Symmetry generalisation of the Euler–Schläfli theorem for multi-shell polyhedra

Abstract

The point-group representations spanned by the vertices, edge vectors, face circulations and cell centres of a wide variety of multi-shelled polyhedral structures are related in a simple equation that generalises the well known Euler and Schläfli theorems. The new theorem can be expected to have applications in spectroscopic and electronic structure theory of a large class of clusters and coordination polyhedra.