Spiral codes and Goldberg representations of icosahedral fullerenes and octahedral analogues

An icosahedral fullerene may be considered as a tessellation of the sphere specified by an ordered pair of integers, or as a tightly wound spiral of faces. Explicit analytical relations for interconverting the two representations are given, enabling the canonical spiral code to be constructed for an icosahedral fullerene of any size. Analogous relations hold for the octahedral square + hexagon polyhedra that have been mentioned as possible candidates for boron-nitride fullerenes.