# Fractal penrose tiles II: Tiles with fractal boundary as duals of penrose triangles

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## Summary

Suppose given a quasi-periodic tiling of some Euclidean space E_{ u } which is self-similar under the linear expansion*g*: E_{μ}→E_{μ}. It is known that there is an embedding of E_{μ} into some higher-dimensional space ℝ^{ N } and a linear automorphism\(\bar g:\mathbb{R}^N \to \mathbb{R}^N \) with integer coefficients such that E_{ u } ⊂ ℝ^{ N } is invariant under\(\bar g\) and*g* is the restriction of\(\bar g\) to E_{ u }.

Let E_{ s } be the\(\bar g\)-invariant complement of E_{ u }, and\(g* = \bar g^{ - 1} \left| {_{E_u } } \right.\). If certain conditions are fulfilled (e.g.\(\bar g\) must be a lattice automorphism,*g* ^{*} is an expansion), we construct a self-similar tiling of E_{ s } whose expansion is*g* ^{*}, using the information contained in the original tiling of E_{μ}. The term “Galois duality” of tilings is motivated by the fact that the eigenvalues of*g* ^{*} are Galois conjugates of those of*g*. Our method can be applied to find the Galois duals which are given by Thurston, obtained in a somewhat other way for the case that dim E_{μ}=1, and*g* is the multiplication by a cubic Pisot unit.

Bandt and Gummelt have found fractally shaped tilings which can be considered as strictly self-similar modifications of the kites-and-darts, and the rhombi tilings of Penrose. As one of the examples, we show that these fractal versions can be constructed by dualizing tilings by Penrose triangles.

## AMS (1991) subject classification

05B45 52C20 52C22## Preview

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## References

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