The topological structure of fractal tilings generated by quadratic number systems

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Abstract

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.

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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. https://doi.org/10.1016/j.camwa.2004.09.008

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