Parallelogram tilings and Jacobi-Perron algorithm
The authors present the following construction of non-periodic tilings of the plane: Consider the face-to-face tiling of 3-space by unit cubes and a plane through the origin, orthogonal to a vector \((1,\alpha, \beta)\) with rationally independent coordinates. The union of those cubes meeting the plane has two boundary surfaces which are the union of (unit) squares (each a translate of some side of the unit cube). The orthogonal projection of one of these surfaces onto the plane yields a tiling by three parallelograms (corresponding to the three types of squares). This tiling is quasi-periodic (in the sense, that every boundary patch occurs in all sufficiently large patches of the tiling), and can be generated by a substitution method related to the Jacobi-Perron algorithm (applied to \((\alpha, \beta)\)). (This corresponds to a previously known construction of one-dimensional tilings related to continued fractions).
scientific article; zbMATH DE number 645684