Abstract
We consider packings of the two Ammann rhombohedra used for tiling the three dimensional space. We define decorations for the facets of the rhombohedra. Using elementary algebraic topology, we prove that any tiling by these rhombohedra with matching decorations is a quasiperiodic Penrose tiling. The proof does not involve any reference to self similarity.
Similar content being viewed by others
References
Penrose, R.: Math. Intel.2, 32–37 (1979)
de Bruijn, N.G.: Nederl. Akad. Wtensh. Proc. Ser. A43, 39–66 (1981)
Garnder, M.: Sci. Am.236, No. 1, 110 (1977)
Grünbaum, B., Shephard, G.C.: Tilings and patterns. Oxford: W. H. Freeman 1987
Mackay, A.L.: Physica114, 66 (1982)
Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Phys. Rev. Lett.53, 1951 (1984)
Les Houches, Workshop on Aperiodic Crystals. J. Phys. France47, C3 (1986)
Kramer, P., Neri, R.: Acta Crystallogr. Sec. A40, 580 (1984)
Katz, A., Duneau, M.: J. Phys. France47, 181–196 (1986)
Oguey, C., Duneau, M., Katz, A.: A geometrical approach of quasiperiodic tilings. Commun. Math. Phys.118, 99–118 (1988)
Bak, P.: Phys. Rev. Lett.56, 861 (1986)
Dress, A.: Private communication
Socolar, J., Steinhardt, P.J.: Phys. Rev. B34, 617 (1986)
Levine, D.: J. Phys. France47, C3, 125 (1986)
Kleman, M., Pavlovitch, A.: J. Phys. France47, C3, 229 (1986)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Katz, A. Theory of matching rules for the 3-dimensional Penrose tilings. Commun.Math. Phys. 118, 263–288 (1988). https://doi.org/10.1007/BF01218580
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01218580