610Accesses
3Altmetric
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.
This is a preview of subscription content,log in via an institution to check access.
Access this article
Subscribe and save
- Get 10 units per month
- Download Article/Chapter or eBook
- 1 Unit = 1 Article or 1 Chapter
- Cancel anytime
Buy Now
Price includes VAT (Japan)
Instant access to the full article PDF.
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. A24 (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 France2 (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,Eureka39 (1978), 16–22.
Sasisekharan, V.: A new method for generation of quasi-periodic structures withn fold axes: Application to five and seven folds,J. Phys. India (Pramana)26 (1986), L283-L293.
Steurer, W.: The Structure of Quasicrystals, Zeitschrift für Kristallographie190 (1990), 179–234.
Senechal M. and Taylor, J.: Quasicrystals: the view from Les Houches,Math. Intelligencer12 (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
Fachbereich Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, Jahnstraße 15a, 17489, Greifswald, Germany
Petra Gummelt
- Petra Gummelt
You can also search for this author inPubMed Google Scholar
Rights and permissions
About this article
Cite this article
Gummelt, P. Penrose tilings as coverings of congruent decagons.Geom Dedicata62, 1–17 (1996). https://doi.org/10.1007/BF00239998
Received:
Revised:
Issue Date:
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative