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Tilings, quasicrystals, discrete planes, generalized substitutions, and multidimensional continued fractions. (English) Zbl 1017.68147
Discrete models: combinatorics, computation, and geometry. Proceedings of the 1st international conference (DM-CCG), Paris, France, July 2-5, 2001. Paris: Maison de l’Informatique et des Mathématiques Discrètes (MIMD), Discrete Math. Theor. Comput. Sci., Proc. AA, 59-78, electronic only (2001).
Summary: The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrational examples by using multidimensional continued fractions. We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.
For the entire collection see [Zbl 0985.00015].

MSC:
68W05 Nonnumerical algorithms
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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