Abstract
This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann–Beenker tilings as the solution of a simple optimization problem.
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At the time we wrote this paper, we unfortunately were unaware of Ref. [5]. There, A. Katz already obtained Theorem 8.1, and moreover showed that the uniform bound on the thickness of the tilings that can be formed is actually one. We, however, think that our proof deserve to be published. Indeed, the proof in [5] relies on purely geometric considerations in the four-dimensional space, which can be hard to follow by the reader (as acknowledged by the author himself). Alternatively, the notions of shadows and subperiods we rely on reduce much of the problem to the more usual three-dimensional space, while the use of Grassmann coordinates points the way towards a purely algebraic way to solve a wide range of similar tiling problems.
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Bédaride, N., Fernique, T. (2013). The Ammann–Beenker Tilings Revisited. In: Schmid, S., Withers, R., Lifshitz, R. (eds) Aperiodic Crystals. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6431-6_8
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DOI: https://doi.org/10.1007/978-94-007-6431-6_8
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