, Volume 15, Issue 2, pp 221-236

A construction of inflation rules based onn-fold symmetry

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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 symmetryD nfor 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.

Communicated by Imre Bárány and János Pach
Editors' note: This paper was accepted for the special issue ofDiscrete & Computational Geometry (Volume 13, Numbers 3–4) devoted to the László Fejes Tóth, Festschrift, but was not received in final form in time to appear in that issue.
Research supported by the DFG and the Fritz Thyssen Stiftung.