Abstract
The open problem of tiling theory whether there is a single aperiodic two-dimensional prototile with corresponding matching rules, is answered for coverings instead of tilings. We introduce admissible overlaps for the regular decagon determining only nonperiodic coverings of the Euclidean plane which are equivalent to tilings by Robinson triangles. Our work is motivated by structural properties of quasicrystals.
Similar content being viewed by others
References
Ammann, R., Grünbaum, B. and Shephard, G. C.: Aperiodic tiles, Discrete Comput. Geom. 8 (1992), 1–25.
Baake, M., Schlottmann, M. and Jarvis, P. D.: Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A 24 (1991), 4637–4654.
Burkov, S. E.: Structure Model of the AL-Cu-Co Decagonal Quasicrystal, Phys. Rev. Lett. 67 (1991), 614–618.
Burkov, S. E.: Modeling decagonal quasicrystals: random assembly of interpenetrating decagonal clusters, J. Phys. I France 2 (1992), 695–706.
Fuijiwara, T. and Ogava, T.: Quasicrystals, Berlin, 1990.
Gardner, M.: Mathematical Games. Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientif. Amer. 236 (1977), 110–121.
Grünbaum, B. and Shephard, G. C.: Tilings and Patterns, Freeman, New York, 1987.
Gummelt, P.: Construction of Penrose tilings by a single aperiodic protoset, Proc. 5th Internat. Conf. on Quasicrystals, Avignon, 1995, 84–87.
Jarić, M.: Aperiodicity and Order, Vol. 1: Introduction to Quasicrystals, Vol. 2: Introduction to the Mathematics of Quasicrystals, San Diego, 1988 and 1989.
Jeong, H. C. and Steinhardt, P. J.: A cluster approach for quasicrystals, Phys. Rev. Lett. 73 (1994), 1943.
Penrose, R.: Pentaplexity, Eureka 39 (1978), 16–22.
Sasisekharan, V.: A new method for generation of quasi-periodic structures with n fold axes: Application to five and seven folds, J. Phys. India (Pramana) 26 (1986), L283-L293.
Steurer, W.: The Structure of Quasicrystals, Zeitschrift für Kristallographie 190 (1990), 179–234.
Senechal M. and Taylor, J.: Quasicrystals: the view from Les Houches, Math. Intelligencer 12 (1990), 54–64.
Senechal, M. The Geometry of Quasicrystals, Cambridge University Press, 1995.
DiVincenzo, D. P. and Steinhardt, P. J.: Quasicrystals: the State of the Art, Directions in Condensed Matter Physics, 11, NJ, 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gummelt, P. Penrose tilings as coverings of congruent decagons. Geom Dedicata 62, 1–17 (1996). https://doi.org/10.1007/BF00239998
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00239998