Discrete & Computational Geometry

, Volume 15, Issue 2, pp 221–236

A construction of inflation rules based onn-fold symmetry

  • K.-P Nischke
  • L. Danzer

DOI: 10.1007/BF02717732

Cite this article as:
Nischke, KP. & Danzer, L. Discrete Comput Geom (1996) 15: 221. doi:10.1007/BF02717732


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.

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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