Some applications of the formula for the volume of an octahedron
Referring to a simplicial complex \(K\), first the authors introduce three types of symmetry: combinatorial symmetry (if there exists an automorphism of the vertex set of \(K\) preserving the occuring incidences), metrical symmetry (if in addition the edge-lengths are preserved), and spatial symmetry of polyhedra in 3-space (if there is a motion taking these polyhedra into themselves, preserving or reversing the orientaion). In earlier contributions, the authors confirmed the existence of polynomials related to the combinatorial structure, squared edge-lengths and volume of a polyhedron in 3-space. Here they give explicit forms of canonical polynomials determining the volume of octahedral polyhedra having certain types of symmetry. These results are also interesting from the viewpoint of crystallography.
scientific article; zbMATH DE number 2121440