The topological structure of fractal tilings generated by quadratic number systems

Abstract

Let α be a root of an irreducible quadratic polynomial x 2 + Ax + B with integer coefficients A, B and assume that α forms a canonical number system, i.e., each x ∈ ℤ[ α] admits a representation of the shape x = a 0 + a 1 α + ⋯ + a h α h , with a i ∈ {0, 1,…,| B| - 1}. It is possible to associate a tiling to such a number system in a natural way. If 2 A < B + 3, then we show that the fractal boundary of the tiles of this tiling is a simple closed curve and its interior is connected. Furthermore, the exact set equation for the boundary of a tile is given. If 2 A ≥ B + 3, then the topological structure of the tiles is quite involved. In this case, we prove that the interior of a tile is disconnected. Furthermore, we are able to construct finite labelled directed graphs which allow to determine the set of “neighbours” of a given tile T, i.e., the set of all tiles which have nonempty intersection with T. In a next step, we give the structure of the set of points, in which T coincides with L other tiles. In this paper, we use two different approaches: geometry of numbers and finite automata theory. Each of these approaches has its advantages and emphasizes different properities of the tiling. In particular, the conjecture in [1], that for A ≠ 0 and 2 A < B + 3 there exist exactly six points where T coincides with two other tiles, is solved in these two ways in Theorems 6.6 and 10.1.