# A construction of inflation rules based on*n*-fold symmetry

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## 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 the*n*-fold symmetry*D* _{n}for every*n* greater than 3 and not divisible by 3. For given*n* 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 of*n* tangents of the well-known deltoid*D*, which form angles with the ζ-axis of type*vπ/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.

## Keywords

Discrete Comput Geom Algebraic Number Theory Symmetric Convex Body Independent Translation Elementary Triangle## References

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