Remarks on Penrose Tilings

  • N. G. de Bruijn
Part of the Algorithms and Combinatorics book series (AC, volume 14)


This paper will cover some details on Penrose tilings presented in lectures over the years but never published in print before. The main topics are: (i) the characterizability of Penrose tilings by means of a local rule that does not refer to arrows on the edges of the tiles, and (ii) the fact that the Ammann quasigrid of the inflation of a Penrose tiling is topologically equivalent to the pentagrid that generates the original tiling.


Common Edge Double Arrow Singular Case Billiard Ball Central Grid 
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  1. 1.
    N. G. de Bruijn, Algebraic theory of Penrose’s non-periodic tilings of the plane. Kon. Nederl. Akad. Wetensch. Proc. Ser. A 84 (=Indagationes Mathematicae 43), 38–52 and 53–66 (1981). Reprinted in: P. J. Steinhardt and Stellan Ostlund: The Physics of Quasicrystals, World Scientific Publ., Singapore, New Jersey, Hong Kong.Google Scholar
  2. 2.
    N.G. de Bruijn, Dualization of multigrids. In: Proceedings of the International Workshop Aperiodic Crystals, Les Houches 1986. Journal de Physique, Vol.47, Colloque C3, supplement to nr. 7, July 1986, pp. 9–18.Google Scholar
  3. 3.
    N. G. de Bruijn, A riffle shuffle card trick and its relation to quasicrystal theory. Nieuw Archief Wiskunde (4) 5 (1987) 285–301.zbMATHGoogle Scholar
  4. 4.
    N. G. de Bruijn, Symmetry and quasisymmetry. In: Symmetrie in Geistes- und Naturwissenschaft. Herausg. R. Wille. Springer Verlag 1988, pp. 215–233.Google Scholar
  5. 5.
    N. G. de Bruijn, Updown generation of Penrose tilings. Indagationes Mathematicae, N.S., 1, pp. 201–219 (1990).zbMATHGoogle Scholar
  6. 6.
    Martin Gardner, Mathematical games Extraordinary nonperiodic tiling that enriches the theory of tiles. Scientific American 236 (1) 110–121 (Jan. 1977).CrossRefGoogle Scholar
  7. 7.
    Branko Grünbaum and G.C. Shephard. Tilings and patterns. New York, W.H. Freeman and Co. 1986.Google Scholar
  8. 8.
    R. Penrose. Pentaplexity. Mathematical Intelligencer vol 2 (1) pp. 32–37 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J. E. S. Socolar and P. J. Steinhardt. Quasicrystals. II. Unit cell configurations. Physical Rev. B Vol. 34 (1986), 617–647. Reprinted in: P.J. Steinhardt and Stellan Ostlund: The Physics of Quasicrystals, World Scientific Publ., Singapore, New Jersey, Hong Kong.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • N. G. de Bruijn
    • 1
  1. 1.Department of Mathematics and Computing ScienceTechnological University EindhovenEindhovenThe Netherlands

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