The topological structure of fractal tilings generated by quadratic number systems

Citations of this article
Mendeley users who have this article in their library.


Let α be a root of an irreducible quadratic polynomial x2 + Ax + B with integer coefficients A, B and assume that α forms a canonical number system, i.e., each x ∈ ∈[α] admits a representation of the shape x=a0+a1α+⋯+ahαh,with a i ∈ {0, 1,...,|B| - 1}. It is possible to associate a tiling to such a number system in a natural way. If 2A < 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 2A < 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 2A < 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. © 2005 Elsevier Ltd. All rights reserved.




Akiyama, S., & Thuswaldner, J. M. (2005). The topological structure of fractal tilings generated by quadratic number systems. Computers and Mathematics with Applications, 49(9–10), 1439–1485.

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free