Maximum Eigenvalue Problem for Escherization

This paper proposes a new and efficient method for “Escherization”, that is, for generating a tile which is close to a given shape and whose copies cover the plane without gaps or overlaps except at their boundaries. In this method, the Escherization problem is reduced to a maximum eigenvalue problem, which can be solved easily, while the existing method requires time consuming heuristic search. Furthermore, we show that the optimal shape corresponds to the orthogonal projection of the vector representing the given shape to the “space of tilable shapes”.