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Combinatorial aspects of Escher tilings

Abstract : In the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell.
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Alexandre Blondin Massé, Srecko Brlek, Sébastien Labbé. Combinatorial aspects of Escher tilings. FPSAC: InternaFormal Power Series and Algebraic Combinatorics, 2010, San Francisco, United States. pp.533-544, ⟨10.46298/dmtcs.2868⟩. ⟨hal-01186297⟩



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