List of aperiodic sets of tiles

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 diagram to the right (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 tiles 2 E2 1999 [4] Tilings MLD from the chair tilings
 Penrose P1 tiles 6 E2 1974[Note 1] [5] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
 Penrose P2 tiles 2 E2 1977[Note 2] [6] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
 Penrose P3 tiles 2 E2 1978[Note 3] [7] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
 Binary tiles 2 E2 1988 [8][9] 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[Note 4] [10] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No image Ammann A1 tiles 6 E2 1977[11] [12] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
 Ammann A2 tiles 2 E2 1986[Note 5] [13]
 Ammann A3 tiles 3 E2 1986[Note 5] [13]
 Ammann A4 tiles 2 E2 1986[Note 5] [13][14] Tilings MLD with Ammann A5.
 Ammann A5 tiles 2 E2 1982[Note 6] [15][16] Tilings MLD with Ammann A4.
No image Penrose Hexagon-Triangle tiles 2 E2 1997[17] [17][18]
No image Golden Triangle tiles 10 E2 2001 [19] [20] date is for discovery of matching rules. Dual to Ammann A2
 Socolar tiles 3 E2 1989[Note 7] [21][22] Tilings MLD from the tilings by the Shield tiles
 Shield tiles 4 E2 1988[Note 8] [23][24] Tilings MLD from the tilings by the Socolar tiles
 Square triangle tiles 5 E2 1986[25] [26]
 Sphinx tiling 91 E2 [27]
 Starfish, ivy leaf and hex tiles 3 E2 [28][29][30] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
 Robinson triangle 4 E2 [12] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
 Danzer triangles 6 E2 1996[31] [32]
 Pinwheel tiles E2 1994[33][34] [35][36] Date is for publication of matching rules.
 Socolar–Taylor tile 1 E2 2010 [37][38] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426 E2 1966 [39]
No image Wang tiles 104 E2 2008 [40]
No image Wang tiles 52 E2 1971[Note 4] [41] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
 Wang tiles 32 E2 1986 [42] Locally derivable from the Penrose tiles.
No image Wang tiles 24 E2 1986 [42] Locally derivable from the A2 tiling
 Wang tiles 16 E2 1986 [43][44] Derived from tiling A2 and its Ammann bars
 Wang tiles 14 E2 1996 [45][46]
 Wang tiles 13 E2 1996 [47][48]
No image Decagonal Sponge tile 1 E2 2002 [49][50] Porous tile consisting of non-overlapping point sets
No image Goodman-Strauss strongly aperiodic tiles 85 H2 2005 [51]
No image Goodman-Strauss strongly aperiodic tiles 26 H2 2005 [52]
 Böröczky hyperbolic tile 1 Hn 1974[53] [54][55] Only weakly aperiodic
No image Schmitt tile 1 E3 1988 [56] Screw-periodic
 Schmitt–Conway–Danzer tile 1 E3 [56] Screw-periodic and convex
 Socolar-Taylor tile 1 E3 2010 [37][38] Periodic in third dimension
No image Penrose rhombohedra 2 E3 1981[57] [58][59][60][61][62][63][64]
 Mackay-Amman rhombohedra 4 E3 1981 [65] Icosahedral symmetry. These are decorated Penrose rhomohedra with a matching rule that force aperiodicity.
No image Wang cubes 21 E3 1996 [66]
No image Wang cubes 18 E3 1999 [67]
No image Danzer tetrahedra 4 E3 1989[68] [69]
 I and L tiles 2 En for all n ≥ 3 1999 [70]

Notes

First published in

1.^ Penrose, R. (1974), "The role of Aesthetics in Pure and Applied Mathematical Research", Bull. Inst. Math. and its Appl. 10: 266-271
2.^ Gardner, M. (January 1977), "Extraordinary nonperiodic tiling that enriches the theory of tiles", Scientific American 236: 110-121
3.^ Penrose, R. (1978), "Pentaplexity", Eureka 39: 16-22
4.^ Robinson, R. (1971), "Undecidability and nonperiodicity of tilings in the plane", Inv. Math. 12: 177-209
5.^ Grünbaum, B.; Shephard, G. C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1194-X .
6.^ Beenker, F. P. M.(1982), "Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus", Eindhoven University of Technology, TH Report 82-WSK04
7.^ Socolar, J. E. S. (1989), "Simple octagonal and dodecagonal quasicrystals", Phys. Rev. A 39: 10519-51
8.^ Gähler, F., "Crystallography of dodecagonal quasicrystals", published in Janot, C.: Quasicrystalline materials : Proceedings of the I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988. Singapore : World Scientific, 1988, 272-284

References

  1. ↑ Grünbaum B., Shephard G. C. (1977), "Tilings by Regular Polygons" (PDF), Math. Mag. 50 (5): 227–247, doi:10.2307/2689529.(archived at WebCite)
  2. ↑ Edwards S., Fundamental Regions and Primitive cells (archived at WebCite)
  3. ↑ Ollinger N. Mathematica in action (see page 268)
  4. ↑ Goodman-Strauss C. (1999), "A Small Aperiodic Set of Planar Tiles", European Journal of Combinatorics 20 (5): 375–384, doi:10.1006/eujc.1998.0281 (preprint available here)
  5. ↑ Mikhael J. Colloidal Monolayers On Quasiperiodic Laser Fields (see page 23) (archived at WebCite)
  6. ↑ Gardner M. Penrose tiles to trapdoor ciphers (see page 86) (archived at WebCite)
  7. ↑ Penrose R. (1979/80), "Pentaplexity", Math. Intell. 2: 32–37, doi:10.1007/bf03024384. Check date values in: |date= (help)(archived at WebCite)
  8. ↑ Lançon F., Billard L. (1988), "Two-dimensional system with a quasi-crystalline ground state" (PDF), J. Phys. France 49 (2): 249–256, doi:10.1051/jphys:01988004902024900. (archived at WebCite)
  9. ↑ Lançon F., Billard L. (1992), "A simple example of a non-Pisot tiling with five-fold symmetry" (PDF), J. Phys. I France 2 (2): 207–220, Bibcode:1992JPhy1...2..207G, doi:10.1051/jp1:1992134.(archived at WebCite)
  10. ↑ Goodman-Strauss C. (1999), "Aperiodic Hierarchical tilings", Proc. of NATO-ASI "Foams, Emulsions, and Cellular Materials" Ser. E 354: 481–496, doi:10.1007/978-94-015-9157-7_28.
  11. ↑ Gardner, Martin (2001). The Colossal Book of Mathematics. W. W. Norton & Company. p. 76.
  12. 1 2 Grünbaum B. and Shephard G. C. Tilings and Patterns., according to ; cf
  13. 1 2 3 Ammann R., Grünbaum B. and Shephard G. C. (1992), "Aperiodic Tiles", Discrete Comp Geom 8: 1–25, doi:10.1007/BF02293033.
  14. ↑ Harris E., Frettlöh D. Ammann A4
  15. ↑ Komatsu K., Nomakuchi K., Sakamoto K., Tokitou T. (2004), "Representation of Ammann-Beenker tilings by an automaton", Nihonkai Math. J. 15: 109–118.(archived at WebCite)
  16. ↑ Harris E., Frettlöh D. Ammann-Beenker
  17. 1 2 Penrose R. (1997), "Remarks on tiling: Details of a (1+ε+ε2) aperiodic set.", Nato Asi Series C, 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.
  18. ↑ Goodman-Strauss C., An aperiodic pair of tiles
  19. ↑ Danzer, Ludwig; van Ophuysen, Gerrit (2001). "A species of planar triangular tilings with inflation factor \sqrt-\tau". Res. Bull. Panjab Univ. Sci. 50 (1-4): 137–175. MR 1914493.
  20. ↑ Gelbrich, G (1997). "Fractal Penrose tiles II. Tiles with fractal boundary as duals of Penrose triangles". Aequationes Math. 54: 108–116. doi:10.1007/bf02755450. MR MR1466298.
  21. ↑ Gähler, F.; Lück, R.; Ben-Abraham, S. I.; Gummelt, P. "Dodecagonal tilings as maximal cluster coverings". CiteSeerX: 10.1.1.26.8116. Retrieved 2013-09-25.
  22. ↑ The Socolar tiling
  23. ↑ Gähler F., Frettlöh D. Shield
  24. ↑ Gähler F. (1993), "Matching rules for quasicrystals: the composition-decomposition method" (PDF), J. of Non-crystalline Solids, 153&154: 160–164, doi:10.1016/0022-3093(93)90335-u.(archived at WebCite)
  25. ↑ Stampfli, P (1986). "A Dodecagonal Quasiperiodic Lattice in Two Dimensions". Helv. Phys. Acta 59: 1260–1263.
  26. ↑ Hermisson J., Richard C., Baake M. A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes (archived at WebCite)
  27. ↑ Goodman-Strauss C., Aperiodic tilings (see page 74)
  28. ↑ Lord E. A. (1991), "Quasicrystals and Penrose patterns", Current Science 61: 315.
  29. ↑ Olamy Z., Kléman M. (1989), "A two dimensional aperiodic dense tiling" (PDF), J. Phys. France 50: 19–33, doi:10.1051/jphys:0198900500101900. (archived at WebCite)
  30. ↑ Mihalkovič M., Henley C. L., Widom M. (2004), "Combined energy-diffraction data refinement of decagonal AlNiCo" (PDF), J. Non-Cryst. Solids, 334&335: 177–183. (archived at WebCite)
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  32. ↑ Hayashi H., Kawachi Y., Komatsu K., Konda A., Kurozoe M., Nakano F., Odawara N., Onda R., Sugio A., Yamauchi M. Abstract:Notes on vertex atlas of planar Danzer tiling
  33. ↑ Radin, C (1994). "The pinwheel tilings of the plane". Annals of Mathematics. Second Series 139 (3): 661–702. doi:10.2307/2118575. JSTOR 2118575. MR 95d:52021. CiteSeerX: 10.1.1.44.9723. Retrieved 2013-09-25.
  34. ↑ Charles Radin (1994). "Symmetry Of Tilings Of The Plane". Annals of Mathematics. doi:10.1090/s0273-0979-1993-00425-7. CiteSeerX: 10.1.1.45.5319. Retrieved 2013-09-25.
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  36. ↑ Radin, C (1997). "Aperiodic tilings, ergodic theory, and rotations". The mathematics of long-range aperiodic order. Kluwer Acad. Publ., Dordrecht. MR 1460035.
  37. 1 2 Socolar J. E. S. and Taylor J. M. An aperiodic hexagonal tile
  38. 1 2 Socolar J. E. S. and Taylor J. M. Forcing nonperiodicity with a single tile
  39. ↑ Burger, R (1966). "The Undecidability of the Domino Problem". Memoirs of the American Mathematical Society 66: 1–72.
  40. ↑ Ollinger, Nicolas (2008). Two-by-two Substitution Systems and the Undecidability of the Domino Problem (pdf). Springer. pp. 476–485. Retrieved 2013-09-25.
  41. ↑ Kari, J.; Papasoglu, P. (1999). "Deterministic Aperiodic Tile Sets". Geometric and Functional Analysis 9: 353–369. doi:10.1007/s000390050090.
  42. 1 2 Lagae, A.; Kari, J.; Dutré, P. (2006), "Aperiodic Sets of Square Tiles with Colored Corners", Report CW 460: 12, CiteSeerX: 10.1.1.89.1294, archived from the original (PDF) on 2010-10-02
  43. ↑ Grünbaum, B.; Shephard, G. C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1194-X
  44. ↑ Carbone, A.; Gromov, M.; Prusinkiewicz, P. (2000), Pattern Formation in Biology, Vision and Dynamics, Singapore: World Scientific Publishing Co. Pte. Ltd., ISBN 981-02-3792-8
  45. ↑ Kari J. A small aperiodic set of Wang tiles". Discrete Mathematics, 160(1-3):259–264
  46. ↑ Lagae A. Tile Based Methods in Computer Graphics Dissertation (see page 149) (archived at WebCite)
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  48. ↑ Culik, K. "An aperiodic set of 13 Wang tiles". CiteSeerX: 10.1.1.53.5421. Archived from the original on 2010-10-02. Retrieved 2013-09-25.
  49. ↑ Zhu F. The Search for a Universal Tile
  50. ↑ Bailey, D. A.; Zhu, F. "A Sponge-Like (Almost) Universal Tile". CiteSeerX: 10.1.1.103.3739. Retrieved 2013-09-25.
  51. ↑ Goodman-Strauss C., A hierarchical strongly aperiodic set of tiles in the hyperbolic plane
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  53. ↑ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben I", Mat. Lapok. 25: 265–306.Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben II", Mat. Lapok. 26: 67–90.
  54. ↑ Goodman-Strauss C. (2005), "A strongly aperiodic set of tiles in the hyperbolic plane", Invent. Math. 159: 120, Bibcode:2004InMat.159..119G, doi:10.1007/s00222-004-0384-1
  55. ↑ Dolbilin N., Frettlöh D. Properties of Böröczky tilings in high dimensional hyperbolic spaces (archived at WebCite)
  56. 1 2 Radin, Charles (1995). "Aperiodic tilings in higher dimensions" (pdf). Proceedings of the American Mathematical Society. American Mathematical Society. pp. 3543–3548. doi:10.2307/2161105. JSTOR 2161105. Retrieved 2013-09-25.
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  58. ↑ Meisterernst G. Experimente zur Wachstumskinetik Dekagonaler Quasikristalle (Experiments on the growth kinetics of decagonal quasicrystals) Dissertation (see page 18-19) (archived at WebCite)
  59. ↑ Jirong S. (1993), "Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field" (PDF), Chinese Phys. Lett., 10, No.8: 449–452, doi:10.1088/0256-307x/10/8/001. (archived at WebCite)
  60. ↑ Inchbald G. A 3-D Quasicrystal Structure
  61. ↑ Lord E. A., Ranganathan S., Kulkarni U. D. (2001), "Quasicrystals: tiling versus clustering" (PDF), Phil. Mag. A 81: 2645–2651, doi:10.1080/01418610108216660. (archived at WebCite)
  62. ↑ Rudhart C. P. Zur numerischen Simulation des Bruchs von Quasikristallen (On the numeric simulation of cracking in quasicrystals) see page 11
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  64. ↑ Katz A. (1988), "Theory of Matching Rules for the 3-Dimensional Penrose Tilings", Commun. Math. Phys. 118 (2): 263–288, doi:10.1007/BF01218580. (archived at WebCite)
  65. ↑ Lord, Eric A. (1991), "Quasicrystals and Penrose patterns", Current Science 61 (5): 313
  66. ↑ Culik, K.; Kari, J. "An aperiodic set of Wang cubes". CiteSeerX: 10.1.1.54.5897. Retrieved 2013-09-25.
  67. ↑ Walther, G.; Selter, C. (1999), Mathematikdidaktik als design science, Leipzig: Ernst Klett Grundschulverlag, ISBN 3122000601
  68. ↑ Danzer, L. (1989), "Three-Dimensional Analogs of the Planar Penrose Tilings and Quasicrystals.", Discrete Mathematics 76: 1–7, doi:10.1016/0012-365X(89)90282-3
  69. ↑ Zerhusen A., Danzer's three dimensional tiling
  70. ↑ Goodman-Strauss C. (1999), "An Aperiodic Pair of Tiles in En for all n ≥ 3", European Journal of Combinatorics 20 (5): 385–395, doi:10.1006/eujc.1998.0282 (preprint available here)

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