Similarity symmetry of a 2D quasi-periodic Rauzy tiling

Abstract

The coordinates and parameters of vertices of a Rauzy tiling are calculated and the fractal character of tiling boundaries is described on the basis of complex and real parametrizations of Rauzy tiles. For a half-group of similarity transformations, which map tiling boundaries into themselves, an infinite system of generatrices is found. It is proven that a punctured Rauzy tiling has a central symmetry. Infinite helical stars with centers at the tiling vertices and rays formed by tiling boundaries are found and described; the set of similarity transformations for such stars forms a group. The set of such infinite stars covers all Rauzy tiling boundaries.