A computational investigation into shapes of polyhedra

In this paper we first derive a measure to evaluate the shape of polyhedra based on the well-known concept of roundness. The resulting roundness index expresses the degree of deformity of a polyhedron measured as the deviation from a hypersphere. Since part of the roundness index requires solving an NP-complete problem, computational issues are discussed, including some necessary approximations. In a series of computational tests we then determine average values of the roundness index for a set of randomly generated polyhedra. We also provide computational results that show the degree of change in deformity when scaling techniques are used.