A quasicrystal is a structure having long-range order but no translational periodicity. Mathematically, quasicrystalline point sets are commonly modeled by Delone
sets with finite local complexity and pure point diffraction, or by cut-and-project
sets obtained by projecting selected points of a higher-dimensional point
lattice.
Unlike a periodic crystal, a quasicrystal may possess rotational symmetries incompatible with a periodic point lattice,
such as fivefold or icosahedral symmetry. One-dimensional substitution sequences
and Penrose tilings provide standard mathematical
examples of quasicrystalline order.
Meyer sets give a useful class of Delone sets: their difference sets also have uniformly separated points
(Moody 1997).
Baake, M. and Grimm, U. Aperiodic Order, Vol. 1: A Mathematical Invitation. Cambridge, England: Cambridge University Press, 2013.Barache,
D.; Champagne, B.; and Gazeau, J.-P. "Pisot-Cyclotomic Quasilattices and Their
Symmetry Semigroups." In Quasicrystals and Discrete Geometry (Ed. J. Patera).
Providence, RI: Amer. Math. Soc., pp. 15-66, 1998.Moody, R. V.
"Meyer Sets and Their Duals." In The Mathematics of Long-Range Aperiodic
Order (Ed. R. V. Moody). Dordrecht, Netherlands: Kluwer, pp. 403-441,
1997. https://doi.org/10.1007/978-94-015-8784-6_16.Moody,
R. V. "Model Sets: A Survey." In From Quasicrystals to More Complex
Systems (Ed. F. Axel, F. Dénoyer, and J.-P. Gazeau). Les
Ulis, France: EDP Sciences and Berlin, Germany: Springer-Verlag, pp. 145-166,
2000. https://arxiv.org/abs/math/0002020.
