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Working paper

A family of non-periodic tilings of the plane by right golden triangles

We consider tilings of the plane by two prototiles which are right triangles. They are called the small and the large tiles. The small tile is similar to the large tile with some similarity coefficient ψ. The large tile can be cut into two pieces so that one piece is a small tile and the other one is similar to the small tile with the same similarity coefficient ψ. Using this cut we define in a standard way the substitution scheme, in which the large tile is replaced by a large and a small tile and the small tile is replaced by a large tile. To every substitution of this kind, there corresponds a family of the so-called substitution tilings of the plane in the sense of [C. Goodman-Strauss, Matching Rules and Substitution Tilings, Annals of Mathematics 147 (1998) 181-223]. All tilings in this family are non-periodic. It was shown in the paper [N. Vereshchagin. Aperiodic Tilings by Right Triangles. In: Proc. of DCFS 2014, LNCS vol. 8614 (2014) 29--41] that this family of substitution tilings is not an SFT. This means that looking at a given tiling trough a bounded window, we cannot determine whether that tiling belongs to the family or not, however large the size of the window is. In the present paper, we prove that this family of substitution tilings is sofic. This means that we can color the prototiles ina finite number of colors and define some local rules for colored prototiles so that the following holds. For any tiling from the family, we can color its tiles so that the resulting tiling (by colored tiles) satisfies local rules. And conversely, for any tiling of the plane satisfying the local rules, by removing colors we obtain a tiling from the family. Besides, the considered substitution can be generalized to colored tiles so that the family of substitution tilings for the resulting substitution coincides with the family of tilings satisfying our local rules.