Some applications of the formula for the volume of an octahedron (Q1889686)

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.