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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 adding missing items with reliable sources.
Image Name Number of tiles Space Publication Date Refs. Comments
Trilobite and cross tiles 2 E2 1999 [4] Tilings MLD from the chair tilings.
Penrose P1 tiles 6 E2 1974[5] [6] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex".
Penrose P2 tiles 2 E2 1977[7] [8] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex".
Penrose P3 tiles 2 E2 1978[9] [10] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex".
Binary tiles 2 E2 1988 [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 6 E2 1971[13] [14] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Ammann A1 tiles 6 E2 1977[15] [16] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2 tiles 2 E2 1986[17] [18]
Ammann A3 tiles 3 E2 1986[17] [18]
Ammann A4 tiles 2 E2 1986[17] [18][19] Tilings MLD with Ammann A5.
Ammann A5 tiles 2 E2 1982[20] [21][22] Tilings MLD with Ammann A4.
No image Penrose hexagon-triangle tiles 3 E2 1997[23] [23][24] Uses mirror images of tiles for tiling.
No image Pegasus tiles 2 E2 2016[25] [25][26] Variant of the Penrose hexagon-triangle tiles. Discovered in 2003 or earlier.
Golden triangle tiles 10 E2 2001[27] [28] Date is for discovery of matching rules. Dual to Ammann A2.
Socolar tiles 3 E2 1989[29] [30][31] Tilings MLD from the tilings by the Shield tiles.
Shield tiles 4 E2 1988[32] [33][34] Tilings MLD from the tilings by the Socolar tiles.
Square triangle tiles 5 E2 1986[35] [36]
Starfish, ivy leaf and hex tiles 3 E2 [37][38][39] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles.
Robinson triangle 4 E2 [17] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles 6 E2 1996[40] [41]
Pinwheel tiles E2 1994[42][43] [44][45] Date is for publication of matching rules.
Socolar–Taylor tile 1 E2 2010 [46][47] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426 E2 1966 [48]
No image Wang tiles 104 E2 2008 [49]
No image Wang tiles 52 E2 1971[13] [50] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices.
Wang tiles 32 E2 1986 [51] Locally derivable from the Penrose tiles.
No image Wang tiles 24 E2 1986 [51] Locally derivable from the A2 tiling.
Wang tiles 16 E2 1986 [17][52] Derived from tiling A2 and its Ammann bars.
Wang tiles 14 E2 1996 [53][54]
Wang tiles 13 E2 1996 [55][56]
Wang tiles 11 E2 2015 [57] Smallest aperiodic set of Wang tiles.
No image Decagonal Sponge tile 1 E2 2002 [58][59] Porous tile consisting of non-overlapping point sets.
No image Goodman-Strauss strongly aperiodic tiles 85 H2 2005 [60]
No image Goodman-Strauss strongly aperiodic tiles 26 H2 2005 [61]
Böröczky hyperbolic tile 1 Hn 1974[62][63] [61][64] Only weakly aperiodic.
No image Schmitt tile 1 E3 1988 [65] Screw-periodic.
Schmitt–Conway–Danzer tile 1 E3 [65] Screw-periodic and convex.
Socolar–Taylor tile 1 E3 2010 [46][47] Periodic in third dimension.
No image Penrose rhombohedra 2 E3 1981[66] [67][68][69][70][71][72][73]
Mackay–Amman rhombohedra 4 E3 1981 [37] Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No image Wang cubes 21 E3 1996 [74]
No image Wang cubes 18 E3 1999 [75]
No image Danzer tetrahedra 4 E3 1989[76] [77]
I and L tiles 2 En for all n ≥ 3 1999 [78]
Aperiodic monotile construction diagram, based on Smith (2023)
Aperiodic monotile construction diagram, based on Smith (2023)
 Smith–Myers–Kaplan–Goodman-Strauss or "Hat" polytile 1 E2 2023 [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" polytile 1 E2 2023 [80] "Strictly chiral" aperiodic monotile, the first example of a real "einstein".
Supertile made of 2 tiles.
 TS1 2 E2 2014 [81]

References

  1. ^ Grünbaum, Branko; Shephard, Geoffrey C. (1977), "Tilings by Regular Polygons", Math. Mag., 50 (5): 227–247, doi:10.2307/2689529, JSTOR 2689529
  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
  3. ^ Wagon, Steve (2010), Mathematica in Action (3rd ed.), Springer Science & Business Media, p. 268, ISBN 9780387754772
  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
  8. ^ Gardner, Martin (1997), Penrose Tiles to Trapdoor Ciphers (Revised ed.), The Mathematical Association of America, p. 86, ISBN 9780883855218
  9. ^ Penrose, Roger (1978), "Pentaplexity" (PDF), Eureka, 39: 16–22
  10. ^ Penrose, Roger (1979), "Pentaplexity", Math. Intell., 2 (1): 32–37, doi:10.1007/bf03024384, S2CID 120305260, archived from the original on 2011-06-07, retrieved 2010-07-26
  11. ^ Lançon, F.; Billard, L. (1988), "Two-dimensional system with a quasi-crystalline ground state" (PDF), Journal de Physique, 49 (2): 249–256, CiteSeerX 10.1.1.700.3611, doi:10.1051/jphys:01988004902024900, archived (PDF) from the original on 2011-05-09
  12. ^ Godrèche, C.; Lançon, F. (1992), "A simple example of a non-Pisot tiling with five-fold symmetry" (PDF), Journal de Physique I, 2 (2): 207–220, Bibcode:1992JPhy1...2..207G, doi:10.1051/jp1:1992134, S2CID 120168483, archived (PDF) from the original on 2012-03-07
  13. ^ a b Robinson, Raphael M. (1971), "Undecidability and nonperiodicity of tilings in the plane", Inventiones Mathematicae, 12 (3): 177–209, Bibcode:1971InMat..12..177R, doi:10.1007/BF01418780, S2CID 14259496
  14. ^ 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
  15. ^ Gardner, Martin (2001), The Colossal Book of Mathematics, W. W. Norton & Company, p. 76, ISBN 978-0393020236
  16. ^ Grünbaum, Branko & Shephard, Geoffrey C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 978-0-7167-1194-0, according to Dutch, Steven (2003), Aperiodic Tilings, University of Wisconsin - Green Bay, archived from the original on 2006-08-30, retrieved 2011-04-02; cf. Savard, John J. G., Aperiodic Tilings Within Conventional Lattices
  17. ^ a b c d e Grünbaum, Branko & Shephard, Geoffrey C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 978-0-7167-1194-0
  18. ^ a b c Ammann, Robert; Grünbaum, Branko; Shephard, Geoffrey C. (July 1992), "Aperiodic tiles", Discrete & Computational Geometry, 8 (1): 1–25, doi:10.1007/BF02293033, S2CID 39158680
  19. ^ Harriss, Edmund; Frettlöh, Dirk, "Ammann A4", Tilings Encyclopedia, Bielefeld University
  20. ^ 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, vol. 82-WSK04, Eindhoven University of Technology
  21. ^ Komatsu, Kazushi; Nomakuchi, Kentaro; Sakamoto, Kuniko; Tokitou, Takashi (2004), "Representation of Ammann-Beenker tilings by an automaton", Nihonkai Math. J., 15 (2): 109–118, archived from the original on 2012-09-22, retrieved 2017-01-12
  22. ^ Harriss, Edmund; Frettlöh, Dirk, "Ammann-Beenker", Tilings Encyclopedia, Bielefeld University
  23. ^ a b Penrose, R. (1997), "Remarks on tiling: Details of a (1+ε+ε2) aperiodic set.", NATO ASI Series, Series C: Mathematical and Physical Sciences, 489 (The Mathematics of Long-Range Aperiodic Order): 467–497, doi:10.1007/978-94-015-8784-6_18, ISBN 978-0-7923-4506-0
  24. ^ Baake, Michael; Gähler, Franz; Grimm, Uwe (2012). "Hexagonal inflation tilings and planar monotiles". arXiv:1210.3967 [math.DS].
  25. ^ a b Goodman-Strauss, Chaim (2016). "The Pegasus Tiles: an aperiodic pair of tiles". arXiv:1608.07166 [math.CO].
  26. ^ Goodman-Strauss, Chaim (2003), An aperiodic pair of tiles (PDF), University of Arkansas
  27. ^ Danzer, Ludwig; van Ophuysen, Gerrit (2001), "A species of planar triangular tilings with inflation factor ", Res. Bull. Panjab Univ. Sci., 50 (1–4): 137–175, MR 1914493
  28. ^ 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, MR 1466298, S2CID 120531480
  29. ^ Socolar, Joshua E. S. (1989), "Simple octagonal and dodecagonal quasicrystals", Physical Review B, 39 (15): 10519–51, Bibcode:1989PhRvB..3910519S, doi:10.1103/PhysRevB.39.10519, PMID 9947860
  30. ^ Gähler, Franz; Lück, Reinhard; Ben-Abraham, Shelomo I.; Gummelt, Petra (2001), "Dodecagonal tilings as maximal cluster coverings", Ferroelectrics, 250 (1): 335–338, Bibcode:2001Fer...250..335G, doi:10.1080/00150190108225095, S2CID 123171399
  31. ^ Savard, John J. G., The Socolar tiling
  32. ^ Gähler, Franz (1988), "Crystallography of dodecagonal quasicrystals"" (PDF), in Janot, Christian (ed.), Quasicrystalline materials: Proceedings of the I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988, Singapore: World Scientific, pp. 272–284
  33. ^ Gähler, Franz; Frettlöh, Dirk, "Shield", Tilings Encyclopedia, Bielefeld University
  34. ^ Gähler, Franz (1993), "Matching rules for quasicrystals: the composition-decomposition method" (PDF), Journal of Non-Crystalline Solids, 153–154 (Proceddings of the Fourth International Conference on Quasicrystals): 160–164, Bibcode:1993JNCS..153..160G, CiteSeerX 10.1.1.69.2823, doi:10.1016/0022-3093(93)90335-u, archived (PDF) from the original on 2011-07-17
  35. ^ Stampfli, P. (1986), "A Dodecagonal Quasiperiodic Lattice in Two Dimensions", Helv. Phys. Acta, 59: 1260–1263
  36. ^ 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, Bibcode:1997JPhy1...7.1003H, CiteSeerX 10.1.1.46.5796, doi:10.1051/jp1:1997200
  37. ^ a b Lord, Eric. A. (1991), "Quasicrystals and Penrose patterns" (PDF), Current Science, 61 (5): 313–319, archived (PDF) from the original on October 24, 2016
  38. ^ Olamy, Z.; Kléman, M. (1989), "A two dimensional aperiodic dense tiling" (PDF), Journal de Physique, 50 (1): 19–33, doi:10.1051/jphys:0198900500101900, archived (PDF) from the original on 2011-05-09
  39. ^ Mihalkovič, M.; Henley, C. L.; Widom, M. (2004), "Combined energy-diffraction data refinement of decagonal AlNiCo", Journal of Non-Crystalline Solids, 334–335 (8th International Conference on Quasicrystals): 177–183, arXiv:cond-mat/0311613, Bibcode:2004JNCS..334..177M, doi:10.1016/j.jnoncrysol.2003.11.034, S2CID 18958430
  40. ^ 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, S2CID 22538367
  41. ^ Hayashi, Hiroko; Kawachi, Yuu; Komatsu, Kazushi; Konda, Aya; Kurozoe, Miho; Nakano, Fumihiko; Odawara, Naomi; Onda, Rika; Sugio, Akinobu; Yamauchi, Masatetsu (2009), "Abstract: Notes on vertex atlas of planar Danzer tiling" (PDF), Japan Conference on Computational Geometry and Graphs, Kanazawa, November 11–13, 2009
  42. ^ Radin, Charles (1994), "The pinwheel tilings of the plane", Annals of Mathematics, Second Series, 139 (3): 661–702, CiteSeerX 10.1.1.44.9723, doi:10.2307/2118575, JSTOR 2118575, MR 1283873
  43. ^ Radin, Charles (1993), "Symmetry of Tilings of the Plane", Bull. Amer. Math. Soc., 29 (2): 213–217, arXiv:math/9310234, Bibcode:1993math.....10234R, CiteSeerX 10.1.1.45.5319, doi:10.1090/s0273-0979-1993-00425-7, S2CID 14935227
  44. ^ Radin, Charles; Wolff, Mayhew (1992), "Space tilings and local isomorphism", Geom. Dedicata, 42 (3): 355–360, doi:10.1007/bf02414073, MR 1164542, S2CID 16334831
  45. ^ Radin, C (1997), "Aperiodic tilings, ergodic theory, and rotations", NATO ASI Series, Series C: Mathematical and Physical Sciences, Kluwer Acad. Publ., Dordrecht, 489 (The mathematics of long-range aperiodic order), MR 1460035
  46. ^ a b Socolar, Joshua E. S.; Taylor, Joan M. (2011), "An aperiodic hexagonal tile", Journal of Combinatorial Theory, Series A, 118 (8): 2207–2231, arXiv:1003.4279v1, doi:10.1016/j.jcta.2011.05.001, S2CID 27912253
  47. ^ a b Socolar, Joshua E. S.; Taylor, Joan M. (2011), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419v1, doi:10.1007/s00283-011-9255-y, S2CID 10747746
  48. ^ 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
  49. ^ Ollinger, Nicolas (2008), "Two-by-two Substitution Systems and the Undecidability of the Domino Problem" (PDF), Logic and Theory of Algorithms, Lecture Notes in Computer Science, vol. 5028, Springer, pp. 476–485, CiteSeerX 10.1.1.371.9357, doi:10.1007/978-3-540-69407-6_51, ISBN 978-3-540-69405-2
  50. ^ Kari, J.; Papasoglu, P. (1999), "Deterministic Aperiodic Tile Sets", Geometric and Functional Analysis, 9 (2): 353–369, doi:10.1007/s000390050090, S2CID 8775966
  51. ^ a b Lagae, Ares; Kari, Jarkko; Dutré, Phillip (2006), Aperiodic Sets of Square Tiles with Colored Corners, Report CW, vol. 460, KU Leuven, p. 15, CiteSeerX 10.1.1.89.1294
  52. ^ Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000), Pattern Formation in Biology, Vision and Dynamics, Singapore: World Scientific, ISBN 978-981-02-3792-9
  53. ^ Kari, Jarkko (1996), "A small aperiodic set of Wang tiles", Discrete Mathematics, 160 (1–3): 259–264, doi:10.1016/0012-365X(95)00120-L
  54. ^ Lagae, Ares (2007), Tile Based Methods in Computer Graphics (PDF) (PhD thesis), KU Leuven, p. 149, ISBN 978-90-5682-789-2, archived from the original (PDF) on 2011-07-20
  55. ^ Culik, Karel; Kari, Jarkko (1997), "On aperiodic sets of Wang tiles", Foundations of Computer Science, Lecture Notes in Computer Science, vol. 1337, pp. 153–162, doi:10.1007/BFb0052084, ISBN 978-3-540-63746-2
  56. ^ Culik, Karel (1996), "An aperiodic set of 13 Wang tiles", Discrete Mathematics, 160 (1–3): 245–251, CiteSeerX 10.1.1.53.5421, doi:10.1016/S0012-365X(96)00118-5
  57. ^ Jeandel, Emmanuel; Rao, Michaël (2021), "An aperiodic set of 11 Wang tiles", Advances in Combinatorics: Paper No. 1, 37, arXiv:1506.06492, doi:10.19086/aic.18614, MR 4210631, S2CID 13261182
  58. ^ Zhu, Feng (2002), The Search for a Universal Tile (PDF) (BA thesis), Williams College
  59. ^ Bailey, Duane A.; Zhu, Feng (2001), A Sponge-Like (Almost) Universal Tile (PDF), CiteSeerX 10.1.1.103.3739
  60. ^ Goodman-Strauss, Chaim (2010), "A hierarchical strongly aperiodic set of tiles in the hyperbolic plane" (PDF), Theoretical Computer Science, 411 (7–9): 1085–1093, doi:10.1016/j.tcs.2009.11.018
  61. ^ a b Goodman-Strauss, Chaim (2005), "A strongly aperiodic set of tiles in the hyperbolic plane", Invent. Math., 159 (1): 130–132, Bibcode:2004InMat.159..119G, CiteSeerX 10.1.1.477.1974, doi:10.1007/s00222-004-0384-1, S2CID 5348203
  62. ^ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben I", Matematikai Lapok, 25: 265–306
  63. ^ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben II", Matematikai Lapok, 26: 67–90
  64. ^ Dolbilin, Nikkolai; Frettlöh, Dirk (2010), "Properties of Böröczky tilings in high dimensional hyperbolic spaces" (PDF), European J. Combin., 31 (4): 1181–1195, arXiv:0705.0291, CiteSeerX 10.1.1.246.9821, doi:10.1016/j.ejc.2009.11.016, S2CID 13607905
  65. ^ a b Radin, Charles (1995), "Aperiodic tilings in higher dimensions" (PDF), Proceedings of the American Mathematical Society, American Mathematical Society, 123 (11): 3543–3548, doi:10.2307/2161105, JSTOR 2161105, retrieved 2013-09-25
  66. ^ Mackay, Alan L. (1981), "De Nive Quinquangula: On the pentagonal snowflake" (PDF), Sov. Phys. Crystallogr., 26 (5): 517–522, archived (PDF) from the original on 2011-07-21
  67. ^ Meisterernst, Götz, Experimente zur Wachstumskinetik Dekagonaler Quasikristalle (PDF) (Dissertation), Ludwig Maximilian University of Munich, pp. 18–19, archived (PDF) from the original on 2011-06-17
  68. ^ Jirong, Sun (1993), "Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field", Chinese Physics Letters, 10 (8): 449–452, Bibcode:1993ChPhL..10..449S, doi:10.1088/0256-307x/10/8/001, S2CID 250911962
  69. ^ Inchbald, Guy (2002), A 3-D Quasicrystal Structure
  70. ^ Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2001), "Quasicrystals: tiling versus clustering" (PDF), Philosophical Magazine A, 81 (11): 2645–2651, Bibcode:2001PMagA..81.2645L, CiteSeerX 10.1.1.487.2640, doi:10.1080/01418610108216660, S2CID 138403519, archived (PDF) from the original on 2011-07-21
  71. ^ Rudhart, Christoph Paul (June 1999), Zur numerischen Simulation des Bruchs von Quasikristallen (Thesis), University of Stuttgart, p. 11, doi:10.18419/opus-4639
  72. ^ Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000), "Tilings, coverings, clusters and quasicrystals" (PDF), Current Science, 78 (1): 64–72, archived (PDF) from the original on 2011-07-21
  73. ^ Katz, A. (1988), "Theory of Matching Rules for the 3-Dimensional Penrose Tilings", Communications in Mathematical Physics, 118 (2): 263–288, Bibcode:1988CMaPh.118..263K, doi:10.1007/BF01218580, S2CID 121086829
  74. ^ Culik, Karel; Kari, Jarkko (1995), "An aperiodic set of Wang cubes", Journal of Universal Computer Science, 1 (10), CiteSeerX 10.1.1.54.5897, doi:10.3217/jucs-001-10-0675
  75. ^ 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
  76. ^ 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
  77. ^ Zerhusen, Aaron (1997), Danzer's three dimensional tiling, University of Kentucky
  78. ^ 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)
  79. ^ Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "An aperiodic monotile". arXiv:2303.10798 [math.CO].
  80. ^ Smith, David; Myers, Joseph Samuel; Kaplan, Craig S.; Goodman-Strauss, Chaim (2023). "A chiral aperiodic monotile". arXiv:2305.17743 [math.CO].
  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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