Discrete & Computational Geometry

, Volume 15, Issue 2, pp 221–236

# A construction of inflation rules based onn-fold symmetry

• K.-P Nischke
• L. Danzer
Article

## Summary

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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