List of aperiodic sets of tiles

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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

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
Trilobite and cross.png
 Trilobite and cross tiles 2 E2 1999 [4] Tilings MLD from the chair tilings.
Penrose P3 arcs.svg
 Penrose P3 tiles 2 E2 1978[5] [6] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex".
Binary tiling arcs.svg
 Binary tiles 2 E2 1988 [7][8] 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.svg
 Robinson tiles 6 E2 1971[9] [10] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Ammann A3.svg
 Ammann A3 tiles 3 E2 1986[11] [12]
Ammann-Beenker-tileset.svg
 Ammann A5 tiles 2 E2 1982[13] [14][15] Tilings MLD with Ammann A4.
No image Pegasus tiles 2 E2 2016[16] [16][17] Variant of the Penrose hexagon-triangle tiles. Discovered in 2003 or earlier.
Goldren Triangle 200px.png
 Golden triangle tiles 10 E2 2001[18] [19] Date is for discovery of matching rules. Dual to Ammann A2.
Socolar.svg
 Socolar tiles 3 E2 1989[20] [21][22] Tilings MLD from the tilings by the Shield tiles.
Shield.svg
 Shield tiles 4 E2 1988[23] [24][25] Tilings MLD from the tilings by the Socolar tiles.
Square triangle tiles.svg
 Square triangle tiles 5 E2 1986[26] [27]
Robinson triangle decompositions.svg
 Robinson triangle 4 E2 [11] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles.svg
 Danzer triangles 6 E2 1996[28] [29]
Socolar-Taylor tile.svg
 Socolar–Taylor tile 1 E2 2010 [30][31] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426 E2 1966 [32]
No image Wang tiles 104 E2 2008 [33]
No image Wang tiles 24 E2 1986 [34] Locally derivable from the A2 tiling.
Wang 16 tiles.svg
 Wang tiles 16 E2 1986 [11][35] Derived from tiling A2 and its Ammann bars.
Wang 11 tiles.svg
 Wang tiles 11 E2 2015 [36] Smallest aperiodic set of Wang tiles.
No image Decagonal Sponge tile 1 E2 2002 [37][38] Porous tile consisting of non-overlapping point sets.
No image Goodman-Strauss strongly aperiodic tiles 85 H2 2005 [39]
No image Goodman-Strauss strongly aperiodic tiles 26 H2 2005 [40]
Goodman-Strauss hyperbolic tile.svg
 Böröczky hyperbolic tile 1 Hn 1974[41][42] [40][43] Only weakly aperiodic.
No image Schmitt tile 1 E3 1988 [44] Screw-periodic.
SCD tile.svg
 Schmitt–Conway–Danzer tile 1 E3 [44] Screw-periodic and convex.
Socolar Taylor 3D.svg
 Socolar–Taylor tile 1 E3 2010 [30][31] Periodic in third dimension.
No image Penrose rhombohedra 2 E3 1981[45] [46][47][48][49][50][51][52]
Nets for icosahedral aperiodic tile set.svg
 Mackay–Amman rhombohedra 4 E3 1981 [53] Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No image Wang cubes 18 E3 1999 [54]
No image Danzer tetrahedra 4 E3 1989[55] [56]
I and L tiles.png
 I and L tiles 2 En for all n ≥ 3 1999 [57]
Aperiodic monotile construction diagram, based on Smith (2023)
 Smith–Myers–Kaplan–Goodman-Strauss or "Hat" polytile 1 E2 2023 [58] Mirrored monotiles, the first example of an "einstein".
Aperiodic monotile construction diagram, based on Smith (2023)
 Smith–Myers–Kaplan–Goodman-Strauss or "Spectre" polytile 1 E2 2023 [59] "Strictly chiral" aperiodic monotile, the first example of a real "einstein".
Supertile made of 2 tiles.
 TS1 2 E2 2014 [60]

References

  1. ↑ Grünbaum, Branko; Shephard, Geoffrey C. (1977), "Tilings by Regular Polygons", Math. Mag. 50 (5): 227–247, doi:10.2307/2689529 
  2. ↑ Edwards, Steve, Fundamental Regions and Primitive cells, Kennesaw State University, http://ksuweb.kennesaw.edu/~sedwar77/tile/defs/fundamental.htm, retrieved 2017-01-11 
  3. ↑ Wagon, Steve (2010), Mathematica in Action (3rd ed.), Springer Science & Business Media, pp. 268, ISBN 9780387754772, https://books.google.com/books?id=EbVrWLNiub4C&q=nonperiodic+tiling&pg=PA268 
  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 (1978), "Pentaplexity", Eureka 39: 16–22, https://www.archim.org.uk/eureka/archive/Eureka-39.pdf 
  6. ↑ Penrose, Roger (1979), "Pentaplexity", Math. Intell. 2 (1): 32–37, doi:10.1007/bf03024384, http://www.ma.utexas.edu/users/radin/pentaplexity.html, retrieved 2010-07-26 
  7. ↑ Lançon, F.; Billard, L. (1988), "Two-dimensional system with a quasi-crystalline ground state", Journal de Physique 49 (2): 249–256, doi:10.1051/jphys:01988004902024900, http://hal.archives-ouvertes.fr/docs/00/21/06/91/PDF/ajp-jphys_1988_49_2_249_0.pdf 
  8. ↑ Godrèche, C.; Lançon, F. (1992), "A simple example of a non-Pisot tiling with five-fold symmetry", Journal de Physique I 2 (2): 207–220, doi:10.1051/jp1:1992134, Bibcode: 1992JPhy1...2..207G, http://hal.archives-ouvertes.fr/docs/00/24/64/73/PDF/ajp-jp1v2p207.pdf 
  9. ↑ Robinson, Raphael M. (1971), "Undecidability and nonperiodicity of tilings in the plane", Inventiones Mathematicae 12 (3): 177–209, doi:10.1007/BF01418780, Bibcode: 1971InMat..12..177R 
  10. ↑ Goodman-Strauss, Chaim (1999), Sadoc, J. F.; Rivier, N., eds., "Aperiodic Hierarchical tilings", NATO ASI Series, Series E: Applied Sciences 354 (Foams and Emulsions): 481–496, doi:10.1007/978-94-015-9157-7_28, ISBN 978-90-481-5180-6 
  11. ↑ 11.0 11.1 11.2 Cite error: Invalid <ref> tag; no text was provided for refs named TilPat
  12. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Aperiodic tiles
  13. ↑ Beenker, F. P. M. (1982), Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus, TH Report, 82-WSK04, Eindhoven University of Technology 
  14. ↑ Komatsu, Kazushi; Nomakuchi, Kentaro; Sakamoto, Kuniko; Tokitou, Takashi (2004), "Representation of Ammann-Beenker tilings by an automaton", Nihonkai Math. J. 15 (2): 109–118, http://projecteuclid.org/euclid.nihmj/1273779794, retrieved 2017-01-12 
  15. ↑ Harriss, Edmund; Frettlöh, Dirk, Ammann-Beenker, Bielefeld University, http://tilings.math.uni-bielefeld.de/substitution/ammann-beenker/ 
  16. ↑ 16.0 16.1 Goodman-Strauss, Chaim (2016). "The Pegasus Tiles: an aperiodic pair of tiles". arXiv:1608.07166 [math.CO].
  17. ↑ Goodman-Strauss, Chaim (2003), An aperiodic pair of tiles, University of Arkansas, http://comp.uark.edu/~strauss/distribution/tilings/penhex.pdf 
  18. ↑ Danzer, Ludwig; van Ophuysen, Gerrit (2001), "A species of planar triangular tilings with inflation factor [math]\displaystyle{ \sqrt{-\tau} }[/math]", Res. Bull. Panjab Univ. Sci. 50 (1–4): 137–175 
  19. ↑ Gelbrich, G (1997), "Fractal Penrose tiles II. Tiles with fractal boundary as duals of Penrose triangles", Aequationes Mathematicae 54 (1–2): 108–116, doi:10.1007/bf02755450 
  20. ↑ Socolar, Joshua E. S. (1989), "Simple octagonal and dodecagonal quasicrystals", Physical Review B 39 (15): 10519–51, doi:10.1103/PhysRevB.39.10519, PMID 9947860, Bibcode: 1989PhRvB..3910519S 
  21. ↑ Gähler, Franz; Lück, Reinhard; Ben-Abraham, Shelomo I.; Gummelt, Petra (2001), "Dodecagonal tilings as maximal cluster coverings", Ferroelectrics 250 (1): 335–338, doi:10.1080/00150190108225095, Bibcode: 2001Fer...250..335G 
  22. ↑ Savard, John J. G., The Socolar tiling, http://www.quadibloc.com/math/dode01.htm 
  23. ↑ Gähler, Franz (1988), "Crystallography of dodecagonal quasicrystals"", in Janot, Christian, Quasicrystalline materials: Proceedings of the I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988, Singapore: World Scientific, pp. 272–284, http://elib.uni-stuttgart.de/opus/volltexte/2009/4662/pdf/gaeh7.pdf 
  24. ↑ Gähler, Franz; Frettlöh, Dirk, Shield, Bielefeld University, http://tilings.math.uni-bielefeld.de/substitution/shield/ 
  25. ↑ Gähler, Franz (1993), "Matching rules for quasicrystals: the composition-decomposition method", Journal of Non-Crystalline Solids 153–154 (Proceddings of the Fourth International Conference on Quasicrystals): 160–164, doi:10.1016/0022-3093(93)90335-u, Bibcode: 1993JNCS..153..160G, http://elib.uni-stuttgart.de/opus/volltexte/2009/3989/pdf/gaeh11.pdf 
  26. ↑ Stampfli, P. (1986), "A Dodecagonal Quasiperiodic Lattice in Two Dimensions", Helv. Phys. Acta 59: 1260–1263 
  27. ↑ Hermisson, Joachim; Richard, Christoph; Baake, Michael (1997), "A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes", Journal de Physique I 7 (8): 1003–1018, doi:10.1051/jp1:1997200, Bibcode: 1997JPhy1...7.1003H, https://hal.archives-ouvertes.fr/jpa-00247374 
  28. ↑ Nischke, K.-P.; Danzer, L. (1996), "A construction of inflation rules based on n-fold symmetry", Discrete & Computational Geometry 15 (2): 221–236, doi:10.1007/bf02717732 
  29. ↑ Hayashi, Hiroko; Kawachi, Yuu; Komatsu, Kazushi; Konda, Aya; Kurozoe, Miho; Nakano, Fumihiko; Odawara, Naomi; Onda, Rika et al. (2009), "Abstract: Notes on vertex atlas of planar Danzer tiling", Japan Conference on Computational Geometry and Graphs, Kanazawa, November 11–13, 2009, http://www.jaist.ac.jp/~uehara/JCCGG09/short/paper_29.pdf 
  30. ↑ 30.0 30.1 Socolar, Joshua E. S.; Taylor, Joan M. (2011), "An aperiodic hexagonal tile", Journal of Combinatorial Theory, Series A 118 (8): 2207–2231, doi:10.1016/j.jcta.2011.05.001 
  31. ↑ 31.0 31.1 Socolar, Joshua E. S.; Taylor, Joan M. (2011), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer 34 (1): 18–28, doi:10.1007/s00283-011-9255-y 
  32. ↑ Berger, Robert (1966), "The Undecidability of the Domino Problem", Memoirs of the American Mathematical Society 66 (66), doi:10.1090/memo/0066, ISBN 978-0-8218-1266-2 
  33. ↑ Ollinger, Nicolas (2008), "Two-by-two Substitution Systems and the Undecidability of the Domino Problem", Logic and Theory of Algorithms, Lecture Notes in Computer Science, 5028, Springer, pp. 476–485, doi:10.1007/978-3-540-69407-6_51, ISBN 978-3-540-69405-2, http://hal.inria.fr/docs/00/26/01/12/PDF/sutica.pdf 
  34. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Aperiodic sets of square tiles
  35. ↑ Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000), Pattern Formation in Biology, Vision and Dynamics, Singapore: World Scientific, ISBN 978-981-02-3792-9 
  36. ↑ Jeandel, Emmanuel; Rao, Michaël (2021), "An aperiodic set of 11 Wang tiles", Advances in Combinatorics: Paper No. 1, 37, doi:10.19086/aic.18614 
  37. ↑ Zhu, Feng (2002), The Search for a Universal Tile, Williams College, http://www.cs.williams.edu/~bailey/Zh02.pdf 
  38. ↑ Bailey, Duane A.; Zhu, Feng (2001), A Sponge-Like (Almost) Universal Tile, http://cs.williams.edu/~bailey/BaileyZhu.pdf 
  39. ↑ Goodman-Strauss, Chaim (2010), "A hierarchical strongly aperiodic set of tiles in the hyperbolic plane", Theoretical Computer Science 411 (7–9): 1085–1093, doi:10.1016/j.tcs.2009.11.018, http://comp.uark.edu/~strauss/papers/STAP.pdf 
  40. ↑ 40.0 40.1 Goodman-Strauss, Chaim (2005), "A strongly aperiodic set of tiles in the hyperbolic plane", Invent. Math. 159 (1): 130–132, doi:10.1007/s00222-004-0384-1, Bibcode: 2004InMat.159..119G 
  41. ↑ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben I", Matematikai Lapok 25: 265–306 
  42. ↑ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben II", Matematikai Lapok 26: 67–90 
  43. ↑ Dolbilin, Nikkolai; Frettlöh, Dirk (2010), "Properties of Böröczky tilings in high dimensional hyperbolic spaces", European J. Combin. 31 (4): 1181–1195, doi:10.1016/j.ejc.2009.11.016, http://www.mathematik.uni-bielefeld.de/baake/frettloe/papers/hypart.pdf 
  44. ↑ 44.0 44.1 Radin, Charles (1995), "Aperiodic tilings in higher dimensions", Proceedings of the American Mathematical Society (American Mathematical Society) 123 (11): 3543–3548, doi:10.2307/2161105, http://www.ma.utexas.edu/users/radin/papers/schmitt.pdf, retrieved 2013-09-25 
  45. ↑ Mackay, Alan L. (1981), "De Nive Quinquangula: On the pentagonal snowflake", Sov. Phys. Crystallogr. 26 (5): 517–522, http://materials.iisc.ernet.in/~lord/webfiles/Alan/CV073eng.pdf 
  46. ↑ Meisterernst, Götz, Experimente zur Wachstumskinetik Dekagonaler Quasikristalle, Ludwig Maximilian University of Munich, pp. 18–19, http://edoc.ub.uni-muenchen.de/7000/1/Meisterernst_Goetz.pdf 
  47. ↑ Jirong, Sun (1993), "Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field", Chinese Physics Letters 10 (8): 449–452, doi:10.1088/0256-307x/10/8/001, Bibcode: 1993ChPhL..10..449S 
  48. ↑ Inchbald, Guy (2002), A 3-D Quasicrystal Structure, https://www.steelpillow.com/polyhedra/quasicr/quasicr.html 
  49. ↑ Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2001), "Quasicrystals: tiling versus clustering", Philosophical Magazine A 81 (11): 2645–2651, doi:10.1080/01418610108216660, Bibcode: 2001PMagA..81.2645L, http://materials.iisc.ernet.in/~lord/webfiles/pmq.pdf 
  50. ↑ Rudhart, Christoph Paul (June 1999), Zur numerischen Simulation des Bruchs von Quasikristallen, University of Stuttgart, p. 11, doi:10.18419/opus-4639, http://elib.uni-stuttgart.de/handle/11682/4656 
  51. ↑ Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000), "Tilings, coverings, clusters and quasicrystals", Current Science 78 (1): 64–72, http://www.currentscience.ac.in/Downloads/article_id_078_01_0064_0072_0.pdf 
  52. ↑ Katz, A. (1988), "Theory of Matching Rules for the 3-Dimensional Penrose Tilings", Communications in Mathematical Physics 118 (2): 263–288, doi:10.1007/BF01218580, Bibcode: 1988CMaPh.118..263K, http://projecteuclid.org/euclid.cmp/1104161989 
  53. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Quasicrystals and Penrose patterns
  54. ↑ Walther. Gerd; Selter, Christoph, eds. (1999), Mathematikdidaktik als design science : Festschrift für Erich Christian Wittmann, Leipzig: Ernst Klett Grundschulverlag, ISBN 978-3-12-200060-8 
  55. ↑ Danzer, L. (1989), "Three-Dimensional Analogs of the Planar Penrose Tilings and Quasicrystals", Discrete Mathematics 76 (1): 1–7, doi:10.1016/0012-365X(89)90282-3 
  56. ↑ Zerhusen, Aaron (1997), Danzer's three dimensional tiling, University of Kentucky, http://www.ms.uky.edu/~lee/zerhusen/quasi.html 
  57. ↑ Goodman-Strauss, Chaim (1999), "An Aperiodic Pair of Tiles in En for all n ≥ 3", European J. Combin. 20 (5): 385–395, doi:10.1006/eujc.1998.0282  (preprint available)
  58. ↑ Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "An aperiodic monotile". arXiv:2303.10798 [math.CO].
  59. ↑ Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "A chiral aperiodic monotile". arXiv:2305.17743 [math.CO].
  60. ↑ Mehta, Chirag (2021-04-03). "The art of what if" (in en). Journal of Mathematics and the Arts 15 (2): 198–200. doi:10.1080/17513472.2021.1919977. ISSN 1751-3472. https://www.tandfonline.com/doi/full/10.1080/17513472.2021.1919977. 

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