On sizes and shapes of lattice figures

In this paper we study the size of a kind of lattice figures in the Euclidean space ℝn, called the L-hyperpolyhedra. We emphasize the relationship between the size and the shape of such figures through the Euler characteristic. We obtain formulae for computing the volume of any L,-hyperpolyhedron. Furthermore, we give plausible arguments for showing that these formulae also work for computing the volume of a more general type of figures (called the Rhyperpolyhedra). The article depicts a gradual evolution in the mathematical solution of the computation of the volume of an L-hyperpolyhedron. We follow a heuristic approach, which shows how some new problems and concepts appear as well as the role of intuition in the statement of conjectures. In this way, it claims to reveal some aspects of the didactical significance of mathematics as a process instead of as a closed set of results. © 1991 Taylor and Francis Group, LLC.