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Arithmetic discrete planes are quasicrystals. (English) Zbl 1261.52013
Brlek, Srečko (ed.) et al., Discrete geometry for computer imagery. 15th IAPR international conference, DGCI 2009, Montréal, Canada, September 30 – October 2, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-04396-3/pbk). Lecture Notes in Computer Science 5810, 1-12 (2009).
Summary: Arithmetic discrete planes can be considered as liftings in the space of quasicrystals and tilings of the plane generated by a cut and project construction. We first give an overview of methods and properties that can be deduced from this viewpoint. Substitution rules are known to be an efficient construction process for tilings. We then introduce a substitution rule acting on discrete planes, which maps faces of unit cubes to unions of faces, and we discuss some applications to discrete geometry.
For the entire collection see [Zbl 1176.68004].

MSC:
52C23 Quasicrystals and aperiodic tilings in discrete geometry
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
68R15 Combinatorics on words
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