A study of Jacobi–Perron boundary words for the generation of discrete planes

The construction of a Sturmian word, and thus of a discrete line, from a continued fraction development generalizes to higher dimensions. Given any vector v∈R3, a list of 6-connected points approximating the line defined by v may be obtained via a generalized continued fraction algorithm. By duality, a discrete plane with normal vector v can also be generated using a related technique. We focus on such discrete planes, more precisely on the finite patterns generated at each step of the process. We show that the choice of the Jacobi–Perron algorithm as a higher dimension generalization of Euclid’s algorithm together with the specific substitutions deduced from it allows us to guaranty the simple connectedness of those patterns.