Abstract
A set of tiles (closed topological disks) is calledaperiodic if there exist tilings of the plane by tiles congruent to those in the set, but no such tiling has any translational symmetry. Several aperiodic sets have been discussed in the literature. We consider a number of aperiodic sets which were briefly described in the recent bookTilings and Patterns, but for which no proofs of their aperiodic character were given. These proofs are presented here in detail, using a technique with goes back to R. M. Robinson and Roger Penrose.
Article PDF
Similar content being viewed by others
References
F. P. M. Beenker, Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus. Technical Report 82-WSK-04, Eindhoven University of Technology, September 1982.
N. G. de Brujin, Algebraic theory of Penrose's non-periodic tilings.Nederl. Akad. Wetensch. Proc. Ser. A 84 (1981), 39–66.
N. G. de Bruijn, Remarks on Beenker patterns. (Mimeographed notes.)
L. Danzer, Three-dimensional analogs of the planar Penrose tilings and quasicrystals.Discrete Math. 76 (1989), 1–7.
M. Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles.Scientific American, January 1977, pp. 110–121.
B. Grünbaum and G. C. Shephard,Tilings and Patterns. Freeman, New York, 1986.
A. Katz, Construction of matching rules for quasiperiodic tilings. Abstract 88A-51-9,Abstracts Amer. Math. Soc. 9 (1988), 437.
A. Katz, Theory of matching rules for the 3-dimensional Penrose tilings.Comm. Math. Phys. 118 (1988), 263–288.
D. Levine and P. J. Steinhardt, Quasicrystals. I. Definition and structure.Phys. Rev. B 34 (1986), 596–616.
A. L. Mackay, Crystallography and the Penrose pattern.Phys. A 114 (1982), 606–613.
G. Y. Onoda, P. J. Steinhardt, D. P. DiVincenzo, and J. E. S. Socolar, Growing perfect quasicrystals.Phys. Rev. Lett. 60 (1988), 2653–2656.
R. Penrose, The role of aesthetics in pure and applied mathematical research.Bull. Inst. Math. Appl. 10 (1974), 266–271.
I. Peterson, Tiling to infinity.Sci. News 134 (1988), 42.
R. M. Robinson, Undecidability and nonperiodicity of tilings of the plane.Invent. Math. 12 (1971), 177–209.
M. Senechal and J. Taylor, Quasicrystals: the view from Les Houches.Math. Intelligencer 12 (1990), 54–64.
J. E. S. Socolar and P. J. Steinhardt, Quasicrystals. II. Unit-cell configurations.Phys. Rev. B 34 (1986), 617–647.
P. J. Steinhardt, Quasicrystals.Amer. Sci. 74 (1986), 586–597.
Y. Watanabe, M. Ito, and T. Soma, Nonperiodic tessellation with eightfold rotational symmetry.Acta Cryst. Sect. A 43 (1987), 133–134.
Author information
Authors and Affiliations
Additional information
The research of Branko Grünbaum was supported in part by NSF Grants MCS8301971 and DMS-8620181.
Rights and permissions
About this article
Cite this article
Ammann, R., Grünbaum, B. & Shephard, G.C. Aperiodic tiles. Discrete Comput Geom 8, 1–25 (1992). https://doi.org/10.1007/BF02293033
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02293033