Baake, Michael (ed.) et al., Directions in mathematical quasicrystals. Providence, RI: AMS, American Mathematical Society. CRM Monogr. Ser. 13, 95-141 (2000).

A cut-and-project set (also called a model set) is a discrete set of the form

$\mathcal{M}=\{{\pi}_{V}\left(l\right)\mid l\in \mathcal{L}$,

${\pi}_{W}\left(l\right)\in {\Omega}\}$, defined by starting from a Euclidean space

${E}^{N}$ decomposed into a pair of complementary spaces

${E}^{N}=V\oplus W$ with

${\pi}_{V}$ and

${\pi}_{W}$ the associated projections of

${E}^{N}$ onto

$V$ and

$W$, a lattice

$\mathcal{L}\subset {E}^{N}$, and a bounded Riemann measurable set

${\Omega}\subset W$, called a window or acceptance domain. The 17 properties concerning a model set (uniformity, diffraction, Ammann bars, symmetry, inflation, local rules, etc.) defined in the first part of the article are useful in quasicrystal modelling, and interesting from a mathematical point of view. The author investigates in detail what conditions on

$\mathcal{L}$,

${\pi}_{V}$,

${\pi}_{W}$, and

${\Omega}$ in the cut-and-project construction are required for various properties of the resulting model set. Based on a deep understanding of the relation between quasiperiodicity and number theory, the author describes a natural way of obtaining a lattice and projections from a module over an algebraic number field and shows that these necessarily satisfy the conditions for all quasicrystal properties. Some interesting examples and comments concerning the model set symmetries are also included.