In this paper we introduce a new method in order to find the Riemann surface M of a fixed topological type with the longest systole; it is based on a cell decomposition of the Teichmüller space of M. The method also works in the Euclidean case and is similar to the so-called Voronoï algorithm for positive definite quadratic forms, or equivalently, for lattice sphere packings. In particular, we give a new proof of Rogers' theorem.