  One-dimensional cut-and-project point sets obtained from the square lattice
in the plane are considered from a unifying point of view and in the
perspective of aperiodic wavelet constructions. We successively examine their
geometrical aspects, combinatorial properties from the point of view of the
theory of languages, and self-similarity with algebraic scaling factor
$\theta$. We explain the relation of the cut-and-project sets to non-standard
numeration systems based on $\theta$. We finally examine the substitutivity, a
weakened version of substitution invariance, which provides us with an
algorithm for symbolic generation of cut-and-project sequences.
