Discrete & Computational Geometry

, Volume 15, Issue 2, pp 221–236 | Cite as

A construction of inflation rules based onn-fold symmetry

  • K.-P Nischke
  • L. Danzer


In analogy to the well-known tilings of the euclidean plane\(\mathbb{E}^2 \) by Penrose rhombs (or, to be more precise, to the equivalent tilings by Robinson triangles) we give a construction of an inflation rule based on then-fold symmetryDnfor everyn greater than 3 and not divisible by 3. For givenn the inflation factor η can be any quotient\(\mu _{n,k} : = \sin \left( {k\pi /n} \right)/\sin \left( {\pi /n} \right)\) as well as any product\(\prod {_{k = 2}^{n/2} \mu _{n,k}^{ak} ,} \) where\(\alpha _2 ,\alpha _3 ,..., \in \mathbb{N} \cup \left\{ 0 \right\}\). The construction is based on the system ofn tangents of the well-known deltoidD, which form angles with the ζ-axis of typevπ/n. None of these tilings permits two linearly independent translations. We conjecture that they have no period at all. For some of them the Fourier transform contains a ℤ-module of Dirac deltas.


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  1. 1.
    E. Bombieri and J.E. Taylor: Quasicrystals, tilings and algebraic number theory.Contemp. Math. 64 (1987).Google Scholar
  2. 2.
    S. I. Borewicz, I. R. Šafarevic:Zahlentheorie, Birkhäuser-Verlag, Basel, 1966.MATHGoogle Scholar
  3. 3.
    L. C. Washington,Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982.MATHGoogle Scholar
  4. 4.
    B. Gruenbaum and G. C. Shephard.Tilings and Patterns, Freeman, New York, 1987.MATHGoogle Scholar
  5. 5.
    E. J. W. Whittaker and R. M. Whittaker: Some generalized Penrose patterns from projections ofn-dimensional lattices,Acta Cryst. Sect A 44 (1988), 105–112.CrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler: The Triangle Pattern—a New Quasiperiodic Tiling with Fivefold Symmetry, Preprint TPT-QC-89-08-1 (August 22, 1989).Google Scholar
  7. 7.
    M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler: Planar patterns with fivefold symmetry as sections of periodic structures in 4-space.Internat. J. Mod. Phys. B,4 (1990), 2217–2268.CrossRefMathSciNetGoogle Scholar
  8. 8.
    R. V. Moody and J. Patera: Quasicrystals and Icosians,J. Phys. A,26 (1993), 2829–2853.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    E. A. Robinson, Jr.: The Dynamical Theory of Tilings and Quasicrystallography, Preprint, July 1993.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • K.-P Nischke
    • 1
  • L. Danzer
    • 1
  1. 1.Institut für MathematikUniversität DortmundDortmundGermany

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