Canonical projection tilings defined by patterns
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Canonical projection tilings defined by patterns Nicolas Bédaride1
· Thomas Fernique2
Received: 18 December 2018 / Accepted: 4 February 2020 © Springer Nature B.V. 2020
Abstract We give a necessary and sufficient condition on a d-dimensional affine subspace of Rn to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of coincidence and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns. Keywords Tilings · Local rules · Cut and project tiling · Ergodic theory Mathematics Subject Classification 37B50
1 Introduction The cut and project scheme is a popular way to define aperiodic tilings (see, e.g., [7] and references therein). A rich subfamily of these tilings is formed by the so-called canonical projection tilings, which are digitizations of affines d-planes of Rn . It includes, e.g., Sturmian words (lines of R2 ), billiard words (lines of R3 ), Ammann–Beenker tilings (2-planes of R4 ), Penrose tilings (2-planes of R5 ) or icosahedral tilings (3-planes of R6 ). In particular, canonical projection tilings with irrational slopes (the d-plane it digitizes) are widely used in condensed matter theory to model quasicrystals. Both are indeed aperiodic but nonetheless “ordered” (in a sense that can be slightly different in condensed matter theory or mathematics). In this context, assuming that the stability of a real material is governed only by finite range energetic interactions, it is important to decide whether such a tiling can be characterized only by its patterns of a given (finite) size—one speaks about local rules. This issue has be tackled by numerous authors and several conditions have been obtained [3–6,8,16,17,19–
Part of this work has been done in the Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, CNRS-UMI 2807, Universitad de Chile, Santiago, Chile.
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Nicolas Bédaride [email protected]
1
CNRS, Centrale Marseille, I2M, UMR 7373, Aix Marseille Univ., 13453 Marseille, France
2
CNRS, Sorbonne Paris Cité, UMR 7030, Univ. Paris 13, 93430 Villetaneuse, France
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Geometriae Dedicata
23,26], but not complete characterization yet exists (except if we allow tiles to be decorated, see [10], but the situation becomes quite different). We shall here focus on generic slopes: Definition 1 An affine d-plane of Rn is said to be generic if it does not contain any integer point ( i.e., it is non-singular) and its direction ( i.e., the associated linear subspace) is not included in a strict rational subspace of Rn . The set of generic d-planes of Rn is denoted by G (n, d). A plane E ∈ G (n, d) is said to be characterized by patterns if there is a finite set of (finite) patterns, called forbidden patterns, such tha
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