Optimum compactness structures derived from the regular octahedron

Many building structures have an overall shape similar to the geometry of a polyhedron. The aim of this research study is to ascertain the most compact shapes (closest to the sphere) that can be obtained from the regular octahedron by truncating its vertex and chamfering its edges, while keeping all its symmetry elements. The results obtained show two shapes with excellent compactness among polyhedra with octahedral symmetry.