The stability of the fullerenes Cn, with n = 24, 28, 32, 36, 50, 60 and 70

It has been proposed1 that the geodesic and chemical properties inherent in a closed, hollow, spheroidal, carbon cage structure with the symmetry of a European football can readily explain the remarkable stability observed for the C60 molecule. Here I present a set of simple, empirical chemical and geodesic rules which relate the stability of carbon cages mainly to the disposition of pentagonal rings, or various directly fused pentagonal ring configurations. The rules yield cluster magic numbers consistent with observation and in particular predict that the fullerenes, Cn for which n = 24, 28, 32, 36, 50, 60 and 70 should have enhanced stability relative to near neighbours. These results provide further evidence for the proposal that closed hollow cages form when carbon nucleates in the vapour phase, and in particular that C60 buckminsterfullerene is indeed a truncated icosahedron as originally proposed1.