Fullerene polyhedra are realisable within exactly 28 point groups: Ih, I, Td, Th, T, Dnh, Dnd, Dn (n=2, 3, 5, 6), S2m, Cmh, Cmv, Cm (m=2, 3), Cs, Ci, C1. Examples of each group can be found within the range C20 to C140, and isolated-pentagon fullerenes can be constructed for all 28 groups. The presence of a fivefold axis implies icosahedral or dihedral symmetry, and a sixfold axis implies dihedral symmetry. For groups with these axes of high order, the ring spiral algorithm is shown to be complete. It is shown that the NMR spectrum of any single fullerene isomer can contain peaks of at most three different intensities. Classification of the 1812 hypothetical C60 fullerene isomers illustrates the overwhelming likelihood that a randomly constructed fullerene cage will be totally asymmetric. © 1993.
