Geodesic
Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001), DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) (2001).

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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 irrationalexamples 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. 
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     author = {Arnoux, Pierre and Berthe, Valerie and Ei, Hiromi and Ito, Shunji},
     title = {Tilings, {Quasicrystals,} {Discrete} {Planes,} {Generalized} {Substitutions,} and {Multidimensional} {Continued} {Fractions}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
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}
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Arnoux, Pierre; Berthe, Valerie; Ei, Hiromi; Ito, Shunji. Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001), DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) (2001). doi : 10.46298/dmtcs.2291. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2291/

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