman-Strauss or "Hat" polytile1E22023[79]Mirrored monotiles, the first example of an "einstein".
Aperiodic monotile construction diagram, based on Smith (2023)
Aperiodic monotile construction diagram, based on Smith (2023)
Smith–Myers–Kaplan–Goodman-Strauss or "Spectre" polytile1E22023[80]"Strictly chiral" aperiodic monotile, the first example of a real "einstein".
Supertile made of 2 tiles.
 TS1 2 E2 2014 [81]

References

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  2. ↑ Edwards, Steve, "Fundamental Regions and Primitive cells", Tiling Plane & Fancy, Kennesaw State University, archived from the original on 2010-07-05, retrieved 2017-01-11
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  4. ↑ Goodman-Strauss, Chaim (1999), "A Small Aperiodic Set of Planar Tiles", European J. Combin., 20 (5): 375–384, doi:10.1006/eujc.1998.0281 (preprint available)
  5. ↑ Penrose, Roger (1974), "The role of Aesthetics in Pure and Applied Mathematical Research", Bulletin of the Institute of Mathematics and Its Applications, 10 (2): 266–271
  6. ↑ Mikhael, Jules (2010), Colloidal Monolayers On Quasiperiodic Laser Fields (PDF) (Dr. rer. nat thesis), p. 23, doi:10.18419/opus-4924, archived (PDF) from the original on 2011-07-17
  7. ↑ Gardner, Martin (January 1977), "Mathematical Games: Extraordinary nonperiodic tiling that enriches the theory of tiles", Scientific American, 236 (1): 110–121, Bibcode:1977SciAm.236a.110G, doi:10.1038/scientificamerican0177-110
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  9. ↑ Penrose, Roger (1978), "Pentaplexity" (PDF), Eureka, 39: 16–22
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  81. ↑ Mehta, Chirag (2021-04-03). "The art of what if". Journal of Mathematics and the Arts. 15 (2): 198–200. doi:10.1080/17513472.2021.1919977. ISSN 1751-3472.
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