Summary
Suppose given a quasi-periodic tiling of some Euclidean space E u which is self-similar under the linear expansiong: Eμ→Eμ. It is known that there is an embedding of Eμ into some higher-dimensional space ℝN and a linear automorphism\(\bar g:\mathbb{R}^N \to \mathbb{R}^N \) with integer coefficients such that E u ⊂ ℝN is invariant under\(\bar g\) andg 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 isg *, using the information contained in the original tiling of Eμ. The term “Galois duality” of tilings is motivated by the fact that the eigenvalues ofg * are Galois conjugates of those ofg. 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 Eμ=1, andg 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.
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