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

Summary

Suppose given a quasi-periodic tiling of some Euclidean space E _u which is self-similar under the linear expansiong: 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$\bar g$ 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 E_μ. 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 E_μ=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.